Subgroups — #30761649

Order 17152.

1126 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863266	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863158	C₂	1	4	yes	maximal
4	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₆₇	—	2	2	no	—
7	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
9	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
10	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	complemented
11	#16863266	—	2	2	no	—
12	#16863266	C₄	1	4	yes	complemented
13	#16863158	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#384×C₆₇	—	2	2	no	—
18	#384×C₆₇	C₄	1	4	yes	complemented
19	C₂³×Dic₆₇	—	2	2	no	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₆₇	—	2	8	no	—
28	C₂×C₄⋊Dic₆₇	—	2	4	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	4	no	—
31	C₂×C₄⋊Dic₆₇	—	4	4	no	—
32	C₂×C₄⋊Dic₆₇	—	4	4	no	—
33	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
34	C₄×C₆₇⋊C₈	—	2	8	no	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	D₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
51	C₂²×Dic₆₇	—	2	4	no	—
52	C₂²×Dic₆₇	—	4	2	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	C₂²×Dic₆₇	—	4	4	no	—
55	C₂²×Dic₆₇	—	4	4	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
63	C₄⋊Dic₆₇	—	2	8	no	—
64	C₄⋊Dic₆₇	—	4	16	no	—
65	C₄⋊Dic₆₇	—	4	4	no	—
66	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	D₈	1	4	yes	—
76	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
77	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
78	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	#41	1	2	yes	—
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄⋊C₄×C₆₇	—	4	16	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄²×C₆₇	C₄²	1	4	yes	—
93	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
95	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₆₇	#124	1	1	yes	characteristic
100	C₂³×C₆₇	#124	1	1	yes	characteristic
101	C₂³×C₆₇	#127	1	1	yes	characteristic
102	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₂³×C₆₇	C₂×D₈	1	2	yes	—
104	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
105	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
106	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	8	no	—
109	C₂×C₄×C₆₇	—	8	4	no	—
110	C₂×C₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	—	8	2	no	—
112	C₂×C₄×C₆₇	#95	1	8	yes	—
113	C₂×C₄×C₆₇	#102	1	8	yes	—
114	C₂×C₄×C₆₇	#102	1	8	yes	—
115	C₂×C₄×C₆₇	#103	1	8	yes	—
116	C₂×C₄×C₆₇	#103	1	8	yes	—
117	C₂×C₄×C₆₇	#106	1	8	yes	—
118	C₂×C₄×C₆₇	#107	1	8	yes	—
119	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	2	yes	—
125	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	#121	1	2	yes	—
130	C₂×C₄×C₆₇	#121	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₆₇	#125	1	2	yes	—
134	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
135	C₂×Dic₆₇	—	4	4	no	—
136	C₂×Dic₆₇	—	4	8	no	—
137	C₂×Dic₆₇	—	8	4	no	—
138	C₂×Dic₆₇	—	8	2	no	—
139	C₂×Dic₆₇	—	8	2	no	—
140	C₆₇⋊C₈	—	2	8	no	—
141	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
142	C₂²×C₆₇	#382	1	1	yes	characteristic
143	C₂²×C₆₇	#384	1	1	yes	characteristic
144	C₂²×C₆₇	#390	1	1	yes	characteristic
145	C₂²×C₆₇	#390	1	1	yes	characteristic
146	C₂²×C₆₇	#391	1	1	yes	characteristic
147	C₂²×C₆₇	#391	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#102	1	2	yes	—
149	C₂²×C₆₇	C₂×#102	1	2	yes	—
150	C₂²×C₆₇	C₂×#103	1	2	yes	—
151	C₂²×C₆₇	C₂×#103	1	2	yes	—
152	C₂²×C₆₇	C₂×#106	1	2	yes	—
153	C₂²×C₆₇	C₂×#107	1	2	yes	—
154	C₂²×C₆₇	#486	1	2	yes	—
