Subgroups — C₆₇×#35103 #30714354

Order 17152.

2346 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2850	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2618	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2601	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2565	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2545	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2544	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2325	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	#529×C₆₇	—	2	16	no	—
16	C₂×#126×C₆₇	C₂²	1	4	yes	—
17	C₂×#125×C₆₇	C₂²	1	2	yes	—
18	C₂×#124×C₆₇	C₂²	1	2	yes	—
19	C₂×#124×C₆₇	C₂²	1	2	yes	—
20	C₂×#123×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#121×C₆₇	C₂²	1	4	yes	—
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
27	C₂×#117×C₆₇	C₂²	1	4	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#398×C₆₇	C₂²	1	8	yes	—
33	#396×C₆₇	C₂²	1	8	yes	—
34	#396×C₆₇	C₂²	1	8	yes	—
35	#394×C₆₇	C₂²	1	8	yes	—
36	#391×C₆₇	C₂²	1	2	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#390×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	4	yes	—
41	#389×C₆₇	C₂²	1	4	yes	—
42	#388×C₆₇	C₂²	1	8	yes	—
43	#388×C₆₇	C₂²	1	8	yes	—
44	#387×C₆₇	C₂²	1	8	yes	—
45	#386×C₆₇	C₂²	1	8	yes	—
46	#386×C₆₇	C₂²	1	8	yes	—
47	#385×C₆₇	C₂²	1	8	yes	—
48	#384×C₆₇	C₂²	1	1	yes	characteristic
49	#384×C₆₇	C₂²	1	1	yes	characteristic
50	#382×C₆₇	C₂²	1	2	yes	—
51	#380×C₆₇	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
55	C₂×#95×C₆₇	C₂²	1	2	yes	—
56	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₆₇	—	4	2	no	—
67	#127×C₆₇	—	4	2	no	—
68	#126×C₆₇	—	2	16	no	—
69	#125×C₆₇	—	2	8	no	—
70	#124×C₆₇	—	2	8	no	—
71	#124×C₆₇	—	4	4	no	—
72	#123×C₆₇	—	2	16	no	—
73	C₂²⋊Q₈×C₆₇	—	2	8	no	—
74	#121×C₆₇	—	2	8	no	—
75	#121×C₆₇	—	2	16	no	—
76	C₄×Q₈×C₆₇	—	2	8	no	—
77	C₄×D₄×C₆₇	—	2	16	no	—
78	C₄×D₄×C₆₇	—	2	8	no	—
79	#117×C₆₇	—	2	16	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
100	C₂×C₄²×C₆₇	—	4	2	no	—
101	C₂×C₄²×C₆₇	—	4	2	no	—
102	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
103	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	#95×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	4	yes	—
113	#95×C₆₇	C₂³	1	8	yes	—
114	#95×C₆₇	C₂³	1	8	yes	—
115	#95×C₆₇	C₂³	1	8	yes	—
116	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
117	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	16	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	8	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	4	4	no	—
128	C₂×D₄×C₆₇	—	4	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	8	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	4	2	no	—
136	C₂²×C₄×C₆₇	—	4	2	no	—
137	C₂²×C₄×C₆₇	—	4	2	no	—
138	C₂²×C₄×C₆₇	—	4	8	no	—
139	C₂²×C₄×C₆₇	—	4	2	no	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₆₇	#41	1	8	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	8	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	2	8	no	—
160	C₄⋊C₄×C₆₇	—	2	8	no	—
161	C₄⋊C₄×C₆₇	—	2	8	no	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	4	4	no	—
164	C₄⋊C₄×C₆₇	—	4	4	no	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	8	no	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	16	no	—
171	C₂²⋊C₄×C₆₇	—	4	8	no	—
172	C₂²⋊C₄×C₆₇	—	4	8	no	—
173	C₄²×C₆₇	—	2	8	no	—
174	C₄²×C₆₇	—	2	4	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	—	4	2	no	—
178	C₄²×C₆₇	—	4	2	no	—
179	C₄²×C₆₇	C₂×D₄	1	8	yes	—
180	C₄²×C₆₇	C₂×D₄	1	8	yes	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	8	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	8	no	—
189	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂³×C₆₇	C₂×#41	1	4	yes	—
191	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
192	Q₈×C₆₇	—	4	8	no	—
193	D₄×C₆₇	—	4	16	no	—
194	D₄×C₆₇	—	4	8	no	—
195	D₄×C₆₇	—	8	8	no	—
196	C₂×C₄×C₆₇	—	2	8	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	8	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	8	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	8	no	—
216	C₂×C₄×C₆₇	—	4	4	no	—
217	C₂×C₄×C₆₇	—	4	8	no	—
