Subgroups — C₆₇×#59628 #30738879

Order 17152.

2376 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2712	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2709	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2600	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2541	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2491	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2491	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2459	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2458	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2445	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2365	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2361	C₂	1	2	yes	maximal, complemented
14	C₄×#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
16	#559×C₆₇	—	2	16	no	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#127×C₆₇	C₂²	1	2	yes	—
19	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#124×C₆₇	C₂²	1	4	yes	—
21	C₂×#124×C₆₇	C₂²	1	4	yes	—
22	C₂×#123×C₆₇	C₂²	1	4	yes	—
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×#123×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#121×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#398×C₆₇	C₂²	1	2	yes	—
35	#397×C₆₇	C₂²	1	2	yes	—
36	#394×C₆₇	C₂²	1	4	yes	—
37	#394×C₆₇	C₂²	1	4	yes	—
38	#394×C₆₇	C₂²	1	4	yes	—
39	#394×C₆₇	C₂²	1	4	yes	—
40	#393×C₆₇	C₂²	1	4	yes	—
41	#393×C₆₇	C₂²	1	4	yes	—
42	#392×C₆₇	C₂²	1	2	yes	—
43	#392×C₆₇	C₂²	1	2	yes	—
44	#391×C₆₇	C₂²	1	4	yes	—
45	#390×C₆₇	C₂²	1	2	yes	—
46	#390×C₆₇	C₂²	1	2	yes	—
47	#390×C₆₇	C₂²	1	2	yes	—
48	#390×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#388×C₆₇	C₂²	1	4	yes	—
52	#388×C₆₇	C₂²	1	2	yes	—
53	#388×C₆₇	C₂²	1	4	yes	—
54	#388×C₆₇	C₂²	1	4	yes	—
55	#388×C₆₇	C₂²	1	4	yes	—
56	#387×C₆₇	C₂²	1	2	yes	—
57	#387×C₆₇	C₂²	1	4	yes	—
58	#387×C₆₇	C₂²	1	4	yes	—
59	#386×C₆₇	C₂²	1	4	yes	—
60	#386×C₆₇	C₂²	1	1	yes	characteristic
61	#386×C₆₇	C₂²	1	4	yes	—
62	#386×C₆₇	C₂²	1	4	yes	—
63	#385×C₆₇	C₂²	1	1	yes	characteristic
64	#385×C₆₇	C₂²	1	4	yes	—
65	#381×C₆₇	C₂²	1	1	yes	characteristic
66	#381×C₆₇	C₂²	1	2	yes	—
67	#380×C₆₇	C₂²	1	2	yes	—
68	#380×C₆₇	C₂²	1	1	yes	characteristic
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
74	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
79	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
81	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₆₇	—	2	8	no	—
90	#127×C₆₇	—	2	8	no	—
91	#127×C₆₇	—	4	2	no	—
92	#124×C₆₇	—	2	16	no	—
93	#124×C₆₇	—	2	16	no	—
94	#123×C₆₇	—	2	8	no	—
95	#123×C₆₇	—	2	16	no	—
96	#123×C₆₇	D₄	1	8	yes	complemented
97	#121×C₆₇	—	2	8	no	—
98	#121×C₆₇	—	2	8	no	—
99	C₄×Q₈×C₆₇	—	2	16	no	—
100	C₄×D₄×C₆₇	—	2	16	no	—
101	C₄×D₄×C₆₇	—	2	16	no	—
102	C₄×D₄×C₆₇	—	4	4	no	—
103	C₄×D₄×C₆₇	—	4	2	no	—
104	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	—	4	2	no	—
133	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	2	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	2	yes	—
150	#95×C₆₇	C₂³	1	2	yes	—
151	#95×C₆₇	C₂³	1	4	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
