Subgroups — #30825072

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22568875	C₂	1	1	yes	maximal, characteristic
3	#22558990	C₂	1	2	yes	maximal, complemented
4	#22558987	C₂	1	2	yes	maximal, complemented
5	#22558981	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558979	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558970	C₂	1	2	yes	maximal, complemented
8	#22558967	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558955	C₂	1	2	yes	maximal, complemented
10	C₂×#16863148	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic
12	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	D₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	D₄×C₆₇⋊C₈	C₄	1	2	yes	—
16	D₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
17	D₄×C₆₇⋊C₈	C₄	1	2	yes	—
18	#16863457	—	2	2	no	—
19	#16863453	C₂²	1	2	yes	—
20	#16863449	C₂²	1	2	yes	—
21	#16863449	C₂²	1	1	yes	characteristic
22	#16863449	C₂²	1	2	yes	—
23	#16863420	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	#16863363	—	2	1	no	—
26	#16863352	—	2	1	no	—
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	1	yes	characteristic
33	#16863148	C₂²	1	1	yes	characteristic
34	#16863148	C₂²	1	2	yes	—
35	#16863146	C₂²	1	2	yes	—
36	#16863146	C₂²	1	2	yes	—
37	#16863146	C₂²	1	2	yes	—
38	#16863146	C₂²	1	2	yes	—
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#433×C₆₇	—	2	2	no	—
41	#432×C₆₇	C₂²	1	2	yes	—
42	#431×C₆₇	C₂²	1	2	yes	—
43	#423×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#408×C₆₇	—	2	1	no	—
46	#405×C₆₇	—	2	1	no	—
47	C₂×#11761612	—	2	1	no	—
48	C₂²×C₆₇⋊C₈	—	2	1	no	—
49	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
51	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
56	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
57	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
59	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
63	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
65	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
67	#14523954	—	2	2	no	—
68	C₄×C₆₇⋊C₈	—	2	2	no	—
69	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
73	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
77	C₂×M₁₆×C₆₇	—	2	1	no	—
78	C₂²×C₈×C₆₇	—	2	1	no	—
79	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	Q₈	1	1	yes	characteristic
81	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×D₄×C₆₇	Q₈	1	2	yes	—
84	C₄×D₄×C₆₇	Q₈	1	1	yes	characteristic
85	#117×C₆₇	D₄	1	1	yes	characteristic
86	#117×C₆₇	D₄	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	—	2	1	no	—
91	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
94	C₂²⋊C₈×C₆₇	—	2	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	2	no	—
96	#97×C₆₇	—	2	2	no	—
97	C₄×C₈×C₆₇	—	2	2	no	—
98	#11761612	—	4	2	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	4	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	1	no	—
104	C₂×C₆₇⋊C₈	—	4	1	no	—
105	C₂×C₆₇⋊C₈	—	4	2	no	—
106	C₂×C₆₇⋊C₈	—	4	2	no	—
107	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	#41×C₆₇	—	2	2	no	—
114	#41×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
117	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
124	M₁₆×C₆₇	—	4	2	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
130	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
134	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
135	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
136	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
137	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
141	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
144	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	Q₈×C₆₇	—	2	2	no	—
146	D₄×C₆₇	—	2	4	no	—
