Subgroups — C₆₇×#34070 #30713321

Order 17152.

3102 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2328	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2327	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2326	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2325	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2297	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2290	C₂	1	8	yes	maximal, complemented
9	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	2	yes	—
12	#532×C₆₇	C₄	1	16	yes	complemented
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₂²	1	2	yes	—
16	C₂×#121×C₆₇	C₂²	1	4	yes	—
17	C₄×#41×C₆₇	—	2	16	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	C₂×#117×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₂²	1	4	yes	—
29	#390×C₆₇	C₂²	1	4	yes	—
30	#388×C₆₇	C₂²	1	16	yes	—
31	#387×C₆₇	C₂²	1	16	yes	—
32	#386×C₆₇	C₂²	1	16	yes	—
33	#386×C₆₇	C₂²	1	8	yes	—
34	#385×C₆₇	C₂²	1	16	yes	—
35	#384×C₆₇	C₂²	1	2	yes	—
36	#382×C₆₇	C₂²	1	2	yes	—
37	#380×C₆₇	C₂²	1	4	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
40	#376×C₆₇	C₂²	1	4	yes	—
41	#376×C₆₇	C₂²	1	2	yes	—
42	#376×C₆₇	C₂²	1	2	yes	—
43	C₂×#95×C₆₇	C₂²	1	4	yes	—
44	C₂×#41×C₆₇	—	2	8	no	—
45	C₂×#41×C₆₇	—	2	8	no	—
46	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
47	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
58	#127×C₆₇	C₂×C₄	1	8	yes	complemented
59	#124×C₆₇	C₂×C₄	1	16	yes	complemented
60	#121×C₆₇	C₂×C₄	1	32	yes	complemented
61	C₄×Q₈×C₆₇	—	2	16	no	—
62	C₄×D₄×C₆₇	—	2	32	no	—
63	C₄×D₄×C₆₇	—	2	16	no	—
64	C₄×D₄×C₆₇	C₂×C₄	1	32	yes	complemented
65	#117×C₆₇	—	2	8	no	—
66	#117×C₆₇	—	2	16	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₄²×C₆₇	—	2	4	no	—
76	C₂×C₄²×C₆₇	—	2	16	no	—
77	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
78	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
79	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
88	#95×C₆₇	C₂³	1	8	yes	—
89	#95×C₆₇	C₂³	1	8	yes	—
90	#95×C₆₇	C₂³	1	16	yes	—
91	#95×C₆₇	C₂³	1	8	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
94	#41×C₆₇	—	2	64	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	4	8	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
105	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	16	no	—
110	C₂²×C₄×C₆₇	—	4	4	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	—	4	16	no	—
113	C₂²×C₄×C₆₇	—	4	4	no	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
116	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
119	C₂²×C₄×C₆₇	#41	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₆₇	—	2	16	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	4	8	no	—
131	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	—	2	16	no	—
134	C₂²⋊C₄×C₆₇	—	4	16	no	—
135	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
136	C₄²×C₆₇	—	2	8	no	—
137	C₄²×C₆₇	—	2	16	no	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	2	4	no	—
140	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
141	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
142	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
143	C₄²×C₆₇	C₂×D₄	1	8	yes	—
144	C₄²×C₆₇	C₂×D₄	1	8	yes	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	16	no	—
149	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
150	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
152	C₂³×C₆₇	C₂×#41	1	2	yes	—
153	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	16	no	—
155	D₄×C₆₇	—	2	32	no	—
156	D₄×C₆₇	—	2	16	no	—
157	D₄×C₆₇	—	4	16	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	32	no	—
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	16	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	8	no	—
170	C₂×C₄×C₆₇	—	4	32	no	—
171	C₂×C₄×C₆₇	—	4	8	no	—
172	C₂×C₄×C₆₇	C₄×D₄	1	16	yes	—
173	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
174	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
175	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
176	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	8	no	—
187	C₂²×C₆₇	—	2	16	no	—
188	C₂²×C₆₇	—	4	8	no	—
189	C₂²×C₆₇	—	4	8	no	—
190	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
191	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
192	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
193	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