218	C₂×C₄×C₆₇	—	4	16	no	—
219	C₂×C₄×C₆₇	—	4	8	no	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
225	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
226	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
227	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	4	no	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	8	no	—
236	C₂²×C₆₇	—	4	8	no	—
237	C₂²×C₆₇	#532	1	4	yes	—
238	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
239	C₂²×C₆₇	C₂²×#41	1	2	yes	—
240	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
241	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂²×C₆₇	#585	1	2	yes	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	4	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	—	2	4	no	—
248	C₄×C₆₇	—	4	4	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	#537	1	4	yes	—
253	#35103	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₆₇	—	2	4	no	—
255	C₂×C₆₇	—	8	4	no	—
256	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
258	C₂×C₆₇	#3449	1	1	yes	characteristic
259	C₂×C₆₇	#3505	1	2	yes	—
260	C₂×C₆₇	#3507	1	2	yes	—
261	C₂×#529	C₂×C₆₇	1	2	yes	complemented
262	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#2856	C₂×C₆₇	1	1	yes	characteristic
264	#2850	C₂×C₆₇	1	2	yes	complemented
265	#2618	C₂×C₆₇	1	4	yes	complemented
266	#2601	C₂×C₆₇	1	8	yes	complemented
267	#2566	C₂×C₆₇	1	2	yes	complemented
268	#2565	C₂×C₆₇	1	2	yes	complemented
269	#2545	C₂×C₆₇	1	2	yes	complemented
270	#2544	C₂×C₆₇	1	2	yes	complemented
271	#2325	C₂×C₆₇	1	4	yes	complemented
272	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₆₇	#35103	1	1	yes	characteristic, direct product factor, Sylow
274	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
276	#529	—	2	16	no	—
277	C₂×#126	C₂²×C₆₇	1	4	yes	—
278	C₂×#125	C₂²×C₆₇	1	2	yes	—
279	C₂×#124	C₂²×C₆₇	1	2	yes	—
280	C₂×#124	C₂²×C₆₇	1	2	yes	—
281	C₂×#123	C₂²×C₆₇	1	4	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
283	C₂×#121	C₂²×C₆₇	1	4	yes	—
284	C₂×#121	C₂²×C₆₇	1	2	yes	—
285	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
286	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
288	C₂×#117	C₂²×C₆₇	1	4	yes	—
289	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
290	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
291	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
292	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
293	#398	C₂²×C₆₇	1	8	yes	—
294	#396	C₂²×C₆₇	1	8	yes	—
295	#396	C₂²×C₆₇	1	8	yes	—
296	#394	C₂²×C₆₇	1	8	yes	—
297	#391	C₂²×C₆₇	1	2	yes	—
298	#391	C₂²×C₆₇	1	2	yes	—
299	#390	C₂²×C₆₇	1	2	yes	—
300	#390	C₂²×C₆₇	1	2	yes	—
301	#390	C₂²×C₆₇	1	4	yes	—
302	#389	C₂²×C₆₇	1	4	yes	—
303	#388	C₂²×C₆₇	1	8	yes	—
304	#388	C₂²×C₆₇	1	8	yes	—
305	#387	C₂²×C₆₇	1	8	yes	—
306	#386	C₂²×C₆₇	1	8	yes	—
307	#386	C₂²×C₆₇	1	8	yes	—
308	#385	C₂²×C₆₇	1	8	yes	—
309	#384	C₂²×C₆₇	1	1	yes	characteristic
310	#384	C₂²×C₆₇	1	1	yes	characteristic
311	#382	C₂²×C₆₇	1	2	yes	—
312	#380	C₂²×C₆₇	1	4	yes	—
313	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
314	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
315	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
316	C₂×#95	C₂²×C₆₇	1	2	yes	—
317	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
318	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
319	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
320	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
321	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
325	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
326	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
327	C₄⋊Q₈	—	4	2	no	—
328	#127	—	4	2	no	—
329	#126	—	2	16	no	—
330	#125	—	2	8	no	—
331	#124	—	2	8	no	—
332	#124	—	4	4	no	—
333	#123	—	2	16	no	—
334	C₂²⋊Q₈	—	2	8	no	—
335	#121	—	2	8	no	—
336	#121	—	2	16	no	—
337	C₄×Q₈	—	2	8	no	—
338	C₄×D₄	—	2	16	no	—
339	C₄×D₄	—	2	8	no	—
340	#117	—	2	16	no	—
341	C₂×C₄⋊C₄	—	2	8	no	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
345	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