147	D₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#135	1	2	yes	—
163	C₂×C₄×C₆₇	#135	1	2	yes	—
164	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
167	C₈×C₆₇	—	4	1	no	—
168	C₈×C₆₇	—	4	1	no	—
169	C₈×C₆₇	—	4	2	no	—
170	C₆₇⋊C₈	—	2	2	no	—
171	C₆₇⋊C₈	—	2	1	no	—
172	C₆₇⋊C₈	—	2	1	no	—
173	C₆₇⋊C₈	—	2	2	no	—
174	C₆₇⋊C₈	—	2	1	no	—
175	C₆₇⋊C₈	—	4	1	no	—
176	C₆₇⋊C₈	—	4	1	no	—
177	C₆₇⋊C₈	—	4	2	no	—
178	C₆₇⋊C₈	#131	1	2	yes	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	#427	1	2	yes	—
185	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	#427	1	2	yes	—
193	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
194	C₄×C₆₇	#568	1	1	yes	characteristic
195	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
197	#17829	—	67	1	no	maximal, Sylow
198	C₂×C₆₇	—	2	2	no	—
199	C₂×C₆₇	—	4	2	no	—
200	C₂×C₆₇	#2906	1	1	yes	characteristic
201	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
202	C₂×C₆₇	#2916	1	1	yes	characteristic
203	C₈×#41	—	67	1	no	—
204	#2925	—	67	1	no	—
205	#2925	D₆₇	1	1	yes	characteristic
206	#1572	—	67	2	no	—
207	#1572	—	67	2	no	—
208	#1570	—	67	1	no	—
209	#1570	—	67	1	no	—
210	#1568	—	67	2	no	—
211	#1567	—	67	1	no	—
212	#1564	—	67	2	no	—
213	C₂×#335	—	67	1	no	—
214	C₆₇	#17829	1	1	yes	characteristic, complemented, Sylow
215	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
216	C₈×Q₈	—	67	1	no	—
217	C₈×Q₈	—	67	1	no	—
218	C₈×D₄	—	67	1	no	—
219	C₈×D₄	—	67	2	no	—
220	C₈×D₄	—	67	1	no	—
221	C₈×D₄	—	67	2	no	—
222	#433	—	2	2	no	—
223	#433	—	134	2	no	—
224	#432	—	67	2	no	—
225	#432	C₂×D₆₇	1	2	yes	—
226	#431	—	67	1	no	—
227	#431	—	67	2	no	—
228	#431	—	67	2	no	—
229	#431	C₂×D₆₇	1	2	yes	—
230	#423	—	67	1	no	—
231	#423	C₂×D₆₇	1	1	yes	characteristic
232	C₂×C₄⋊C₈	—	67	1	no	—
233	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
234	#408	—	2	1	no	—
235	#408	—	134	1	no	—
236	#405	—	2	1	no	—
237	#405	—	134	1	no	—
238	C₂×C₄×C₈	—	67	1	no	—
239	C₂×C₄×C₈	—	67	2	no	—
240	#335	—	67	1	no	—
241	#335	—	67	1	no	—
242	#335	—	67	1	no	—
243	#335	—	67	1	no	—
244	#335	—	67	2	no	—
245	#335	—	67	2	no	—
246	#334	—	67	2	no	—
247	#334	—	67	2	no	—
248	#334	—	67	2	no	—
249	#334	—	67	2	no	—
250	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	—	134	1	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	67	1	no	—
255	C₂²×C₈	—	67	2	no	—
256	C₂²×C₈	—	67	1	no	—
257	C₂²×C₈	—	67	2	no	—
258	C₂²×C₈	—	134	1	no	—
259	C₄×Q₈	Dic₁₃₄	1	1	yes	characteristic
260	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
261	C₄×D₄	Dic₁₃₄	1	2	yes	—
262	C₄×D₄	Dic₁₃₄	1	1	yes	characteristic
263	C₄×D₄	C₄×D₆₇	1	2	yes	—
264	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
265	#117	C₂²⋊D₆₇	1	1	yes	characteristic
266	#117	C₂²⋊D₆₇	1	2	yes	—
267	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
268	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	67	1	no	—
272	C₄⋊C₈	—	67	1	no	—
273	C₄⋊C₈	—	67	2	no	—
274	C₄⋊C₈	—	67	2	no	—
275	C₄⋊C₈	—	67	1	no	—
276	C₄⋊C₈	—	67	1	no	—
277	C₄⋊C₈	—	67	2	no	—
278	C₄⋊C₈	—	134	1	no	—
279	C₄⋊C₈	—	134	1	no	—
280	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
281	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
282	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	67	2	no	—
286	C₂²⋊C₈	—	67	2	no	—
287	C₂²⋊C₈	—	67	2	no	—
288	C₂²⋊C₈	—	134	2	no	—
289	C₂²⋊C₈	—	134	2	no	—
290	#97	—	2	2	no	—
291	#97	—	134	2	no	—
292	C₄×C₈	—	2	2	no	—
293	C₄×C₈	—	67	1	no	—
294	C₄×C₈	—	67	2	no	—
295	C₄×C₈	—	67	1	no	—
296	C₄×C₈	—	67	2	no	—
297	C₄×C₈	—	67	2	no	—
298	C₄×C₈	—	67	1	no	—
299	C₄×C₈	—	67	1	no	—
300	C₄×C₈	—	134	2	no	—
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	Dic₆₇⋊C₄	1	1	yes	characteristic