194	C₂²×C₆₇	#585	1	4	yes	—
195	C₄×C₆₇	—	2	8	no	—
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	2	8	no	—
199	C₄×C₆₇	—	4	8	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
202	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
203	#34070	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	4	no	—
205	C₂×C₆₇	—	4	8	no	—
206	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
207	C₂×C₆₇	#3428	1	2	yes	—
208	C₂×C₆₇	#3449	1	2	yes	—
209	C₂×C₆₇	#3455	1	2	yes	—
210	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
211	C₂×C₄×#41	C₂×C₆₇	1	2	yes	complemented
212	#2328	C₂×C₆₇	1	2	yes	complemented
213	#2327	C₂×C₆₇	1	2	yes	complemented
214	#2326	C₂×C₆₇	1	4	yes	complemented
215	#2325	C₂×C₆₇	1	8	yes	complemented
216	#2297	C₂×C₆₇	1	2	yes	complemented
217	#2290	C₂×C₆₇	1	8	yes	complemented
218	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#34070	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²×#41	C₂²×C₆₇	1	2	yes	—
222	#532	C₄×C₆₇	1	16	yes	complemented
223	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#124	C₂²×C₆₇	1	2	yes	—
226	C₂×#121	C₂²×C₆₇	1	4	yes	—
227	C₄×#41	—	2	16	no	—
228	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
229	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
230	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
231	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
232	C₂×#117	C₂²×C₆₇	1	2	yes	—
233	C₂×#117	C₂²×C₆₇	1	4	yes	—
234	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
235	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
236	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
237	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
238	#391	C₂²×C₆₇	1	4	yes	—
239	#390	C₂²×C₆₇	1	4	yes	—
240	#388	C₂²×C₆₇	1	16	yes	—
241	#387	C₂²×C₆₇	1	16	yes	—
242	#386	C₂²×C₆₇	1	8	yes	—
243	#386	C₂²×C₆₇	1	16	yes	—
244	#385	C₂²×C₆₇	1	16	yes	—
245	#384	C₂²×C₆₇	1	2	yes	—
246	#382	C₂²×C₆₇	1	2	yes	—
247	#380	C₂²×C₆₇	1	4	yes	—
248	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
249	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
250	#376	C₂²×C₆₇	1	2	yes	—
251	#376	C₂²×C₆₇	1	2	yes	—
252	#376	C₂²×C₆₇	1	4	yes	—
253	C₂×#95	C₂²×C₆₇	1	4	yes	—
254	C₂×#41	—	2	8	no	—
255	C₂×#41	—	2	8	no	—
256	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
257	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
258	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
259	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
260	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
261	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
262	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
263	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
267	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
268	#127	C₂×C₄×C₆₇	1	8	yes	complemented
269	#124	C₂×C₄×C₆₇	1	16	yes	complemented
270	#121	C₂×C₄×C₆₇	1	32	yes	complemented
271	C₄×Q₈	—	2	16	no	—
272	C₄×D₄	—	2	32	no	—
273	C₄×D₄	—	2	16	no	—
274	C₄×D₄	C₂×C₄×C₆₇	1	32	yes	complemented
275	#117	—	2	8	no	—
276	#117	—	2	16	no	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
281	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
282	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
285	C₂×C₄²	—	2	4	no	—
286	C₂×C₄²	—	2	16	no	—
287	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
288	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
289	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
290	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
291	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
293	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
298	#95	C₂³×C₆₇	1	8	yes	—
299	#95	C₂³×C₆₇	1	8	yes	—
300	#95	C₂³×C₆₇	1	16	yes	—
301	#95	C₂³×C₆₇	1	8	yes	—
302	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
303	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
304	#41	—	2	64	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
308	C₂×D₄	—	2	8	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	4	8	no	—
313	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
314	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
315	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	16	no	—
320	C₂²×C₄	—	4	4	no	—
321	C₂²×C₄	—	4	4	no	—
322	C₂²×C₄	—	4	16	no	—
323	C₂²×C₄	—	4	4	no	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