153	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₂×Q₈×C₆₇	—	2	8	no	—
156	C₂×Q₈×C₆₇	—	2	8	no	—
157	C₂×Q₈×C₆₇	—	4	4	no	—
158	C₂×D₄×C₆₇	—	2	8	no	—
159	C₂×D₄×C₆₇	—	2	8	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	8	no	—
162	C₂×D₄×C₆₇	—	2	8	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	4	1	no	—
167	C₂×D₄×C₆₇	—	4	1	no	—
168	C₂×D₄×C₆₇	—	4	4	no	—
169	C₂×D₄×C₆₇	—	4	4	no	—
170	C₂×D₄×C₆₇	—	4	2	no	—
171	C₂×D₄×C₆₇	—	4	2	no	—
172	C₂×D₄×C₆₇	—	4	2	no	—
173	C₂×D₄×C₆₇	—	4	4	no	—
174	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	8	no	—
181	C₂²×C₄×C₆₇	—	4	1	no	—
182	C₂²×C₄×C₆₇	—	4	1	no	—
183	C₂²×C₄×C₆₇	—	4	1	no	—
184	C₂²×C₄×C₆₇	—	4	2	no	—
185	C₂²×C₄×C₆₇	—	4	2	no	—
186	C₂²×C₄×C₆₇	—	4	4	no	—
187	C₂²×C₄×C₆₇	—	4	4	no	—
188	C₂²×C₄×C₆₇	—	4	2	no	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
202	C₄⋊C₄×C₆₇	—	2	8	no	—
203	C₄⋊C₄×C₆₇	—	2	8	no	—
204	C₄⋊C₄×C₆₇	—	2	8	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	4	2	no	—
208	C₄⋊C₄×C₆₇	—	4	2	no	—
209	C₄⋊C₄×C₆₇	—	4	1	no	—
210	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
211	C₄⋊C₄×C₆₇	#41	1	8	yes	—
212	C₂²⋊C₄×C₆₇	—	2	8	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	8	no	—
215	C₂²⋊C₄×C₆₇	—	2	8	no	—
216	C₂²⋊C₄×C₆₇	—	2	8	no	—
217	C₂²⋊C₄×C₆₇	—	2	8	no	—
218	C₂²⋊C₄×C₆₇	—	2	8	no	—
219	C₂²⋊C₄×C₆₇	—	4	1	no	—
220	C₂²⋊C₄×C₆₇	—	4	4	no	—
221	C₂²⋊C₄×C₆₇	—	4	2	no	—
222	C₂²⋊C₄×C₆₇	—	4	4	no	—
223	C₂²⋊C₄×C₆₇	—	4	4	no	—
224	C₂²⋊C₄×C₆₇	—	4	4	no	—
225	C₂²⋊C₄×C₆₇	—	4	1	no	—
226	C₂²⋊C₄×C₆₇	—	4	1	no	—
227	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
228	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
229	C₄²×C₆₇	—	2	8	no	—
230	C₄²×C₆₇	—	2	8	no	—
231	C₄²×C₆₇	—	2	4	no	—
232	C₄²×C₆₇	—	2	4	no	—
233	C₄²×C₆₇	—	2	4	no	—
234	C₄²×C₆₇	—	2	2	no	—
235	C₄²×C₆₇	—	4	2	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	1	no	—
239	C₂³×C₆₇	—	2	1	no	—
240	C₂³×C₆₇	—	2	1	no	—
241	C₂³×C₆₇	—	2	2	no	—
242	C₂³×C₆₇	—	2	1	no	—
243	C₂³×C₆₇	—	2	2	no	—
244	C₂³×C₆₇	—	2	4	no	—
245	C₂³×C₆₇	—	2	4	no	—
246	C₂³×C₆₇	—	2	4	no	—
247	C₂³×C₆₇	—	2	4	no	—
248	C₂³×C₆₇	—	2	4	no	—
249	C₂³×C₆₇	—	2	4	no	—
250	C₂³×C₆₇	—	2	4	no	—
251	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
252	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
253	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
254	Q₈×C₆₇	—	4	16	no	—
255	D₄×C₆₇	—	4	4	no	—
256	D₄×C₆₇	—	4	16	no	—
257	D₄×C₆₇	—	4	16	no	—
258	D₄×C₆₇	—	4	8	no	—
259	D₄×C₆₇	—	8	4	no	—
260	D₄×C₆₇	—	8	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	1	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	1	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	1	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	4	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	4	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	4	4	no	—
288	C₂×C₄×C₆₇	—	4	4	no	—
289	C₂×C₄×C₆₇	—	4	4	no	—
290	C₂×C₄×C₆₇	—	4	4	no	—
291	C₂×C₄×C₆₇	—	8	8	no	—
292	C₂×C₄×C₆₇	—	8	4	no	—
293	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
294	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