304	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
305	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
306	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
307	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
309	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
311	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
312	M₁₆	—	4	2	no	—
313	M₁₆	—	268	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	4	2	no	—
318	C₂×C₈	—	268	2	no	—
319	C₂×C₈	—	268	1	no	—
320	C₂×C₈	—	268	2	no	—
321	C₂×C₈	—	67	1	no	—
322	C₂×C₈	—	67	1	no	—
323	C₂×C₈	—	67	2	no	—
324	C₂×C₈	—	67	1	no	—
325	C₂×C₈	—	67	2	no	—
326	C₂×C₈	—	67	1	no	—
327	C₂×C₈	—	134	1	no	—
328	C₂×C₈	—	134	2	no	—
329	C₂×C₈	—	134	1	no	—
330	C₂×C₈	—	134	2	no	—
331	C₂×C₈	—	134	4	no	—
332	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
333	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
334	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
335	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
336	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
337	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
338	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
339	C₄²	C₂×C₄×D₆₇	1	2	yes	—
340	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
341	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
342	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
343	C₄²	C₂×Dic₁₃₄	1	2	yes	—
344	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
345	C₄²	C₂³×D₆₇	1	1	yes	characteristic
346	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
347	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
348	Q₈	—	2	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
360	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
361	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
362	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
363	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
364	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
365	C₂×C₄	#14524063	1	2	yes	—
366	C₂×C₄	#14524063	1	2	yes	—
367	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
370	C₈	—	4	1	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	268	1	no	—
374	C₈	—	268	1	no	—
375	C₈	—	268	2	no	—
376	C₈	—	67	2	no	—
377	C₈	—	134	2	no	—
378	C₈	—	134	1	no	—
379	C₈	—	134	1	no	—
380	C₈	—	134	2	no	—
381	C₈	—	134	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	#16863432	1	2	yes	—
388	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	2	2	no	—
391	C₄	—	2	2	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	2	no	—
395	C₄	#16863432	1	2	yes	—
396	C₄	#131×D₆₇	1	1	yes	characteristic
397	C₄	#16864161	1	1	yes	characteristic
398	C₄	C₂×#14524063	1	1	yes	characteristic
399	C₄	C₂×#14524063	1	1	yes	characteristic
400	C₂	—	2	2	no	—
401	C₂	—	4	2	no	—
402	C₂	#22568775	1	1	yes	characteristic
403	C₂	C₂×#16863432	1	1	yes	characteristic
404	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×#2925	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2925	D₆₇
hypocenter	C₆₇	#17829
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2925

Complements

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

N	H ≅ G/N	type
3. #22558990	C₂	semidirect
4. #22558987	C₂	semidirect
5. #22558981	C₂	semidirect
6. #22558979	C₂	semidirect
7. #22558970	C₂	semidirect
8. #22558967	C₂	semidirect
9. #22558955	C₂	semidirect
10. C₂×#16863148	C₂	semidirect
214. C₆₇	#17829	semidirect

Maximal subgroups

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

subgroup	index
#41×C₆₇⋊C₈	2
#22568875	2
#22558990	2
#22558987	2
#22558981	2
#22558979	2
#22558970	2
#22558967	2
#22558955	2
C₂×#16863148	2
C₆₇×#2925	2
#17829	67