325	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
326	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
329	C₂²×C₄	#41×C₆₇	1	4	yes	—
330	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
331	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
338	C₄⋊C₄	—	2	16	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	4	8	no	—
341	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
342	C₂²⋊C₄	—	2	8	no	—
343	C₂²⋊C₄	—	2	16	no	—
344	C₂²⋊C₄	—	4	16	no	—
345	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
346	C₄²	—	2	8	no	—
347	C₄²	—	2	16	no	—
348	C₄²	—	2	4	no	—
349	C₄²	—	2	4	no	—
350	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
351	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
352	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
353	C₄²	C₂×D₄×C₆₇	1	8	yes	—
354	C₄²	C₂×D₄×C₆₇	1	8	yes	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	16	no	—
359	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
360	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
362	C₂³	C₂×#41×C₆₇	1	2	yes	—
363	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
364	Q₈	—	2	16	no	—
365	D₄	—	2	32	no	—
366	D₄	—	2	16	no	—
367	D₄	—	4	16	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	32	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	16	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	32	no	—
381	C₂×C₄	—	4	8	no	—
382	C₂×C₄	C₄×D₄×C₆₇	1	16	yes	—
383	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
384	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
385	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
386	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
387	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
388	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
390	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
391	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
392	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
393	C₂²	—	2	4	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	8	no	—
397	C₂²	—	2	16	no	—
398	C₂²	—	4	8	no	—
399	C₂²	—	4	8	no	—
400	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
401	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
402	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
403	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
404	C₂²	#585×C₆₇	1	4	yes	—
405	C₄	—	2	8	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	4	no	—
408	C₄	—	2	8	no	—
409	C₄	—	4	8	no	—
410	C₄	—	4	4	no	—
411	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
412	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
413	C₂	—	2	4	no	—
414	C₂	—	4	8	no	—
415	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
416	C₂	C₆₇×#3428	1	2	yes	—
417	C₂	C₆₇×#3449	1	2	yes	—
418	C₂	C₆₇×#3455	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₆₇	C₂	semidirect
3. C₆₇×#2328	C₂	semidirect
4. C₆₇×#2327	C₂	semidirect
5. C₆₇×#2326	C₂	semidirect
6. C₆₇×#2325	C₂	semidirect
7. C₆₇×#2297	C₂	semidirect
8. C₆₇×#2290	C₂	semidirect
9. C₆₇×#2285	C₂	semidirect
10. C₂×#376×C₆₇	C₂	semidirect
12. #532×C₆₇	C₄	semidirect
46. C₂×#41×C₆₇	C₂×C₄	semidirect
57. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
58. #127×C₆₇	C₂×C₄	semidirect
59. #124×C₆₇	C₂×C₄	semidirect
60. #121×C₆₇	C₂×C₄	semidirect
64. C₄×D₄×C₆₇	C₂×C₄	semidirect
77. C₂×C₄²×C₆₇	C₂×C₄	semidirect
78. C₂×C₄²×C₆₇	D₄	semidirect
203. #34070	C₆₇	direct (N × H)
211. C₂×C₄×#41	C₂×C₆₇	semidirect
212. #2328	C₂×C₆₇	semidirect
213. #2327	C₂×C₆₇	semidirect
214. #2326	C₂×C₆₇	semidirect
215. #2325	C₂×C₆₇	semidirect
216. #2297	C₂×C₆₇	semidirect
217. #2290	C₂×C₆₇	semidirect
218. #2285	C₂×C₆₇	semidirect
219. C₂×#376	C₂×C₆₇	semidirect
220. C₆₇	#34070	direct (N × H)
222. #532	C₄×C₆₇	semidirect
256. C₂×#41	C₂×C₄×C₆₇	semidirect
267. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
268. #127	C₂×C₄×C₆₇	semidirect
269. #124	C₂×C₄×C₆₇	semidirect
270. #121	C₂×C₄×C₆₇	semidirect
274. C₄×D₄	C₂×C₄×C₆₇	semidirect
287. C₂×C₄²	C₂×C₄×C₆₇	semidirect
288. 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×#532×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2328	2
C₆₇×#2327	2
C₆₇×#2326	2
C₆₇×#2325	2
C₆₇×#2297	2
C₆₇×#2290	2
C₆₇×#2285	2
C₂×#376×C₆₇	2
#34070	67

Subgroups — C67×#34070 #30713321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34070 #30713321