Subgroups — #30869386

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22561146	C₂	1	1	yes	maximal, characteristic
4	#22561146	C₂	1	1	yes	maximal, characteristic
5	#22560775	C₂	1	4	yes	maximal
6	#22560768	C₂	1	4	yes	maximal
7	#22560326	C₂	1	1	yes	maximal, characteristic
8	#22559955	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#16863527	—	2	4	no	—
11	Q₈×C₆₇⋊C₈	—	2	4	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#14523954	C₂²	1	2	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#16863151	C₂²	1	2	yes	—
21	#16863151	C₂²	1	4	yes	—
22	#16863151	C₂²	1	4	yes	—
23	#16863151	C₂²	1	4	yes	—
24	#16863151	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₄	1	1	yes	characteristic
29	#391×C₆₇	C₄	1	1	yes	characteristic
30	#389×C₆₇	C₄	1	4	yes	—
31	#384×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
33	C₄³×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
40	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	#14523954	—	2	4	no	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	—	2	2	no	—
46	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	—	2	2	no	—
51	C₄×Q₈×C₆₇	—	2	2	no	—
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#95×C₆₇	C₂×C₄	1	4	yes	—
68	#95×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	8	no	—
100	C₄⋊C₄×C₆₇	—	2	8	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	4	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	C₂×D₄	1	2	yes	—
111	C₄²×C₆₇	#41	1	2	yes	—
112	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	8	no	—
114	Q₈×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
133	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₆₇	#131	1	4	yes	—
135	C₂×C₄×C₆₇	#131	1	2	yes	—
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
139	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
140	C₆₇⋊C₈	—	2	2	no	—
141	C₆₇⋊C₈	—	2	2	no	—
142	C₆₇⋊C₈	—	4	2	no	—
143	C₆₇⋊C₈	—	4	2	no	—
144	C₆₇⋊C₈	—	4	2	no	—
145	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₆₇	#519	1	2	yes	—
147	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
148	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×#131	1	2	yes	—
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	4	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	#435	1	4	yes	—
158	C₄×C₆₇	#536	1	1	yes	characteristic
159	C₄×C₆₇	#536	1	1	yes	characteristic
160	C₄×C₆₇	#541	1	2	yes	—
161	#22240	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	#2327	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
164	C₂×C₆₇	#2931	1	1	yes	characteristic
165	C₂×C₆₇	#2984	1	2	yes	—
166	C₂×C₆₇	#2986	1	2	yes	—
167	C₂×#446	—	67	1	no	—
168	C₂×C₈×Q₈	—	67	1	no	—
169	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
170	#1984	—	67	1	no	—
171	#1984	—	67	1	no	—
172	#1917	—	67	4	no	—
173	#1916	—	67	4	no	—
174	#1831	—	67	1	no	—
175	#1744	—	67	1	no	—
176	C₆₇	#22240	1	1	yes	characteristic, complemented, Sylow
177	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
180	#446	—	134	4	no	—
181	C₈×Q₈	—	134	4	no	—
182	C₂×C₄⋊C₈	—	67	1	no	—
183	C₂×C₄⋊C₈	—	67	1	no	—
184	C₂×C₄⋊C₈	—	67	2	no	—
185	C₂×C₄⋊C₈	—	67	2	no	—
186	C₂×#97	—	67	2	no	—
187	C₂×C₄×C₈	—	67	1	no	—
188	C₂×C₄×C₈	—	67	2	no	—
189	C₂×C₄×C₈	—	67	1	no	—
190	#391	Dic₆₇	1	1	yes	characteristic
191	#391	Dic₆₇	1	1	yes	characteristic
192	#389	Dic₆₇	1	4	yes	—
193	#384	Dic₆₇	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
195	C₄³	C₂×D₆₇	1	1	yes	characteristic
196	#336	—	67	4	no	—
197	#336	—	67	2	no	—
198	#336	—	67	4	no	—
199	#336	—	67	4	no	—
200	#336	—	67	2	no	—
201	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
202	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
203	C₂²×C₈	—	67	1	no	—
204	C₂²×C₈	—	67	1	no	—
205	C₂²×C₈	—	67	2	no	—
206	C₂²×C₈	—	67	2	no	—
207	C₂²×C₈	—	67	2	no	—
208	C₄⋊Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
219	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
221	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
223	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
224	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
225	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
226	C₄⋊C₈	—	134	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄⋊C₈	—	134	2	no	—
230	#97	—	134	4	no	—
231	C₄×C₈	—	67	4	no	—
232	C₄×C₈	—	134	4	no	—
233	C₄×C₈	—	134	2	no	—
234	#95	C₂×Dic₆₇	1	4	yes	—
235	#95	C₂×Dic₆₇	1	4	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	#11761612	1	4	yes	—
242	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	C₂×C₈	—	67	2	no	—
252	C₂×C₈	—	67	2	no	—
253	C₂×C₈	—	134	2	no	—
254	C₂×C₈	—	134	2	no	—
255	C₂×C₈	—	134	2	no	—
256	C₂×C₈	—	134	2	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	4	no	—
259	C₂×C₈	—	134	4	no	—
260	C₂×C₈	—	134	2	no	—
261	C₂×C₈	—	134	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	D₄×D₆₇	1	2	yes	—
278	C₄²	#11761627	1	2	yes	—
279	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
298	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
299	C₂×C₄	C₂×#11761612	1	4	yes	—
300	C₂×C₄	C₂×#11761612	1	2	yes	—
301	C₂×C₄	#14524050	1	4	yes	—
302	C₂×C₄	#14524050	1	2	yes	—
303	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
305	C₂×C₄	#14524091	1	2	yes	—
306	C₂×C₄	#14524091	1	2	yes	—
307	C₈	—	268	2	no	—
308	C₈	—	268	2	no	—
309	C₈	—	268	2	no	—
310	C₈	—	134	2	no	—
311	C₈	—	134	2	no	—
312	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
313	C₂²	#16863874	1	2	yes	—
314	C₂²	C₂²×#11761612	1	1	yes	characteristic
315	C₂²	C₂×#14524050	1	2	yes	—
316	C₂²	C₂×#14524050	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	#16863468	1	4	yes	—
325	C₄	#16863971	1	1	yes	characteristic
326	C₄	#16863971	1	1	yes	characteristic
327	C₄	#16864007	1	2	yes	—
328	C₂	#22562929	1	1	yes	characteristic
329	C₂	C₂×#16863468	1	1	yes	characteristic
330	C₂	#22568921	1	1	yes	characteristic
331	C₂	#22569298	1	2	yes	—
332	C₂	#22569311	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#22240
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
176. C₆₇	#22240	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
C₂×Q₈×C₆₇⋊C₈	2
#22561146	2
#22560775	2
#22560768	2
#22560326	2
#22559955	2
C₄×C₄⋊Q₈×C₆₇	2
#22240	67