295	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
296	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
297	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
298	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
299	C₂²×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	2	4	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	4	1	no	—
304	C₂²×C₆₇	—	4	1	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	4	1	no	—
308	C₂²×C₆₇	—	4	1	no	—
309	C₂²×C₆₇	—	4	4	no	—
310	C₂²×C₆₇	—	4	4	no	—
311	C₂²×C₆₇	—	4	4	no	—
312	C₂²×C₆₇	—	4	4	no	—
313	C₂²×C₆₇	—	4	4	no	—
314	C₂²×C₆₇	—	4	4	no	—
315	C₂²×C₆₇	—	4	4	no	—
316	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
317	C₂²×C₆₇	C₂²×#41	1	2	yes	—
318	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
319	C₂²×C₆₇	#585	1	2	yes	—
320	C₂²×C₆₇	#585	1	1	yes	characteristic
321	C₄×C₆₇	—	2	4	no	—
322	C₄×C₆₇	—	2	8	no	—
323	C₄×C₆₇	—	2	4	no	—
324	C₄×C₆₇	—	2	2	no	—
325	C₄×C₆₇	—	4	4	no	—
326	C₄×C₆₇	—	4	1	no	—
327	C₄×C₆₇	—	4	1	no	—
328	C₄×C₆₇	—	4	4	no	—
329	C₄×C₆₇	—	4	2	no	—
330	C₄×C₆₇	—	4	2	no	—
331	C₄×C₆₇	—	4	1	no	—
332	C₄×C₆₇	#559	1	4	yes	complemented
333	#59628	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
334	C₂×C₆₇	—	4	2	no	—
335	C₂×C₆₇	—	4	2	no	—
336	C₂×C₆₇	—	8	4	no	—
337	C₂×C₆₇	D₄×#41	1	2	yes	—
338	C₂×C₆₇	C₂×#559	1	1	yes	characteristic
339	C₂×C₆₇	#3496	1	1	yes	characteristic
340	C₂×C₆₇	#3524	1	2	yes	—
341	C₂×C₆₇	#3540	1	1	yes	characteristic
342	C₂×#559	C₂×C₆₇	1	2	yes	complemented
343	#2712	C₂×C₆₇	1	2	yes	complemented
344	#2709	C₂×C₆₇	1	2	yes	complemented
345	#2600	C₂×C₆₇	1	2	yes	complemented
346	#2541	C₂×C₆₇	1	1	yes	characteristic
347	#2491	C₂×C₆₇	1	2	yes	complemented
348	#2491	C₂×C₆₇	1	2	yes	complemented
349	#2486	C₂×C₆₇	1	1	yes	characteristic, complemented
350	#2459	C₂×C₆₇	1	4	yes	complemented
351	#2458	C₂×C₆₇	1	4	yes	complemented
352	#2445	C₂×C₆₇	1	2	yes	complemented
353	#2365	C₂×C₆₇	1	2	yes	complemented
354	#2361	C₂×C₆₇	1	2	yes	complemented
355	C₄×#123	C₂×C₆₇	1	1	yes	characteristic, complemented
356	C₄×#121	C₂×C₆₇	1	2	yes	complemented
357	C₆₇	#59628	1	1	yes	characteristic, direct product factor, Sylow
358	#559	—	2	16	no	—
359	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
360	C₂×#127	C₂²×C₆₇	1	2	yes	—
361	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×#124	C₂²×C₆₇	1	4	yes	—
363	C₂×#124	C₂²×C₆₇	1	4	yes	—
364	C₂×#123	C₂²×C₆₇	1	4	yes	—
365	C₂×#123	C₂²×C₆₇	1	2	yes	—
366	C₂×#123	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×#121	C₂²×C₆₇	1	2	yes	—
368	C₂×#121	C₂²×C₆₇	1	2	yes	—
369	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
371	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
372	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
373	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
374	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
375	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
376	#398	C₂²×C₆₇	1	2	yes	—
377	#397	C₂²×C₆₇	1	2	yes	—
378	#394	C₂²×C₆₇	1	4	yes	—
379	#394	C₂²×C₆₇	1	4	yes	—
380	#394	C₂²×C₆₇	1	4	yes	—
381	#394	C₂²×C₆₇	1	4	yes	—
382	#393	C₂²×C₆₇	1	4	yes	—
