Subgroups — #30824871

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569326	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524050	C₂	1	1	yes	maximal, characteristic
3	#22558977	C₂	1	2	yes	maximal, complemented
4	#22558974	C₂	1	2	yes	maximal, complemented
5	#22558970	C₂	1	2	yes	maximal, complemented
6	#22558964	C₂	1	2	yes	maximal, complemented
7	#22558962	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558959	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2989	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₄	1	1	yes	characteristic
13	C₂×#14524050	C₄	1	1	yes	characteristic
14	#16863527	—	2	1	no	—
15	#16863474	—	2	1	no	—
16	#16863468	—	2	2	no	—
17	#16863453	C₂²	1	2	yes	—
18	#16863449	C₂²	1	1	yes	characteristic
19	#16863420	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#16863352	C₄	1	2	yes	—
22	#16863352	C₄	1	2	yes	—
23	#16863352	C₄	1	1	yes	characteristic
24	#16863352	C₄	1	1	yes	characteristic
25	C₄×#11761612	C₂²	1	2	yes	—
26	C₄×#11761612	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
29	#16863148	C₂²	1	1	yes	characteristic
30	#16863148	C₂²	1	2	yes	—
31	#16863148	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	2	yes	—
33	#16863146	C₂²	1	2	yes	—
34	#16863146	C₂²	1	2	yes	—
35	#16863146	C₂²	1	2	yes	—
36	#16863146	C₂²	1	1	yes	characteristic
37	#16863146	C₂²	1	1	yes	characteristic
38	#16863146	C₂²	1	2	yes	—
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	—	2	1	no	—
41	#436×C₆₇	—	2	1	no	—
42	#435×C₆₇	—	2	2	no	—
43	#432×C₆₇	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#14524050	—	2	4	no	—
48	#14524050	—	2	4	no	—
49	C₂×#11761612	—	2	2	no	—
50	C₂×#11761612	C₂×C₄	1	2	yes	—
51	C₂×#11761612	C₂×C₄	1	2	yes	—
52	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
56	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
57	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
63	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
64	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
66	#14523954	—	2	2	no	—
67	#14523954	C₂×C₄	1	2	yes	—
68	#14523954	C₂×C₄	1	2	yes	—
69	#14523954	C₂×C₄	1	2	yes	—
70	C₄×C₆₇⋊C₈	—	2	1	no	—
71	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
72	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
78	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×M₁₆×C₆₇	—	2	2	no	—
80	C₂²×C₈×C₆₇	—	2	1	no	—
81	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
83	C₄×D₄×C₆₇	D₄	1	2	yes	—
84	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
85	C₄×D₄×C₆₇	Q₈	1	2	yes	—
86	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
87	#117×C₆₇	C₂×C₄	1	2	yes	—
88	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
91	C₄⋊C₈×C₆₇	—	2	1	no	—
92	C₄⋊C₈×C₆₇	—	2	2	no	—
93	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
94	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	C₂²⋊C₈×C₆₇	—	2	2	no	—
98	#97×C₆₇	—	2	2	no	—
99	C₄×C₈×C₆₇	—	2	1	no	—
100	#11761612	—	2	2	no	—
101	#11761612	—	2	4	no	—
102	#11761612	—	2	4	no	—
103	#11761612	—	8	2	no	—
104	#11761612	M₁₆	1	4	yes	complemented
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	4	no	—
107	C₂×C₆₇⋊C₈	—	2	4	no	—
108	C₂×C₆₇⋊C₈	—	2	1	no	—
109	C₂×C₆₇⋊C₈	—	2	1	no	—
110	C₂×C₆₇⋊C₈	—	2	2	no	—
111	C₂×C₆₇⋊C₈	—	8	1	no	—
112	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	complemented
113	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
119	#41×C₆₇	—	2	2	no	—
120	#41×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
122	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
123	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
130	M₁₆×C₆₇	—	8	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	8	1	no	—
135	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
136	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
137	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
140	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
141	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
