Subgroups — #30856274

Order 17152.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863438	C₂	1	2	yes	maximal
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
4	#22562832	C₂	1	2	yes	maximal
5	#22560377	C₂	1	4	yes	maximal, complemented
6	#22560373	C₂	1	2	yes	maximal, complemented
7	#22560366	C₂	1	2	yes	maximal, complemented
8	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524002	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	#16863438	—	2	8	no	—
12	#16863434	—	2	4	no	—
13	C₂×#14523977	C₂²	1	2	yes	—
14	C₂×#14523977	C₂²	1	4	yes	—
15	C₂×#14523975	—	2	4	no	—
16	C₂×#14523975	C₂²	1	2	yes	—
17	C₂×#14523975	C₂²	1	2	yes	—
18	C₂×#14523975	C₂²	1	1	yes	characteristic
19	C₂×#14523975	C₂²	1	1	yes	characteristic
20	C₂×#14523975	C₂²	1	4	yes	—
21	#16863295	—	2	4	no	—
22	#16863271	—	2	4	no	—
23	#16863266	C₂²	1	4	yes	—
24	#16863262	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	#405×C₆₇	C₄	1	8	yes	complemented
29	C₂×#97×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂³×Dic₆₇	—	2	2	no	—
32	#14524002	—	2	4	no	—
33	#14524002	—	2	4	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	—	2	4	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	8	no	—
38	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
42	C₂×C₄×Dic₆₇	—	2	2	no	—
43	#14523977	—	2	4	no	—
44	#14523977	—	2	8	no	—
45	#14523975	—	2	4	no	—
46	#14523975	—	2	4	no	—
47	#14523975	—	2	2	no	—
48	#14523975	—	2	2	no	—
49	#14523975	—	2	8	no	—
50	C₆₇⋊#95	—	2	8	no	—
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
54	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₆₇	D₄	1	4	yes	—
57	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	#117×C₆₇	C₂×C₄	1	4	yes	—
62	#117×C₆₇	Q₈	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	#97×C₆₇	C₂×C₄	1	8	yes	complemented
67	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
68	C₂²×Dic₆₇	—	2	2	no	—
69	C₂²×Dic₆₇	—	2	4	no	—
70	C₂²×Dic₆₇	—	2	2	no	—
71	C₂²×Dic₆₇	—	4	4	no	—
72	C₂²×Dic₆₇	—	8	4	no	—
73	C₄⋊Dic₆₇	—	2	4	no	—
74	C₄⋊Dic₆₇	—	2	2	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	—	2	2	no	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	4	4	no	—
80	C₄⋊Dic₆₇	—	8	8	no	—
81	C₂²⋊Dic₆₇	—	4	4	no	—
82	C₄×Dic₆₇	—	4	4	no	—
83	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
94	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
95	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
104	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
105	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
106	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
107	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
108	C₂³×C₆₇	—	2	4	no	—
109	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₆₇	#127	1	2	yes	—
111	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
115	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	#127	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	4	4	no	—
132	C₂×Dic₆₇	—	4	8	no	—
133	C₂×Dic₆₇	—	4	4	no	—
134	C₂×Dic₆₇	—	8	8	no	—
135	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
137	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	2	4	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	#384	1	4	yes	—
142	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₆₇	#577	1	4	yes	—
146	C₄×C₆₇	—	2	8	no	—
147	C₄×C₆₇	#384	1	4	yes	—
148	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	Dic₆₇	—	8	2	no	—
153	Dic₆₇	—	8	2	no	—
154	Dic₆₇	—	8	4	no	—
155	#20802	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	—	2	4	no	—
157	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
158	C₂×C₆₇	#2912	1	2	yes	—
159	C₂×C₆₇	#3154	1	2	yes	—
160	C₂×C₆₇	#3163	1	2	yes	—
161	C₂×#428	—	67	2	no	—
162	C₂×#427	—	67	1	no	—
163	C₂²×#106	—	67	1	no	—
164	C₂×#405	D₆₇	1	1	yes	characteristic
165	#2317	—	67	2	no	—
166	#1847	—	67	4	no	—
167	#1846	—	67	2	no	—
168	#1844	—	67	2	no	—
169	C₆₇	#20802	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
172	C₂×#117	—	67	2	no	—
173	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	—	67	2	no	—
175	#428	—	134	8	no	—
176	#427	—	134	4	no	—
177	C₂×#107	—	67	4	no	—
178	C₂×#107	—	67	2	no	—
179	C₂×#106	—	67	1	no	—
180	C₂×#106	—	67	2	no	—
181	C₂×#106	—	67	4	no	—
182	C₂×#106	—	67	1	no	—
183	C₂×#106	—	67	2	no	—
184	C₂×#106	—	134	4	no	—
185	#405	Dic₆₇	1	8	yes	complemented
186	C₂×#97	C₂×D₆₇	1	2	yes	—
187	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
188	#389	—	134	4	no	—
189	#385	—	134	4	no	—
190	#384	—	67	4	no	—
191	#382	—	67	4	no	—
192	C₂³×C₄	—	134	2	no	—
193	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
194	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
195	C₂×M₁₆	D₂₆₈	1	4	yes	—
196	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
197	C₂²×C₈	Dic₁₃₄	1	4	yes	—
