Subgroups — #30864517

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#22560933	C₂	1	4	yes	maximal
6	#22560361	C₂	1	1	yes	maximal, characteristic
7	#22560084	C₂	1	4	yes	maximal
8	C₄×#14523954	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	4	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×#14523954	C₂²	1	2	yes	—
18	C₂×#14523954	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	2	yes	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	#16863151	C₂²	1	4	yes	—
24	#16863151	C₂²	1	4	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	1	yes	characteristic
37	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
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	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
44	#14523954	—	2	4	no	—
45	#14523954	—	2	4	no	—
46	#14523954	Q₈	1	8	yes	complemented
47	C₄×C₆₇⋊C₈	—	2	4	no	—
48	C₄×C₆₇⋊C₈	—	2	2	no	—
49	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	#95×C₆₇	C₂×C₄	1	4	yes	—
71	#95×C₆₇	C₂×C₄	1	4	yes	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
80	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
81	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
82	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	M₁₆	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₄⋊C₄×C₆₇	—	4	4	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	—	4	2	no	—
111	C₄²×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
113	C₄²×C₆₇	#41	1	2	yes	—
114	C₄²×C₆₇	#41	1	4	yes	—
115	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	8	no	—
117	Q₈×C₆₇	—	4	4	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	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
135	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
136	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₆₇	#131	1	2	yes	—
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
143	C₆₇⋊C₈	—	2	2	no	—
144	C₆₇⋊C₈	—	2	4	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	—	4	2	no	—
147	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	complemented
148	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
150	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
152	C₂²×C₆₇	#568	1	2	yes	—
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	—	2	4	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	#435	1	4	yes	—
161	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₄×C₆₇	#532	1	1	yes	characteristic
163	C₄×C₆₇	#533	1	2	yes	—
164	#21714	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
167	C₂×C₆₇	#2932	1	1	yes	characteristic
168	C₂×C₆₇	#2962	1	2	yes	—
169	C₂×C₆₇	Q₈×M₁₆	1	2	yes	—
170	C₂×#446	—	67	1	no	—
171	C₂×C₈×Q₈	—	67	1	no	—
172	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
173	#1984	—	67	1	no	—
174	#1984	—	67	1	no	—
175	#1940	—	67	4	no	—
176	#1842	—	67	1	no	—
177	#1771	—	67	4	no	—
178	C₄×#97	—	67	1	no	—
179	C₆₇	#21714	1	1	yes	characteristic, complemented, Sylow
180	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
181	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
182	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
183	#446	—	134	4	no	—
184	C₈×Q₈	—	134	4	no	—
185	C₂×C₄⋊C₈	—	67	1	no	—
186	C₂×C₄⋊C₈	—	67	1	no	—
187	C₂×C₄⋊C₈	—	67	4	no	—
188	C₂×C₄⋊C₈	—	67	2	no	—
189	C₂×C₄⋊C₈	—	67	2	no	—
190	C₂×#97	—	67	2	no	—
191	C₂×#97	—	67	2	no	—
192	C₂×#97	—	67	2	no	—
193	C₂×C₄×C₈	—	67	1	no	—
194	C₂×C₄×C₈	—	67	2	no	—
195	C₂×C₄×C₈	—	67	1	no	—
196	#391	Dic₆₇	1	1	yes	characteristic
197	#391	Dic₆₇	1	1	yes	characteristic
198	#389	Dic₆₇	1	4	yes	—
199	#384	Dic₆₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
201	C₄³	C₂×D₆₇	1	1	yes	characteristic
202	#336	—	67	4	no	—
203	#336	—	67	4	no	—
204	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
205	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	2	no	—
208	C₂²×C₈	—	67	1	no	—
209	C₂²×C₈	—	67	2	no	—
210	C₂²×C₈	—	67	2	no	—
211	C₄⋊Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
222	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
225	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
226	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
227	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
229	C₄⋊C₈	—	134	4	no	—
230	C₄⋊C₈	—	134	4	no	—
231	C₄⋊C₈	—	134	2	no	—
232	C₄⋊C₈	—	134	2	no	—
233	C₄⋊C₈	—	134	8	no	—
234	#97	—	67	8	no	—
235	#97	—	134	4	no	—
236	#97	—	134	4	no	—
237	C₄×C₈	—	67	4	no	—
238	C₄×C₈	—	134	4	no	—
239	C₄×C₈	—	134	2	no	—
240	#95	C₂×Dic₆₇	1	4	yes	—
241	#95	C₂×Dic₆₇	1	4	yes	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	#11761612	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
257	C₂×C₈	—	67	2	no	—
258	C₂×C₈	—	67	4	no	—
259	C₂×C₈	—	67	2	no	—
260	C₂×C₈	—	67	4	no	—
261	C₂×C₈	—	134	2	no	—
262	C₂×C₈	—	134	2	no	—
263	C₂×C₈	—	134	4	no	—
264	C₂×C₈	—	134	4	no	—
265	C₂×C₈	—	134	2	no	—
266	C₂×C₈	—	134	2	no	—
267	C₂×C₈	—	134	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	D₄×D₆₇	1	2	yes	—
282	C₄²	Q₈×D₆₇	1	4	yes	—
283	C₄²	#11761627	1	2	yes	—
284	C₄²	#11761629	1	4	yes	—
285	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	Q₈	—	4	4	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	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
303	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
304	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
305	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
306	C₂×C₄	C₂×#11761612	1	4	yes	—
307	C₂×C₄	C₂×#11761612	1	2	yes	—
308	C₂×C₄	#14524050	1	2	yes	—
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
311	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
312	C₂×C₄	C₂×#11761629	1	2	yes	—
313	C₈	—	268	2	no	—
314	C₈	—	268	2	no	—
315	C₈	—	67	4	no	—
316	C₈	—	134	2	no	—
317	C₈	—	134	4	no	—
318	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
319	C₂²	C₂×Q₈×Dic₆₇	1	2	yes	—
320	C₂²	C₂²×#11761612	1	1	yes	characteristic
321	C₂²	C₂×#14524050	1	1	yes	characteristic
322	C₂²	#16864163	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	#16863468	1	4	yes	—
331	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
332	C₄	#16863947	1	1	yes	characteristic
333	C₄	#16863951	1	2	yes	—
334	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
335	C₂	C₂×#16863468	1	1	yes	characteristic
336	C₂	#22568927	1	1	yes	characteristic
337	C₂	#22569158	1	2	yes	—
338	C₂	Q₈×#11761612	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	#21714
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
46. #14523954	Q₈	semidirect
147. C₆₇⋊C₈	C₄⋊Q₈	semidirect
179. C₆₇	#21714	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
#22560933	2
#22560361	2
#22560084	2
C₄×#14523954	2
C₄×C₄⋊Q₈×C₆₇	2
#21714	67