142	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
144	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₆₇	C₂×D₄	1	2	yes	—
146	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
147	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
148	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
150	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	4	no	—
153	D₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₆₇⋊C₈	—	2	1	no	—
177	C₆₇⋊C₈	—	2	1	no	—
178	C₆₇⋊C₈	—	2	2	no	—
179	C₆₇⋊C₈	—	4	1	no	—
180	C₆₇⋊C₈	—	4	2	no	—
181	C₆₇⋊C₈	—	4	1	no	—
182	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
183	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
184	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
185	C₂²×C₆₇	—	2	1	no	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	#423	1	2	yes	—
191	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	2	1	no	—
196	C₄×C₆₇	—	2	1	no	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	#423	1	2	yes	—
199	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₆₇	#577	1	1	yes	characteristic
202	C₄×C₆₇	#577	1	1	yes	characteristic
203	#17798	—	67	1	no	maximal, Sylow
204	C₂×C₆₇	—	2	2	no	—
205	C₂×C₆₇	—	4	2	no	—
206	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
207	C₂×C₆₇	#2912	1	1	yes	characteristic
208	C₂×C₆₇	#2916	1	1	yes	characteristic
209	#2989	—	67	1	no	—
210	#2989	D₆₇	1	1	yes	characteristic
211	C₄×#131	—	67	1	no	—
212	#1569	—	67	2	no	—
213	#1569	—	67	2	no	—
214	#1568	—	67	2	no	—
215	#1566	—	67	2	no	—
216	#1565	—	67	1	no	—
217	#1565	—	67	1	no	—
218	#1564	—	67	1	no	—
219	C₂×#334	—	67	1	no	—
220	C₆₇	#17798	1	1	yes	characteristic, complemented, Sylow
221	C₂×#131	—	67	1	no	—
222	C₂×#131	—	67	1	no	—
223	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
224	#446	—	2	1	no	—
225	#446	—	134	1	no	—
226	#436	—	2	1	no	—
227	#436	—	134	1	no	—
228	#435	—	2	2	no	—
229	#435	—	134	2	no	—
230	#432	—	67	2	no	—
231	#432	C₂×D₆₇	1	2	yes	—
232	#431	—	67	1	no	—
233	#431	C₂×D₆₇	1	1	yes	characteristic
234	#423	—	67	2	no	—
235	#423	C₂×D₆₇	1	2	yes	—
236	C₂×C₄⋊C₈	—	67	1	no	—
237	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
238	#405	—	67	1	no	—
239	#405	—	67	2	no	—
240	#405	—	67	2	no	—
241	#405	—	67	1	no	—
242	C₄×M₁₆	—	67	1	no	—
243	C₄×M₁₆	—	67	2	no	—
244	C₂×C₄×C₈	—	67	1	no	—
245	C₂×C₄×C₈	—	67	2	no	—
246	#335	—	67	1	no	—
247	#335	—	67	1	no	—
248	#335	—	67	2	no	—
249	#335	—	67	2	no	—
250	#334	—	67	2	no	—
251	#334	—	67	2	no	—
252	#334	—	67	1	no	—
253	#334	—	67	1	no	—
254	#334	—	67	2	no	—
255	#334	—	67	2	no	—
256	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
257	#131	—	134	4	no	—
258	#131	—	134	4	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	67	1	no	—
261	C₂×M₁₆	—	67	2	no	—
262	C₂×M₁₆	—	67	2	no	—
263	C₂×M₁₆	—	67	1	no	—
264	C₂×M₁₆	—	134	2	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	67	1	no	—
267	C₂²×C₈	—	67	2	no	—
268	C₂²×C₈	—	67	2	no	—
269	C₂²×C₈	—	67	1	no	—
270	C₂²×C₈	—	134	1	no	—
271	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
272	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
273	C₄×D₄	Dic₁₃₄	1	2	yes	—
274	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
275	C₄×D₄	C₂²⋊D₆₇	1	2	yes	—
276	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
277	#117	C₄×D₆₇	1	2	yes	—
278	#117	C₄×D₆₇	1	1	yes	characteristic
279	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
280	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	67	2	no	—
284	C₄⋊C₈	—	67	1	no	—
285	C₄⋊C₈	—	67	1	no	—
286	C₄⋊C₈	—	134	1	no	—
287	C₄⋊C₈	—	134	2	no	—
288	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
289	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	134	1	no	—
294	C₂²⋊C₈	—	134	2	no	—
295	#97	—	2	2	no	—
296	#97	—	67	2	no	—
297	#97	—	67	2	no	—
298	#97	—	67	2	no	—
299	#97	—	134	2	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	67	2	no	—
302	C₄×C₈	—	67	1	no	—
303	C₄×C₈	—	67	1	no	—
304	C₄×C₈	—	67	2	no	—
305	C₄×C₈	—	67	1	no	—
306	C₄×C₈	—	67	1	no	—
