Subgroups — #30825123

Order 17152.

616 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	#22569334	C₂	1	2	yes	maximal
2	#22558998	C₂	1	2	yes	maximal, complemented
3	#22558994	C₂	1	4	yes	maximal, complemented
4	#22558972	C₂	1	2	yes	maximal, complemented
5	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863147	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	2	no	—
10	#16863468	—	2	4	no	—
11	D₄×C₆₇⋊C₈	—	2	2	no	—
12	#16863453	C₂²	1	4	yes	—
13	#16863449	C₂²	1	2	yes	—
14	#16863420	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
16	#16863149	C₂²	1	2	yes	—
17	#16863149	C₂²	1	4	yes	—
18	#16863147	C₂²	1	2	yes	—
19	#16863147	C₂²	1	4	yes	—
20	#16863147	C₂²	1	1	yes	characteristic
21	#16863147	C₂²	1	2	yes	—
22	#16863147	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
25	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
26	C₈×D₄×C₆₇	C₄	1	2	yes	—
27	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
28	C₈×D₄×C₆₇	C₄	1	2	yes	—
29	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
30	#431×C₆₇	C₂²	1	2	yes	—
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	2	no	—
35	C₂²×C₆₇⋊C₈	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
43	#14523954	—	2	4	no	—
44	C₄×C₆₇⋊C₈	—	2	2	no	—
45	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
52	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
55	C₄×D₄×C₆₇	D₄	1	2	yes	—
56	#117×C₆₇	Q₈	1	1	yes	characteristic
57	#117×C₆₇	Q₈	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	2	yes	—
74	#11761612	—	8	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	—	8	4	no	—
79	#41×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	4	2	no	—
92	C₂×C₈×C₆₇	—	4	1	no	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
106	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	2	yes	—
110	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
111	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
115	Q₈×C₆₇	—	2	2	no	—
116	D₄×C₆₇	—	2	4	no	—
117	D₄×C₆₇	#108	1	4	yes	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
135	C₈×C₆₇	—	2	2	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	#131	1	2	yes	—
141	C₆₇⋊C₈	—	4	2	no	—
142	C₆₇⋊C₈	—	4	2	no	—
143	C₆₇⋊C₈	—	4	4	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	C₂×#108	1	4	yes	—
150	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	C₂×#108	1	2	yes	—
158	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
159	C₄×C₆₇	#568	1	1	yes	characteristic
160	C₄×C₆₇	#577	1	2	yes	—
161	#17834	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	2	4	no	—
163	C₂×C₆₇	—	4	1	no	—
164	C₂×C₆₇	#2906	1	1	yes	characteristic
165	C₂×C₆₇	#2912	1	1	yes	characteristic
166	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
167	#2990	—	67	2	no	—
168	C₈×#41	D₆₇	1	1	yes	characteristic
169	#1574	—	67	2	no	—
170	#1573	—	67	4	no	—
171	#1568	—	67	2	no	—
172	#1566	—	67	1	no	—
173	#1564	—	67	1	no	—
174	C₂×#334	—	67	2	no	—
175	C₆₇	#17834	1	1	yes	characteristic, complemented, Sylow
176	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
177	#446	—	134	2	no	—
178	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
179	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
180	#435	—	134	4	no	—
181	C₈×D₄	—	134	2	no	—
182	C₈×D₄	Dic₆₇	1	2	yes	—
183	C₈×D₄	Dic₆₇	1	1	yes	characteristic
184	C₈×D₄	Dic₆₇	1	2	yes	—
185	C₈×D₄	Dic₆₇	1	1	yes	characteristic
186	#432	—	67	4	no	—
187	#431	—	67	2	no	—
188	#431	C₂×D₆₇	1	2	yes	—
189	#431	C₂×D₆₇	1	1	yes	characteristic
190	#423	—	67	2	no	—
191	C₂×C₄⋊C₈	—	67	4	no	—
192	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
193	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
194	#335	—	67	2	no	—
195	#335	—	67	4	no	—
196	#334	—	67	1	no	—
197	#334	—	67	2	no	—
198	#334	—	67	4	no	—
199	#334	—	67	1	no	—
200	#334	—	67	2	no	—
201	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
202	C₂×M₁₆	—	134	2	no	—
203	C₂²×C₈	—	134	4	no	—
204	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
207	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
208	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
210	C₄×D₄	D₂₆₈	1	1	yes	characteristic
211	C₄×D₄	D₂₆₈	1	2	yes	—
212	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
213	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
214	#117	Dic₁₃₄	1	1	yes	characteristic
215	#117	Dic₁₃₄	1	2	yes	—
216	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
217	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
218	C₄⋊C₈	—	67	2	no	—
219	C₄⋊C₈	—	67	4	no	—
220	C₄⋊C₈	—	67	2	no	—
221	C₄⋊C₈	—	134	4	no	—
222	C₄⋊C₈	—	134	2	no	—
223	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
224	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
225	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
226	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
227	C₂²⋊C₈	—	134	2	no	—
228	C₂²⋊C₈	—	134	4	no	—
229	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
230	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
231	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
232	#97	—	134	4	no	—
233	C₄×C₈	—	134	2	no	—
234	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
235	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
237	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₆₇	1	2	yes	—
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	#41	—	2	4	no	—
242	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
243	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
244	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
247	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
248	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
250	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
251	M₁₆	—	536	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	536	4	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	4	no	—
260	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
262	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
264	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
267	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
269	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
270	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
271	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
272	C₂²⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
273	C₄²	C₂²×Dic₆₇	1	2	yes	—
274	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
275	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
276	C₄²	C₂×D₂₆₈	1	2	yes	—
277	C₄²	C₂×Dic₁₃₄	1	2	yes	—
278	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
279	C₄²	C₂³×D₆₇	1	1	yes	characteristic
280	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
281	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
282	Q₈	—	2	2	no	—
283	D₄	—	2	4	no	—
284	D₄	#14523979	1	4	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	—	268	2	no	—
308	C₈	—	268	4	no	—
309	C₈	—	268	2	no	—
310	C₈	#14524050	1	2	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×#14523979	1	4	yes	—
317	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	C₂×#14523979	1	2	yes	—
325	C₄	C₂×#14524050	1	1	yes	characteristic
326	C₄	#16864163	1	1	yes	characteristic
327	C₄	#16864208	1	2	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	4	1	no	—
330	C₂	#22568777	1	1	yes	characteristic
331	C₂	#22568804	1	1	yes	characteristic
332	C₂	C₂²×#14523979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	Q₈
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#17834
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22558998	C₂	semidirect
3. #22558994	C₂	semidirect
4. #22558972	C₂	semidirect
5. #22558965	C₂	semidirect
6. #22558957	C₂	semidirect
7. C₂×#16863147	C₂	semidirect
175. C₆₇	#17834	semidirect

Maximal subgroups

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

subgroup	index
#22569334	2
#22558998	2
#22558994	2
#22558972	2
#22558965	2
#22558957	2
C₂×#16863147	2
C₈×#41×C₆₇	2
#17834	67