Subgroups — C₆₇×#59659 #30738910

Order 17152.

2244 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#555×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2731	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2545	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2538	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2491	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2439	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2426	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2362	C₂	1	4	yes	maximal, complemented
10	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
12	Q₈²×C₆₇	—	2	8	no	—
13	#555×C₆₇	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	4	yes	—
18	C₂×#124×C₆₇	C₂²	1	2	yes	—
19	C₂×#124×C₆₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂×#121×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	4	yes	—
27	#396×C₆₇	C₂²	1	8	yes	—
28	#396×C₆₇	C₂²	1	8	yes	—
29	#394×C₆₇	C₂²	1	8	yes	—
30	#393×C₆₇	C₂²	1	8	yes	—
31	#391×C₆₇	C₂²	1	4	yes	—
32	#391×C₆₇	C₂²	1	2	yes	—
33	#391×C₆₇	C₂²	1	2	yes	—
34	#390×C₆₇	C₂²	1	4	yes	—
35	#390×C₆₇	C₂²	1	4	yes	—
36	#390×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	4	yes	—
38	#389×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	8	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#388×C₆₇	C₂²	1	8	yes	—
43	#387×C₆₇	C₂²	1	8	yes	—
44	#384×C₆₇	C₂²	1	1	yes	characteristic
45	#382×C₆₇	C₂²	1	2	yes	—
46	#380×C₆₇	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
51	C₄³×C₆₇	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₄⋊Q₈×C₆₇	—	2	16	no	—
59	C₄⋊Q₈×C₆₇	—	2	16	no	—
60	C₄⋊Q₈×C₆₇	D₄	1	8	yes	—
61	#124×C₆₇	—	2	16	no	—
62	#124×C₆₇	—	2	8	no	—
63	#124×C₆₇	—	4	8	no	—
64	C₂²⋊Q₈×C₆₇	—	2	16	no	—
65	#121×C₆₇	—	2	16	no	—
66	C₄×Q₈×C₆₇	—	2	16	no	—
67	C₄×Q₈×C₆₇	—	2	8	no	—
68	C₄×Q₈×C₆₇	—	4	4	no	—
69	C₄×D₄×C₆₇	—	2	8	no	—
70	#117×C₆₇	—	2	16	no	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	#95×C₆₇	C₂³	1	4	yes	—
98	#95×C₆₇	C₂³	1	4	yes	—
99	#95×C₆₇	C₂³	1	4	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	—	2	8	no	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	—	4	2	no	—
108	C₂×Q₈×C₆₇	—	4	2	no	—
109	C₂×Q₈×C₆₇	—	4	4	no	—
110	C₂×Q₈×C₆₇	—	4	4	no	—
111	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
112	C₂×Q₈×C₆₇	#41	1	8	yes	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	4	4	no	—
118	C₂×D₄×C₆₇	—	4	4	no	—
119	C₂²×C₄×C₆₇	—	2	8	no	—
120	C₂²×C₄×C₆₇	—	2	8	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	16	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	8	no	—
137	C₄⋊C₄×C₆₇	—	2	16	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	4	4	no	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
141	C₂²⋊C₄×C₆₇	—	2	16	no	—
142	C₂²⋊C₄×C₆₇	—	2	8	no	—
143	C₂²⋊C₄×C₆₇	—	2	16	no	—
144	C₂²⋊C₄×C₆₇	—	2	8	no	—
145	C₂²⋊C₄×C₆₇	—	2	8	no	—
146	C₂²⋊C₄×C₆₇	—	2	8	no	—
147	C₂²⋊C₄×C₆₇	—	4	4	no	—
148	C₂²⋊C₄×C₆₇	—	4	8	no	—
149	C₄²×C₆₇	—	2	8	no	—
150	C₄²×C₆₇	—	2	8	no	—
151	C₄²×C₆₇	—	2	8	no	—
152	C₄²×C₆₇	—	2	4	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	8	no	—
155	C₄²×C₆₇	—	2	4	no	—
156	C₄²×C₆₇	—	4	1	no	—
157	C₄²×C₆₇	C₂×D₄	1	4	yes	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	2	4	no	—
161	C₂³×C₆₇	—	2	4	no	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	2	4	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
166	Q₈×C₆₇	—	2	16	no	—
167	Q₈×C₆₇	—	4	8	no	—
168	Q₈×C₆₇	—	8	8	no	—
169	D₄×C₆₇	—	4	8	no	—
170	D₄×C₆₇	—	8	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	4	16	no	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
196	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
197	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
198	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
199	C₂²×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