383	#393	C₂²×C₆₇	1	4	yes	—
384	#392	C₂²×C₆₇	1	2	yes	—
385	#392	C₂²×C₆₇	1	2	yes	—
386	#391	C₂²×C₆₇	1	4	yes	—
387	#390	C₂²×C₆₇	1	2	yes	—
388	#390	C₂²×C₆₇	1	2	yes	—
389	#390	C₂²×C₆₇	1	2	yes	—
390	#390	C₂²×C₆₇	1	2	yes	—
391	#389	C₂²×C₆₇	1	2	yes	—
392	#389	C₂²×C₆₇	1	2	yes	—
393	#388	C₂²×C₆₇	1	4	yes	—
394	#388	C₂²×C₆₇	1	4	yes	—
395	#388	C₂²×C₆₇	1	4	yes	—
396	#388	C₂²×C₆₇	1	4	yes	—
397	#388	C₂²×C₆₇	1	2	yes	—
398	#387	C₂²×C₆₇	1	4	yes	—
399	#387	C₂²×C₆₇	1	4	yes	—
400	#387	C₂²×C₆₇	1	2	yes	—
401	#386	C₂²×C₆₇	1	4	yes	—
402	#386	C₂²×C₆₇	1	4	yes	—
403	#386	C₂²×C₆₇	1	1	yes	characteristic
404	#386	C₂²×C₆₇	1	4	yes	—
405	#385	C₂²×C₆₇	1	4	yes	—
406	#385	C₂²×C₆₇	1	1	yes	characteristic
407	#381	C₂²×C₆₇	1	1	yes	characteristic
408	#381	C₂²×C₆₇	1	2	yes	—
409	#380	C₂²×C₆₇	1	1	yes	characteristic
410	#380	C₂²×C₆₇	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
412	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
413	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
414	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
415	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
416	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
417	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
418	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
419	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
420	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
421	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
422	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
423	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
424	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
425	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
428	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
429	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
430	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
431	C₄⋊Q₈	—	2	8	no	—
432	#127	—	2	8	no	—
433	#127	—	4	2	no	—
434	#124	—	2	16	no	—
435	#124	—	2	16	no	—
436	#123	—	2	8	no	—
437	#123	—	2	16	no	—
438	#123	D₄×C₆₇	1	8	yes	complemented
439	#121	—	2	8	no	—
440	#121	—	2	8	no	—
441	C₄×Q₈	—	2	16	no	—
442	C₄×D₄	—	2	16	no	—
443	C₄×D₄	—	2	16	no	—
444	C₄×D₄	—	4	4	no	—
445	C₄×D₄	—	4	2	no	—
446	C₂×C₄⋊C₄	—	4	2	no	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
465	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
466	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
468	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
469	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
470	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
471	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
472	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
474	C₂×C₄²	—	4	2	no	—
475	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
476	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
477	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
478	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
479	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
480	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
