Subgroups — C₆₇×#11104 #30690355

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#421×C₆₇	C₂	1	2	yes	maximal
3	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	#427×C₆₇	—	2	4	no	—
12	C₂×#107×C₆₇	—	2	1	no	—
13	C₂×#106×C₆₇	—	2	1	no	—
14	#421×C₆₇	—	2	4	no	—
15	#421×C₆₇	—	2	4	no	—
16	C₂×#104×C₆₇	—	2	4	no	—
17	#408×C₆₇	—	2	1	no	—
18	#408×C₆₇	—	2	1	no	—
19	#408×C₆₇	C₄	1	4	yes	complemented
20	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
21	#344×C₆₇	—	2	4	no	—
22	#341×C₆₇	—	2	2	no	—
23	C₂×#41×C₆₇	—	2	1	no	—
24	C₂×#41×C₆₇	—	2	1	no	—
25	C₂×#41×C₆₇	—	2	2	no	—
26	C₂×#41×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×#41×C₆₇	D₄	1	2	yes	—
31	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	—	2	1	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	D₄	1	2	yes	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	D₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	—	2	2	no	—
61	#107×C₆₇	—	2	4	no	—
62	#106×C₆₇	—	2	4	no	—
63	#104×C₆₇	—	2	16	no	—
64	C₂²⋊C₈×C₆₇	—	2	4	no	—
65	C₂²⋊C₈×C₆₇	—	2	4	no	—
66	C₂⁴×C₆₇	—	2	1	no	—
67	C₂⁴×C₆₇	#41	1	1	yes	characteristic
68	#41×C₆₇	—	2	4	no	—
69	#41×C₆₇	—	2	4	no	—
70	#41×C₆₇	—	2	4	no	—
71	#41×C₆₇	—	2	4	no	—
72	#41×C₆₇	—	4	1	no	—
73	#41×C₆₇	—	4	1	no	—
74	#41×C₆₇	—	4	2	no	—
75	#41×C₆₇	—	4	1	no	—
76	#41×C₆₇	—	4	2	no	—
77	#41×C₆₇	—	4	1	no	—
78	C₂×Q₈×C₆₇	—	2	1	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	4	2	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	2	yes	—
104	C₂²×C₄×C₆₇	#41	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
106	M₁₆×C₆₇	—	2	4	no	—
107	M₁₆×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	4	2	no	—
110	M₁₆×C₆₇	—	4	2	no	—
111	M₁₆×C₆₇	—	4	1	no	—
112	M₁₆×C₆₇	—	4	1	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	4	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	4	no	—
121	C₂×C₈×C₆₇	—	4	2	no	—
122	C₂×C₈×C₆₇	—	4	2	no	—
123	C₂×C₈×C₆₇	—	4	2	no	—
124	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₂²⋊C₄×C₆₇	—	2	4	no	—
129	C₂²⋊C₄×C₆₇	—	4	2	no	—
130	C₄²×C₆₇	—	2	4	no	—
131	C₄²×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	—	4	4	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	#117	1	1	yes	characteristic
137	C₂³×C₆₇	#121	1	1	yes	characteristic
138	C₂³×C₆₇	#121	1	2	yes	—
139	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂³×C₆₇	#126	1	2	yes	—
141	Q₈×C₆₇	—	2	4	no	—
142	Q₈×C₆₇	—	4	1	no	—
143	Q₈×C₆₇	—	4	1	no	—
144	D₄×C₆₇	—	2	4	no	—
145	D₄×C₆₇	—	4	4	no	—
146	D₄×C₆₇	—	4	2	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	4	1	no	—
149	D₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	4	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	—	8	4	no	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₆₇	#120	1	2	yes	—
162	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#121	1	2	yes	—
164	C₂×C₄×C₆₇	#121	1	2	yes	—
165	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#123	1	2	yes	—
170	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#123	1	2	yes	—
172	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#124	1	2	yes	—
174	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	4	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	4	2	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	8	4	no	—
187	C₂²×C₆₇	#380	1	1	yes	characteristic
188	C₂²×C₆₇	#388	1	2	yes	—
189	C₂²×C₆₇	#392	1	1	yes	characteristic
190	C₂²×C₆₇	#396	1	2	yes	—
191	C₂²×C₆₇	#397	1	1	yes	characteristic
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	—	8	4	no	—
194	C₄×C₆₇	#385	1	2	yes	—
195	C₄×C₆₇	#386	1	1	yes	characteristic
196	C₄×C₆₇	#390	1	1	yes	characteristic
197	C₄×C₆₇	#393	1	1	yes	characteristic
198	C₄×C₆₇	#394	1	2	yes	—
199	C₄×C₆₇	#399	1	1	yes	characteristic
200	#11104	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	2	no	—
203	C₂×C₆₇	—	2	2	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	8	2	no	—
206	C₂×C₆₇	#1439	1	1	yes	characteristic
207	C₂×C₆₇	#1814	1	2	yes	—
208	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₂×#421	C₂×C₆₇	1	2	yes	—
210	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
211	C₂×#344	C₂×C₆₇	1	2	yes	complemented
212	C₂×#341	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₆₇	#11104	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
215	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
217	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#117	C₂²×C₆₇	1	2	yes	—
219	#427	—	2	4	no	—
220	C₂×#107	—	2	1	no	—