361	C₂×C₄²	—	4	2	no	—
362	C₂×C₄²	—	4	2	no	—
363	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
364	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
365	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
366	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
369	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
370	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
372	#95	C₂³×C₆₇	1	4	yes	—
373	#95	C₂³×C₆₇	1	4	yes	—
374	#95	C₂³×C₆₇	1	8	yes	—
375	#95	C₂³×C₆₇	1	8	yes	—
376	#95	C₂³×C₆₇	1	8	yes	—
377	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
378	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	16	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
404	C₂²×C₄	#41×C₆₇	1	8	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
412	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
414	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	4	4	no	—
425	C₄⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	16	no	—
432	C₂²⋊C₄	—	4	8	no	—
433	C₂²⋊C₄	—	4	8	no	—
434	C₄²	—	2	8	no	—
435	C₄²	—	2	4	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	4	2	no	—
439	C₄²	—	4	2	no	—
440	C₄²	C₂×D₄×C₆₇	1	8	yes	—
441	C₄²	C₂×D₄×C₆₇	1	8	yes	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	8	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	8	no	—
450	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
451	C₂³	C₂×#41×C₆₇	1	4	yes	—
452	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
453	Q₈	—	4	8	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	8	no	—
456	D₄	—	8	8	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	16	no	—
480	C₂×C₄	—	4	8	no	—
481	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
482	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
483	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
485	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
486	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
487	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
488	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	8	no	—
497	C₂²	—	4	8	no	—
498	C₂²	#532×C₆₇	1	4	yes	—
499	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
500	C₂²	C₂²×#41×C₆₇	1	2	yes	—
501	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
502	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
503	C₂²	#585×C₆₇	1	2	yes	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	4	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	4	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	#537×C₆₇	1	4	yes	—
514	C₂	—	2	4	no	—
515	C₂	—	8	4	no	—
516	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
517	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
518	C₂	C₆₇×#3449	1	1	yes	characteristic
519	C₂	C₆₇×#3505	1	2	yes	—
520	C₂	C₆₇×#3507	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₆₇×#2856	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×#529×C₆₇	C₂	semidirect
2. C₆₇×#2857	C₂	semidirect
4. C₆₇×#2850	C₂	semidirect
5. C₆₇×#2618	C₂	semidirect
6. C₆₇×#2601	C₂	semidirect
7. C₆₇×#2566	C₂	semidirect
8. C₆₇×#2565	C₂	semidirect
9. C₆₇×#2545	C₂	semidirect
10. C₆₇×#2544	C₂	semidirect
11. C₆₇×#2325	C₂	semidirect
12. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
102. C₂×C₄²×C₆₇	D₄	semidirect
253. #35103	C₆₇	direct (N × H)
261. C₂×#529	C₂×C₆₇	semidirect
262. #2857	C₂×C₆₇	semidirect
264. #2850	C₂×C₆₇	semidirect
265. #2618	C₂×C₆₇	semidirect
266. #2601	C₂×C₆₇	semidirect
267. #2566	C₂×C₆₇	semidirect
268. #2565	C₂×C₆₇	semidirect
269. #2545	C₂×C₆₇	semidirect
270. #2544	C₂×C₆₇	semidirect
271. #2325	C₂×C₆₇	semidirect
272. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
273. C₆₇	#35103	direct (N × H)
363. C₂×C₄²	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₆₇×#2857	2
C₆₇×#2856	2
C₆₇×#2850	2
C₆₇×#2618	2
C₆₇×#2601	2
C₆₇×#2566	2
C₆₇×#2565	2
C₆₇×#2545	2
C₆₇×#2544	2
C₆₇×#2325	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#35103	67

Subgroups — C67×#35103 #30714354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35103 #30714354