Subgroups — #27085942

Order 12032.

1890 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	2	yes	maximal
2	#20253940	C₂	1	8	yes	maximal, complemented
3	#20253934	C₂	1	8	yes	maximal, complemented
4	#20253912	C₂	1	4	yes	maximal, complemented
5	#20253902	C₂	1	4	yes	maximal, complemented
6	#20251978	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689680	—	2	4	no	—
9	#15689680	—	2	4	no	—
10	#15689680	C₂²	1	4	yes	—
11	#15689680	C₂²	1	4	yes	—
12	#15689680	C₂²	1	4	yes	—
13	#15689680	C₂²	1	4	yes	—
14	C₂×#13094024	C₂²	1	4	yes	—
15	C₂×#13094016	C₂²	1	4	yes	—
16	C₂×#13094003	—	2	4	no	—
17	C₂×C₄×Dic₉₄	—	2	4	no	—
18	C₂×#13093983	C₂²	1	4	yes	—
19	#15689532	C₂²	1	32	yes	complemented
20	#15689528	C₂²	1	8	yes	complemented
21	#15689525	C₂²	1	16	yes	complemented
22	#15689521	C₂²	1	8	yes	complemented
23	#15689165	C₂²	1	32	yes	—
24	#15689161	C₂²	1	16	yes	—
25	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₇	C₂²	1	4	yes	—
28	C₄×M₁₆×C₄₇	C₂²	1	4	yes	—
29	C₂×#97×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₂²×Dic₉₄	—	2	4	no	—
32	#13094024	—	2	4	no	—
33	#13094024	—	2	4	no	—
34	#13094024	C₂³	1	8	yes	—
35	#13094024	C₂³	1	8	yes	—
36	#13094016	—	2	8	no	—
37	#13094016	C₂³	1	8	yes	—
38	#13094016	C₂³	1	8	yes	—
39	#13094003	—	2	8	no	—
40	#13094003	—	2	8	no	—
41	#13094003	—	2	8	no	—
42	#13094003	—	2	8	no	—
43	C₄×Dic₉₄	—	2	8	no	—
44	C₄×Dic₉₄	—	2	8	no	—
45	C₄×Dic₉₄	—	2	8	no	—
46	C₄×Dic₉₄	—	2	8	no	—
47	#13093983	—	2	4	no	—
48	#13093983	—	2	4	no	—
49	#13093983	C₂³	1	8	yes	—
50	#13093983	C₂³	1	8	yes	—
51	C₂×C₄⋊Dic₄₇	—	2	4	no	—
52	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
53	C₂×Dic₄₇⋊C₄	—	2	8	no	—
54	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
55	C₂×C₄×Dic₄₇	—	2	4	no	—
56	#13093958	C₂³	1	32	yes	—
57	#13093956	C₂³	1	32	yes	—
58	C₂³×C₄×C₄₇	D₄	1	2	yes	—
59	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₄₇	Q₈	1	16	yes	complemented
61	C₂×M₁₆×C₄₇	C₂³	1	8	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	D₄	1	4	yes	—
64	C₂×C₄²×C₄₇	D₄	1	2	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	#97×C₄₇	C₂³	1	8	yes	—
70	C₄×C₈×C₄₇	C₂³	1	4	yes	—
71	C₄×C₈×C₄₇	C₂³	1	4	yes	—
72	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	—	4	4	no	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₇	—	2	8	no	—
84	C₂×Dic₉₄	—	2	8	no	—
85	C₂×Dic₉₄	—	2	8	no	—
86	C₂×Dic₉₄	—	4	16	no	—
87	M₁₆×C₄₇	C₂×Q₈	1	32	yes	complemented
88	C₂×C₈×C₄₇	C₂×Q₈	1	16	yes	complemented
89	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
90	C₄²×C₄₇	C₂×D₄	1	2	yes	—
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	4	yes	—
93	C₄²×C₄₇	C₂×D₄	1	4	yes	—
94	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄²×C₄₇	C₂⁴	1	2	yes	—
96	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	—	2	8	no	—
98	C₂²⋊Dic₄₇	—	2	8	no	—
99	C₄⋊Dic₄₇	—	2	8	no	—