481	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
482	#95	C₂³×C₆₇	1	4	yes	—
483	#95	C₂³×C₆₇	1	2	yes	—
484	#95	C₂³×C₆₇	1	2	yes	—
485	#95	C₂³×C₆₇	1	4	yes	—
486	#95	C₂³×C₆₇	1	2	yes	—
487	#95	C₂³×C₆₇	1	2	yes	—
488	#95	C₂³×C₆₇	1	4	yes	—
489	#95	C₂³×C₆₇	1	2	yes	—
490	#95	C₂³×C₆₇	1	4	yes	—
491	#95	C₂³×C₆₇	1	2	yes	—
492	#95	C₂³×C₆₇	1	2	yes	—
493	#95	C₂³×C₆₇	1	4	yes	—
494	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
495	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂×Q₈	—	2	8	no	—
498	C₂×Q₈	—	2	8	no	—
499	C₂×Q₈	—	4	4	no	—
500	C₂×D₄	—	2	8	no	—
501	C₂×D₄	—	2	8	no	—
502	C₂×D₄	—	2	4	no	—
503	C₂×D₄	—	2	8	no	—
504	C₂×D₄	—	2	8	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	4	1	no	—
509	C₂×D₄	—	4	1	no	—
510	C₂×D₄	—	4	4	no	—
511	C₂×D₄	—	4	4	no	—
512	C₂×D₄	—	4	2	no	—
513	C₂×D₄	—	4	2	no	—
514	C₂×D₄	—	4	2	no	—
515	C₂×D₄	—	4	4	no	—
516	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	4	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	8	no	—
523	C₂²×C₄	—	4	1	no	—
524	C₂²×C₄	—	4	1	no	—
525	C₂²×C₄	—	4	1	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	—	4	2	no	—
528	C₂²×C₄	—	4	4	no	—
529	C₂²×C₄	—	4	4	no	—
530	C₂²×C₄	—	4	2	no	—
531	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
532	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
537	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
538	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
539	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
540	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
541	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
542	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
543	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
544	C₄⋊C₄	—	2	8	no	—
545	C₄⋊C₄	—	2	8	no	—
546	C₄⋊C₄	—	2	8	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	4	2	no	—
550	C₄⋊C₄	—	4	1	no	—
551	C₄⋊C₄	—	4	2	no	—
552	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
553	C₄⋊C₄	#41×C₆₇	1	8	yes	—
554	C₂²⋊C₄	—	2	8	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	8	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	8	no	—
561	C₂²⋊C₄	—	4	4	no	—
562	C₂²⋊C₄	—	4	1	no	—
563	C₂²⋊C₄	—	4	1	no	—
564	C₂²⋊C₄	—	4	1	no	—
565	C₂²⋊C₄	—	4	4	no	—
566	C₂²⋊C₄	—	4	2	no	—
567	C₂²⋊C₄	—	4	4	no	—
568	C₂²⋊C₄	—	4	4	no	—
569	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
570	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
571	C₄²	—	2	8	no	—
572	C₄²	—	2	8	no	—
573	C₄²	—	2	4	no	—
574	C₄²	—	2	4	no	—
575	C₄²	—	2	4	no	—
576	C₄²	—	2	2	no	—
577	C₄²	—	4	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	1	no	—
581	C₂³	—	2	1	no	—
582	C₂³	—	2	1	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	1	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	4	no	—
587	C₂³	—	2	4	no	—
588	C₂³	—	2	4	no	—
589	C₂³	—	2	4	no	—
590	C₂³	—	2	4	no	—
591	C₂³	—	2	4	no	—
592	C₂³	—	2	4	no	—
593	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
594	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
595	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
596	Q₈	—	4	16	no	—