155	C₂²×C₆₇	#486	1	2	yes	—
156	C₂²×C₆₇	#487	1	2	yes	—
157	C₂²×C₆₇	#487	1	2	yes	—
158	C₂²×C₆₇	#492	1	1	yes	characteristic
159	C₂²×C₆₇	#492	1	1	yes	characteristic
160	C₂²×C₆₇	#493	1	1	yes	characteristic
161	C₂²×C₆₇	#494	1	1	yes	characteristic
162	C₂²×C₆₇	#498	1	2	yes	—
163	C₂²×C₆₇	#500	1	2	yes	—
164	C₄×C₆₇	—	8	4	no	—
165	C₄×C₆₇	#340	1	16	yes	complemented
166	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
167	C₄×C₆₇	#380	1	2	yes	—
168	C₄×C₆₇	#389	1	2	yes	—
169	C₄×C₆₇	#389	1	2	yes	—
170	Dic₆₇	—	8	4	no	—
171	#10841	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
173	C₂×C₆₇	C₂×#340	1	2	yes	—
174	C₂×C₆₇	#1834	1	2	yes	—
175	C₂×C₆₇	#1844	1	1	yes	characteristic
176	C₂×C₆₇	#1845	1	1	yes	characteristic
177	C₂×C₆₇	#1956	1	1	yes	characteristic
178	C₂×C₆₇	#1956	1	1	yes	characteristic
179	C₂×C₆₇	#1957	1	1	yes	characteristic
180	C₂×C₆₇	#1957	1	1	yes	characteristic
181	C₂×C₆₇	#1990	1	2	yes	—
182	C₂×C₆₇	#1990	1	2	yes	—
183	C₂²×C₄×C₈	—	67	1	no	—
184	C₂×#384	—	67	1	no	—
185	C₂×#384	D₆₇	1	1	yes	characteristic
186	C₂×#340	—	67	4	no	—
187	C₆₇	#10841	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₈	—	67	2	no	—
189	C₂²×C₄⋊C₄	—	2	2	no	—
190	C₂²×C₄⋊C₄	—	67	2	no	—
191	C₂²×C₄⋊C₄	—	134	2	no	—
192	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
193	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	—	67	4	no	—
195	C₂×C₄×C₈	—	134	2	no	—
196	C₂×C₄×C₈	—	134	2	no	—
197	#384	—	2	2	no	—
198	#384	—	67	4	no	—
199	#384	—	134	2	no	—
200	#384	Dic₆₇	1	4	yes	complemented
201	#340	—	134	8	no	—
202	C₂³×C₄	—	2	2	no	—
203	C₂³×C₄	—	134	2	no	—
204	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
205	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
206	C₂²×C₈	—	67	4	no	—
207	C₂²×C₈	—	67	4	no	—
208	C₂²×C₈	—	67	4	no	—
209	C₂²×C₈	—	134	2	no	—
210	C₂²×C₈	—	134	2	no	—
211	C₂²×C₈	—	134	2	no	—
212	C₂²×C₈	—	134	2	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	4	4	no	—
218	C₂×C₄⋊C₄	—	4	4	no	—
219	C₂×C₄⋊C₄	—	268	4	no	—
220	C₂×C₄⋊C₄	—	268	4	no	—
221	C₂×C₄⋊C₄	—	67	4	no	—
222	C₂×C₄⋊C₄	—	134	8	no	—
223	C₂×C₄⋊C₄	—	134	2	no	—
224	C₂×C₄⋊C₄	—	134	4	no	—
225	C₂×C₄⋊C₄	—	134	4	no	—
226	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
227	C₂×C₄²	Dic₁₃₄	1	2	yes	—
228	C₂×C₄²	C₄×D₆₇	1	2	yes	—
229	C₂×C₄²	C₄×D₆₇	1	2	yes	—
230	C₂×C₄²	D₂₆₈	1	2	yes	—
231	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
232	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
233	C₄×C₈	—	134	8	no	—
234	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	268	2	no	—
241	C₂²×C₄	—	268	2	no	—
242	C₂²×C₄	—	268	4	no	—
243	C₂²×C₄	—	268	4	no	—
244	C₂²×C₄	—	134	4	no	—
245	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
246	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
247	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
248	C₂²×C₄	#11761613	1	4	yes	—
249	C₂²×C₄	#11761615	1	4	yes	—
250	C₂²×C₄	#11761616	1	4	yes	—
251	C₂²×C₄	#11761618	1	4	yes	—
252	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₆₇	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
258	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
259	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
260	C₂²×C₄	#11761627	1	2	yes	—
261	C₂²×C₄	#11761629	1	2	yes	—
262	C₂×C₈	—	67	8	no	—
263	C₂×C₈	—	134	4	no	—
264	C₂×C₈	—	134	4	no	—
265	C₂×C₈	—	134	4	no	—
266	C₂×C₈	—	134	4	no	—
267	C₂×C₈	—	134	4	no	—
268	C₂×C₈	—	134	4	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	4	16	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₄⋊C₄	—	268	16	no	—