207	C₂²×C₆₇	C₂²×#41	1	2	yes	—
208	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
210	C₂²×C₆₇	#585	1	1	yes	characteristic
211	C₄×C₆₇	—	2	8	no	—
212	C₄×C₆₇	—	2	8	no	—
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	4	4	no	—
216	C₄×C₆₇	—	4	1	no	—
217	C₄×C₆₇	—	4	4	no	—
218	C₄×C₆₇	—	4	4	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	#548	1	8	yes	—
221	#59659	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂×C₆₇	—	8	4	no	—
223	C₂×C₆₇	C₂×#548	1	2	yes	—
224	C₂×C₆₇	#3483	1	1	yes	characteristic
225	C₂×C₆₇	#3496	1	1	yes	characteristic
226	C₂×C₆₇	#3518	1	2	yes	—
227	C₂×C₆₇	#3544	1	1	yes	characteristic
228	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×#555	C₂×C₆₇	1	1	yes	characteristic
230	#2731	C₂×C₆₇	1	4	yes	complemented
231	#2545	C₂×C₆₇	1	4	yes	complemented
232	#2538	C₂×C₆₇	1	2	yes	complemented
233	#2491	C₂×C₆₇	1	4	yes	complemented
234	#2439	C₂×C₆₇	1	4	yes	complemented
235	#2426	C₂×C₆₇	1	4	yes	complemented
236	#2362	C₂×C₆₇	1	4	yes	complemented
237	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₄×#124	C₂×C₆₇	1	2	yes	complemented
239	C₆₇	#59659	1	1	yes	characteristic, direct product factor, Sylow
240	Q₈²	—	2	8	no	—
241	#555	—	2	8	no	—
242	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
243	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
244	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#124	C₂²×C₆₇	1	2	yes	—
246	C₂×#124	C₂²×C₆₇	1	4	yes	—
247	C₂×#124	C₂²×C₆₇	1	4	yes	—
248	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
249	C₂×#121	C₂²×C₆₇	1	4	yes	—
250	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
251	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
252	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
254	C₂×#117	C₂²×C₆₇	1	4	yes	—
255	#396	C₂²×C₆₇	1	8	yes	—
256	#396	C₂²×C₆₇	1	8	yes	—
257	#394	C₂²×C₆₇	1	8	yes	—
258	#393	C₂²×C₆₇	1	8	yes	—
259	#391	C₂²×C₆₇	1	2	yes	—
260	#391	C₂²×C₆₇	1	2	yes	—
261	#391	C₂²×C₆₇	1	4	yes	—
262	#390	C₂²×C₆₇	1	4	yes	—
263	#390	C₂²×C₆₇	1	4	yes	—
264	#390	C₂²×C₆₇	1	2	yes	—
265	#389	C₂²×C₆₇	1	4	yes	—
266	#389	C₂²×C₆₇	1	4	yes	—
267	#388	C₂²×C₆₇	1	8	yes	—
268	#388	C₂²×C₆₇	1	8	yes	—
269	#388	C₂²×C₆₇	1	4	yes	—
270	#388	C₂²×C₆₇	1	4	yes	—
271	#387	C₂²×C₆₇	1	8	yes	—
272	#384	C₂²×C₆₇	1	1	yes	characteristic
273	#382	C₂²×C₆₇	1	2	yes	—
274	#380	C₂²×C₆₇	1	4	yes	—
275	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
276	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
277	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
278	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
279	C₄³	C₂²×C₆₇	1	1	yes	characteristic
280	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
281	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
282	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
283	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
284	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
285	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
286	C₄⋊Q₈	—	2	16	no	—
287	C₄⋊Q₈	—	2	16	no	—
288	C₄⋊Q₈	D₄×C₆₇	1	8	yes	—
289	#124	—	2	16	no	—
290	#124	—	2	8	no	—
291	#124	—	4	8	no	—
292	C₂²⋊Q₈	—	2	16	no	—
293	#121	—	2	16	no	—
294	C₄×Q₈	—	2	16	no	—
295	C₄×Q₈	—	2	8	no	—
296	C₄×Q₈	—	4	4	no	—
297	C₄×D₄	—	2	8	no	—
298	#117	—	2	16	no	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
314	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
315	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
316	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
317	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
318	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
321	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
322	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
323	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