597	D₄	—	4	4	no	—
598	D₄	—	4	16	no	—
599	D₄	—	4	16	no	—
600	D₄	—	4	8	no	—
601	D₄	—	8	4	no	—
602	D₄	—	8	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	4	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	4	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	—	4	4	no	—
631	C₂×C₄	—	4	4	no	—
632	C₂×C₄	—	4	4	no	—
633	C₂×C₄	—	8	8	no	—
634	C₂×C₄	—	8	4	no	—
635	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
636	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
637	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
638	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
639	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
640	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
641	C₂²	—	2	4	no	—
642	C₂²	—	2	4	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	4	no	—
652	C₂²	—	4	4	no	—
653	C₂²	—	4	4	no	—
654	C₂²	—	4	4	no	—
655	C₂²	—	4	4	no	—
656	C₂²	—	4	4	no	—
657	C₂²	—	4	4	no	—
658	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
659	C₂²	C₂²×#41×C₆₇	1	2	yes	—
660	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
661	C₂²	#585×C₆₇	1	1	yes	characteristic
662	C₂²	#585×C₆₇	1	2	yes	—
663	C₄	—	2	4	no	—
664	C₄	—	2	8	no	—
665	C₄	—	2	4	no	—
666	C₄	—	2	2	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	2	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	4	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	4	no	—
674	C₄	#559×C₆₇	1	4	yes	complemented
675	C₂	—	4	2	no	—
676	C₂	—	4	2	no	—
677	C₂	—	8	4	no	—
678	C₂	D₄×#41×C₆₇	1	2	yes	—
679	C₂	C₂×#559×C₆₇	1	1	yes	characteristic
680	C₂	C₆₇×#3496	1	1	yes	characteristic
681	C₂	C₆₇×#3524	1	2	yes	—
682	C₂	C₆₇×#3540	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₆₇×#2541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#559×C₆₇	C₂	semidirect
2. C₆₇×#2712	C₂	semidirect
3. C₆₇×#2709	C₂	semidirect
4. C₆₇×#2600	C₂	semidirect
6. C₆₇×#2491	C₂	semidirect
7. C₆₇×#2491	C₂	semidirect
8. C₆₇×#2486	C₂	semidirect
9. C₆₇×#2459	C₂	semidirect
10. C₆₇×#2458	C₂	semidirect
11. C₆₇×#2445	C₂	semidirect
12. C₆₇×#2365	C₂	semidirect
13. C₆₇×#2361	C₂	semidirect
14. C₄×#123×C₆₇	C₂	semidirect
15. C₄×#121×C₆₇	C₂	semidirect
96. #123×C₆₇	D₄	semidirect
332. C₄×C₆₇	#559	semidirect
333. #59628	C₆₇	direct (N × H)
342. C₂×#559	C₂×C₆₇	semidirect
343. #2712	C₂×C₆₇	semidirect
344. #2709	C₂×C₆₇	semidirect
345. #2600	C₂×C₆₇	semidirect
347. #2491	C₂×C₆₇	semidirect
348. #2491	C₂×C₆₇	semidirect
349. #2486	C₂×C₆₇	semidirect
350. #2459	C₂×C₆₇	semidirect
351. #2458	C₂×C₆₇	semidirect
352. #2445	C₂×C₆₇	semidirect
353. #2365	C₂×C₆₇	semidirect
354. #2361	C₂×C₆₇	semidirect
355. C₄×#123	C₂×C₆₇	semidirect
356. C₄×#121	C₂×C₆₇	semidirect
357. C₆₇	#59628	direct (N × H)
438. #123	D₄×C₆₇	semidirect
674. C₄	#559×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#559×C₆₇	2
C₆₇×#2712	2
C₆₇×#2709	2
C₆₇×#2600	2
C₆₇×#2541	2
C₆₇×#2491	2
C₆₇×#2486	2
C₆₇×#2459	2
C₆₇×#2458	2
C₆₇×#2445	2
C₆₇×#2365	2
C₆₇×#2361	2
C₄×#123×C₆₇	2
C₄×#121×C₆₇	2
#59628	67

Subgroups — C67×#59628 #30738879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59628 #30738879