198	C₂²×C₈	D₂₆₈	1	4	yes	—
199	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
200	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
203	#117	—	134	4	no	—
204	#117	—	134	4	no	—
205	#117	Dic₁₃₄	1	4	yes	—
206	#117	C₂×Dic₆₇	1	4	yes	—
207	C₂×C₄⋊C₄	—	67	4	no	—
208	C₂×C₄⋊C₄	—	67	2	no	—
209	C₂×C₄⋊C₄	—	67	2	no	—
210	C₂×C₄⋊C₄	—	134	2	no	—
211	C₂×C₄⋊C₄	—	134	4	no	—
212	C₂×C₄⋊C₄	—	134	4	no	—
213	C₂×C₄⋊C₄	—	134	8	no	—
214	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
215	C₂×C₂²⋊C₄	—	134	2	no	—
216	C₂×C₂²⋊C₄	D₂₆₈	1	2	yes	—
217	C₂×C₄²	—	134	2	no	—
218	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
219	#107	—	134	4	no	—
220	#107	—	134	8	no	—
221	#106	—	134	4	no	—
222	#106	—	134	4	no	—
223	#106	—	134	2	no	—
224	#106	—	134	2	no	—
225	#106	—	134	8	no	—
226	#97	C₂×Dic₆₇	1	8	yes	complemented
227	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
228	#95	—	134	8	no	—
229	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
230	C₂²×C₄	—	268	4	no	—
231	C₂²×C₄	—	536	4	no	—
232	C₂²×C₄	—	134	2	no	—
233	C₂²×C₄	—	134	4	no	—
234	C₂²×C₄	—	134	2	no	—
235	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
236	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
237	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
239	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
240	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
241	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
242	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
243	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
244	M₁₆	C₄⋊Dic₆₇	1	16	yes	complemented
245	C₂×C₈	C₄⋊Dic₆₇	1	16	yes	complemented
246	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
247	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
248	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
249	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
250	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
251	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
252	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
253	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
254	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
255	C₄⋊C₄	—	268	4	no	—
256	C₄⋊C₄	—	536	8	no	—
257	C₄⋊C₄	—	134	4	no	—
258	C₄⋊C₄	—	134	2	no	—
259	C₄⋊C₄	—	134	4	no	—
260	C₄⋊C₄	—	134	2	no	—
261	C₄⋊C₄	—	134	2	no	—
262	C₄⋊C₄	—	134	2	no	—
263	C₄⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
264	C₂²⋊C₄	—	268	4	no	—
265	C₂²⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
266	C₄²	—	268	4	no	—
267	C₄²	C₂²×Dic₆₇	1	4	yes	—
268	C₄²	C₂×Dic₁₃₄	1	4	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
271	C₂³	#14524040	1	2	yes	—
272	C₂³	#14524043	1	2	yes	—
273	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	268	2	no	—
276	C₂×C₄	—	268	2	no	—
277	C₂×C₄	—	268	4	no	—
278	C₂×C₄	—	268	4	no	—
279	C₂×C₄	—	268	8	no	—
280	C₂×C₄	—	268	4	no	—
281	C₂×C₄	—	536	8	no	—
282	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
283	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
286	C₂×C₄	#14524040	1	2	yes	—
287	C₂×C₄	#14524043	1	2	yes	—
288	C₂×C₄	#14524043	1	4	yes	—
289	C₂×C₄	#14524043	1	4	yes	—
290	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
296	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
297	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
298	C₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#16863266	1	4	yes	—
303	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂²	C₂×#14524040	1	1	yes	characteristic
305	C₂²	C₂×#14524043	1	1	yes	characteristic
306	C₂²	#16864208	1	4	yes	—
307	C₄	—	2	8	no	—
308	C₄	—	536	4	no	—
309	C₄	—	536	2	no	—
310	C₄	—	536	2	no	—
311	C₄	#16863266	1	4	yes	—
312	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
313	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₄	C₂×#14524043	1	1	yes	characteristic
315	C₄	C₂×#14524043	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#16863266	1	1	yes	characteristic
318	C₂	#22568804	1	2	yes	—
319	C₂	#22570685	1	2	yes	—
320	C₂	#22570740	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#20802
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #22560377	C₂	semidirect
6. #22560373	C₂	semidirect
7. #22560366	C₂	semidirect
28. #405×C₆₇	C₄	semidirect
66. #97×C₆₇	C₂×C₄	semidirect
67. C₄×C₈×C₆₇	C₂×C₄	semidirect
93. M₁₆×C₆₇	C₄⋊C₄	semidirect
94. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
169. C₆₇	#20802	semidirect
185. #405	Dic₆₇	semidirect
226. #97	C₂×Dic₆₇	semidirect
227. C₄×C₈	C₂×Dic₆₇	semidirect
244. M₁₆	C₄⋊Dic₆₇	semidirect
245. C₂×C₈	C₄⋊Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863438	2
C₂×#16863434	2
C₂²×#14523975	2
#22562832	2
#22560377	2
#22560373	2
#22560366	2
C₂×#405×C₆₇	2
#20802	67