221	C₂×#106	—	2	1	no	—
222	#421	—	2	4	no	—
223	#421	—	2	4	no	—
224	C₂×#104	—	2	4	no	—
225	#408	—	2	1	no	—
226	#408	—	2	1	no	—
227	#408	C₄×C₆₇	1	4	yes	complemented
228	C₂×C₂²⋊C₈	—	2	2	no	—
229	#344	—	2	4	no	—
230	#341	—	2	2	no	—
231	C₂×#41	—	2	1	no	—
232	C₂×#41	—	2	1	no	—
233	C₂×#41	—	2	2	no	—
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	2	2	no	—
236	C₂×#41	—	2	2	no	—
237	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
238	C₂×#41	D₄×C₆₇	1	2	yes	—
239	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
240	C₂²×D₄	—	2	1	no	—
241	C₂²×D₄	Q₈×C₆₇	1	1	yes	characteristic
242	C₂³×C₄	—	2	1	no	—
243	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
244	C₂×M₁₆	—	2	1	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
259	C₂²×C₈	D₄×C₆₇	1	2	yes	—
260	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
262	#117	—	2	2	no	—
263	#117	—	2	2	no	—
264	#117	—	2	2	no	—
265	#117	D₄×C₆₇	1	4	yes	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₄²	—	2	2	no	—
269	#107	—	2	4	no	—
270	#106	—	2	4	no	—
271	#104	—	2	16	no	—
272	C₂²⋊C₈	—	2	4	no	—
273	C₂²⋊C₈	—	2	4	no	—
274	C₂⁴	—	2	1	no	—
275	C₂⁴	#41×C₆₇	1	1	yes	characteristic
276	#41	—	2	4	no	—
277	#41	—	2	4	no	—
278	#41	—	2	4	no	—
279	#41	—	2	4	no	—
280	#41	—	4	1	no	—
281	#41	—	4	1	no	—
282	#41	—	4	2	no	—
283	#41	—	4	1	no	—
284	#41	—	4	2	no	—
285	#41	—	4	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	4	2	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	#41×C₆₇	1	2	yes	—
312	C₂²×C₄	#41×C₆₇	1	2	yes	—
313	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
314	M₁₆	—	2	4	no	—
315	M₁₆	—	2	4	no	—
316	M₁₆	—	2	4	no	—
317	M₁₆	—	4	2	no	—
318	M₁₆	—	4	2	no	—
319	M₁₆	—	4	1	no	—
320	M₁₆	—	4	1	no	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	4	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
333	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₂²⋊C₄	—	2	4	no	—
337	C₂²⋊C₄	—	4	2	no	—
338	C₄²	—	2	4	no	—
339	C₄²	—	4	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	4	no	—
343	C₂³	—	4	2	no	—
344	C₂³	#117×C₆₇	1	1	yes	characteristic
345	C₂³	#121×C₆₇	1	1	yes	characteristic
346	C₂³	#121×C₆₇	1	2	yes	—
347	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
348	C₂³	#126×C₆₇	1	2	yes	—
349	Q₈	—	2	4	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	4	1	no	—
352	D₄	—	2	4	no	—
353	D₄	—	4	4	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	1	no	—
357	D₄	—	4	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	8	4	no	—
365	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
368	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
369	C₂×C₄	#120×C₆₇	1	2	yes	—
370	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
371	C₂×C₄	#121×C₆₇	1	2	yes	—
372	C₂×C₄	#121×C₆₇	1	2	yes	—
373	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
375	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
376	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
377	C₂×C₄	#123×C₆₇	1	2	yes	—
378	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
379	C₂×C₄	#123×C₆₇	1	2	yes	—
380	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
381	C₂×C₄	#124×C₆₇	1	2	yes	—
382	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	4	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	8	4	no	—
395	C₂²	#380×C₆₇	1	1	yes	characteristic
396	C₂²	#388×C₆₇	1	2	yes	—
397	C₂²	#392×C₆₇	1	1	yes	characteristic
398	C₂²	#396×C₆₇	1	2	yes	—
399	C₂²	#397×C₆₇	1	1	yes	characteristic
400	C₄	—	4	2	no	—
401	C₄	—	8	4	no	—
402	C₄	#385×C₆₇	1	2	yes	—
403	C₄	#386×C₆₇	1	1	yes	characteristic
404	C₄	#390×C₆₇	1	1	yes	characteristic
405	C₄	#393×C₆₇	1	1	yes	characteristic
406	C₄	#394×C₆₇	1	2	yes	—
407	C₄	#399×C₆₇	1	1	yes	characteristic
408	C₂	—	2	1	no	—
409	C₂	—	2	2	no	—
410	C₂	—	2	2	no	—
411	C₂	—	2	1	no	—
412	C₂	—	8	2	no	—
413	C₂	C₆₇×#1439	1	1	yes	characteristic
414	C₂	C₆₇×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#380
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#427×C₆₇	C₂	semidirect
4. C₂×#344×C₆₇	C₂	semidirect
5. C₂×#341×C₆₇	C₂	semidirect
19. #408×C₆₇	C₄	semidirect
200. #11104	C₆₇	direct (N × H)
208. C₂×#427	C₂×C₆₇	semidirect
211. C₂×#344	C₂×C₆₇	semidirect
212. C₂×#341	C₂×C₆₇	semidirect
213. C₆₇	#11104	direct (N × H)
227. #408	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₆₇	2
C₂×#421×C₆₇	2
C₂×#408×C₆₇	2
C₂×#344×C₆₇	2
C₂×#341×C₆₇	2
#11104	67

Subgroups — C67×#11104 #30690355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11104 #30690355