307	C₄×C₈	—	67	2	no	—
308	C₄×C₈	—	134	1	no	—
309	#41	—	2	2	no	—
310	#41	—	2	2	no	—
311	C₂×Q₈	Dic₆₇⋊C₄	1	1	yes	characteristic
312	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
313	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
314	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
315	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
318	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
319	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
320	M₁₆	—	8	2	no	—
321	M₁₆	—	536	2	no	—
322	M₁₆	—	67	4	no	—
323	M₁₆	—	134	4	no	—
324	M₁₆	—	134	4	no	—
325	M₁₆	—	134	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	8	1	no	—
330	C₂×C₈	—	536	1	no	—
331	C₂×C₈	—	67	4	no	—
332	C₂×C₈	—	67	1	no	—
333	C₂×C₈	—	67	1	no	—
334	C₂×C₈	—	67	2	no	—
335	C₂×C₈	—	67	1	no	—
336	C₂×C₈	—	67	2	no	—
337	C₂×C₈	—	67	1	no	—
338	C₂×C₈	—	134	1	no	—
339	C₂×C₈	—	134	1	no	—
340	C₂×C₈	—	134	2	no	—
341	C₂×C₈	—	134	4	no	—
342	C₂×C₈	—	134	4	no	—
343	C₂×C₈	—	134	2	no	—
344	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
345	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
346	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
347	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
348	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
349	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
350	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
351	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
352	C₄²	C₂×C₄×D₆₇	1	2	yes	—
353	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
354	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
355	C₄²	C₂×Dic₁₃₄	1	2	yes	—
356	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
357	C₄²	C₂³×D₆₇	1	1	yes	characteristic
358	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
359	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
372	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
373	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
374	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
375	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
376	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
377	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
378	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
379	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	268	2	no	—
386	C₈	—	268	1	no	—
387	C₈	—	268	1	no	—
388	C₈	—	67	2	no	—
389	C₈	—	67	2	no	—
390	C₈	—	67	4	no	—
391	C₈	—	134	1	no	—
392	C₈	—	134	2	no	—
393	C₈	—	134	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	#16863419	1	2	yes	—
400	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
401	C₄	—	2	2	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	2	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	#16863419	1	2	yes	—
408	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
409	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
410	C₄	#16864204	1	1	yes	characteristic
411	C₄	#16864207	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	4	2	no	—
414	C₂	C₂×#16863419	1	1	yes	characteristic
415	C₂	#22568802	1	1	yes	characteristic
416	C₂	#22568818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2989	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2989	D₆₇
hypocenter	C₆₇	#17798
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2989

Complements

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

N	H ≅ G/N	type
3. #22558977	C₂	semidirect
4. #22558974	C₂	semidirect
5. #22558970	C₂	semidirect
6. #22558964	C₂	semidirect
7. #22558962	C₂	semidirect
8. #22558959	C₂	semidirect
9. #22558955	C₂	semidirect
10. C₂×#16863146	C₂	semidirect
104. #11761612	M₁₆	semidirect
112. C₂×C₆₇⋊C₈	M₁₆	semidirect
220. C₆₇	#17798	semidirect

Maximal subgroups

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

subgroup	index
#22569326	2
C₄×#14524050	2
#22558977	2
#22558974	2
#22558970	2
#22558964	2
#22558962	2
#22558959	2
#22558955	2
C₂×#16863146	2
C₆₇×#2989	2
#17798	67