273	C₄⋊C₄	—	268	4	no	—
274	C₄⋊C₄	—	134	8	no	—
275	C₄²	C₄×Dic₆₇	1	4	yes	—
276	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
277	C₄²	Dic₆₇⋊C₄	1	4	yes	—
278	C₄²	C₄⋊Dic₆₇	1	4	yes	—
279	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
280	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
281	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
282	C₂³	#14524030	1	1	yes	characteristic
283	C₂³	#14524032	1	1	yes	characteristic
284	C₂³	#14524039	1	1	yes	characteristic
285	C₂³	#14524041	1	1	yes	characteristic
286	C₂³	C₂×#11761613	1	2	yes	—
287	C₂³	C₂×#11761615	1	2	yes	—
288	C₂³	C₂×#11761616	1	2	yes	—
289	C₂³	C₂×#11761618	1	2	yes	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	268	4	no	—
296	C₂×C₄	—	268	8	no	—
297	C₂×C₄	—	536	4	no	—
298	C₂×C₄	—	536	2	no	—
299	C₂×C₄	—	536	2	no	—
300	C₂×C₄	C₆₇⋊#95	1	8	yes	—
301	C₂×C₄	#14523963	1	8	yes	—
302	C₂×C₄	#14523964	1	8	yes	—
303	C₂×C₄	#14523966	1	8	yes	—
304	C₂×C₄	#14523967	1	8	yes	—
305	C₂×C₄	#14523974	1	8	yes	—
306	C₂×C₄	#14523976	1	8	yes	—
307	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
310	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
312	C₂×C₄	#14524001	1	2	yes	—
313	C₂×C₄	#14524005	1	2	yes	—
314	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
315	C₂×C₄	#14524008	1	2	yes	—
316	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
317	C₂×C₄	#14524014	1	2	yes	—
318	C₂×C₄	#14524018	1	2	yes	—
319	C₂×C₄	#14524019	1	2	yes	—
320	C₂×C₄	#14524023	1	2	yes	—
321	C₂×C₄	#14524034	1	2	yes	—
322	C₂×C₄	#14524042	1	2	yes	—
323	C₈	—	134	8	no	—
324	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
325	C₂²	#16863261	1	1	yes	characteristic
326	C₂²	#16863265	1	1	yes	characteristic
327	C₂²	#16863297	1	1	yes	characteristic
328	C₂²	#16863299	1	1	yes	characteristic
329	C₂²	#16863301	1	1	yes	characteristic
330	C₂²	#16863303	1	1	yes	characteristic
331	C₂²	C₂×#14523963	1	2	yes	—
332	C₂²	C₂×#14523964	1	2	yes	—
333	C₂²	C₂×#14523966	1	2	yes	—
334	C₂²	C₂×#14523967	1	2	yes	—
335	C₂²	C₂×#14523974	1	2	yes	—
336	C₂²	C₂×#14523976	1	2	yes	—
337	C₂²	#16863742	1	2	yes	—
338	C₂²	#16863744	1	2	yes	—
339	C₂²	#16863746	1	2	yes	—
340	C₂²	#16863748	1	2	yes	—
341	C₂²	#16863768	1	1	yes	characteristic
342	C₂²	#16863770	1	1	yes	characteristic
343	C₂²	#16863772	1	1	yes	characteristic
344	C₂²	#16863775	1	1	yes	characteristic
345	C₂²	#16863792	1	2	yes	—
346	C₂²	#16863799	1	2	yes	—
347	C₄	—	8	4	no	—
348	C₄	—	536	4	no	—
349	C₄	#16863158	1	16	yes	complemented
350	C₄	C₄⋊C₄×Dic₆₇	1	2	yes	—
351	C₄	#16863253	1	2	yes	—
352	C₄	#16863292	1	2	yes	—
353	C₄	#16863295	1	2	yes	—
354	C₂	#22558438	1	1	yes	characteristic
355	C₂	C₂×#16863158	1	2	yes	—
356	C₂	#22560336	1	2	yes	—
357	C₂	#22560365	1	1	yes	characteristic
358	C₂	#22560368	1	1	yes	characteristic
359	C₂	#22561010	1	1	yes	characteristic
360	C₂	#22561012	1	1	yes	characteristic
361	C₂	#22561014	1	1	yes	characteristic
362	C₂	#22561016	1	1	yes	characteristic
363	C₂	#22561169	1	2	yes	—
364	C₂	#22561171	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#384×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#384	D₆₇
hypocenter	C₆₇	#10841
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
12. #16863266	C₄	semidirect
18. #384×C₆₇	C₄	semidirect
62. C₂×C₆₇⋊C₈	C₄⋊C₄	semidirect
165. C₄×C₆₇	#340	semidirect
187. C₆₇	#10841	semidirect
200. #384	Dic₆₇	semidirect
349. C₄	#16863158	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₆₇⋊C₈	2
C₂×#16863266	2
C₂×#16863158	2
C₂×#384×C₆₇	2
#10841	67