325	#95	C₂³×C₆₇	1	4	yes	—
326	#95	C₂³×C₆₇	1	4	yes	—
327	#95	C₂³×C₆₇	1	4	yes	—
328	#95	C₂³×C₆₇	1	4	yes	—
329	#95	C₂³×C₆₇	1	8	yes	—
330	#95	C₂³×C₆₇	1	8	yes	—
331	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
332	C₂×Q₈	—	2	4	no	—
333	C₂×Q₈	—	2	8	no	—
334	C₂×Q₈	—	2	4	no	—
335	C₂×Q₈	—	4	2	no	—
336	C₂×Q₈	—	4	2	no	—
337	C₂×Q₈	—	4	4	no	—
338	C₂×Q₈	—	4	4	no	—
339	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
340	C₂×Q₈	#41×C₆₇	1	8	yes	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	8	no	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
355	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
356	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
358	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	16	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	8	no	—
365	C₄⋊C₄	—	2	16	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
369	C₂²⋊C₄	—	2	16	no	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₂²⋊C₄	—	2	16	no	—
372	C₂²⋊C₄	—	2	8	no	—
373	C₂²⋊C₄	—	2	8	no	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	4	8	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₄²	—	2	8	no	—
378	C₄²	—	2	8	no	—
379	C₄²	—	2	8	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	8	no	—
383	C₄²	—	2	4	no	—
384	C₄²	—	4	1	no	—
385	C₄²	C₂×D₄×C₆₇	1	4	yes	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	2	4	no	—
393	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
394	Q₈	—	2	16	no	—
395	Q₈	—	4	8	no	—
396	Q₈	—	8	8	no	—
397	D₄	—	4	8	no	—
398	D₄	—	8	8	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	8	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	16	no	—
420	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
421	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
423	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
424	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
425	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
426	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	4	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
435	C₂²	C₂²×#41×C₆₇	1	2	yes	—
436	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
437	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
438	C₂²	#585×C₆₇	1	1	yes	characteristic
439	C₄	—	2	2	no	—
440	C₄	—	2	8	no	—
441	C₄	—	2	8	no	—
442	C₄	—	2	2	no	—
443	C₄	—	4	4	no	—
444	C₄	—	4	1	no	—
445	C₄	—	4	4	no	—
446	C₄	—	4	4	no	—
447	C₄	—	4	2	no	—
448	C₄	#548×C₆₇	1	8	yes	—
449	C₂	—	8	4	no	—
450	C₂	C₂×#548×C₆₇	1	2	yes	—
451	C₂	C₆₇×#3483	1	1	yes	characteristic
452	C₂	C₆₇×#3496	1	1	yes	characteristic
453	C₂	C₆₇×#3518	1	2	yes	—
454	C₂	C₆₇×#3544	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#555×C₆₇	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×Q₈²×C₆₇	C₂	semidirect
3. C₆₇×#2731	C₂	semidirect
4. C₆₇×#2545	C₂	semidirect
5. C₆₇×#2538	C₂	semidirect
6. C₆₇×#2491	C₂	semidirect
7. C₆₇×#2439	C₂	semidirect
8. C₆₇×#2426	C₂	semidirect
9. C₆₇×#2362	C₂	semidirect
10. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
11. C₄×#124×C₆₇	C₂	semidirect
221. #59659	C₆₇	direct (N × H)
228. C₂×Q₈²	C₂×C₆₇	semidirect
230. #2731	C₂×C₆₇	semidirect
231. #2545	C₂×C₆₇	semidirect
232. #2538	C₂×C₆₇	semidirect
233. #2491	C₂×C₆₇	semidirect
234. #2439	C₂×C₆₇	semidirect
235. #2426	C₂×C₆₇	semidirect
236. #2362	C₂×C₆₇	semidirect
237. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
238. C₄×#124	C₂×C₆₇	semidirect
239. C₆₇	#59659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₆₇	2
C₂×#555×C₆₇	2
C₆₇×#2731	2
C₆₇×#2545	2
C₆₇×#2538	2
C₆₇×#2491	2
C₆₇×#2439	2
C₆₇×#2426	2
C₆₇×#2362	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#124×C₆₇	2
#59659	67

Subgroups — C67×#59659 #30738910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59659 #30738910