100	C₄⋊Dic₄₇	—	2	8	no	—
101	C₄⋊Dic₄₇	—	2	8	no	—
102	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
103	Dic₄₇⋊C₄	—	2	16	no	—
104	Dic₄₇⋊C₄	—	2	16	no	—
105	C₄×Dic₄₇	—	2	8	no	—
106	C₄×Dic₄₇	—	2	8	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
109	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
124	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	16	no	—
126	C₂×Dic₄₇	—	4	16	no	—
127	Dic₉₄	—	4	16	no	—
128	Dic₉₄	—	4	16	no	—
129	C₈×C₄₇	C₂²×Q₈	1	16	yes	—
130	#60663	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
135	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
136	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
140	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
141	C₄×C₄₇	#575	1	4	yes	—
142	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
143	Dic₄₇	—	4	16	no	—
144	C₂×#533	—	47	2	no	—
145	#3166	—	47	8	no	—
146	#3165	—	47	8	no	—
147	#3162	—	47	4	no	—
148	#3160	—	47	4	no	—
149	#2911	—	47	4	no	—
150	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
151	C₂×C₄₇	—	2	4	no	—
152	C₂×C₄₇	—	2	2	no	—
153	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#575	1	2	yes	—
155	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
156	#533	—	47	4	no	—
157	#533	—	47	4	no	—
158	#533	—	47	4	no	—
159	#533	—	47	4	no	—
160	#533	—	94	4	no	—
161	#533	—	94	4	no	—
162	C₂×C₄⋊Q₈	—	47	4	no	—
163	C₂×#125	—	47	4	no	—
164	C₂×C₂²⋊Q₈	—	94	4	no	—
165	C₂×C₄×Q₈	—	94	4	no	—
166	C₂×#117	—	47	4	no	—
167	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
168	#501	—	47	32	no	—
169	#500	—	47	8	no	—
170	#499	—	47	16	no	—
171	#498	—	47	8	no	—
172	#428	—	47	32	no	—
173	#427	—	47	16	no	—
174	C₄×M₁₆	C₂×D₄₇	1	4	yes	—
175	C₄×M₁₆	C₂×D₄₇	1	4	yes	—
176	C₂×#97	C₂×D₄₇	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
178	C₄₇	#60663	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	—	94	4	no	—
180	C₂³×C₄	D₁₈₈	1	2	yes	—
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	Dic₉₄	1	16	yes	complemented
183	C₂×M₁₆	C₂²×D₄₇	1	8	yes	—
184	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
185	C₄⋊Q₈	—	47	8	no	—
186	C₄⋊Q₈	—	47	8	no	—
187	C₄⋊Q₈	—	94	4	no	—
188	C₄⋊Q₈	—	94	4	no	—
189	#125	—	47	8	no	—
190	#125	—	47	8	no	—
191	#125	—	94	8	no	—
192	C₂²⋊Q₈	—	94	8	no	—
193	C₂²⋊Q₈	—	94	8	no	—
194	C₂²⋊Q₈	—	94	8	no	—
195	C₂²⋊Q₈	—	94	8	no	—
196	C₄×Q₈	—	94	8	no	—
197	C₄×Q₈	—	94	8	no	—
198	C₄×Q₈	—	94	8	no	—
199	C₄×Q₈	—	94	8	no	—
200	#117	—	47	8	no	—
201	#117	—	47	8	no	—
202	#117	—	94	4	no	—
203	#117	—	94	4	no	—
204	C₂×C₄⋊C₄	—	47	8	no	—
205	C₂×C₄⋊C₄	—	94	8	no	—
206	C₂×C₄⋊C₄	—	94	4	no	—
207	C₂×C₂²⋊C₄	—	94	4	no	—
208	C₂×C₄²	—	94	4	no	—
209	C₂×C₄²	D₁₈₈	1	4	yes	—
210	C₂×C₄²	D₁₈₈	1	2	yes	—
211	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	#107	—	47	32	no	—
216	#106	—	47	32	no	—
217	#97	C₂²×D₄₇	1	8	yes	—
218	C₄×C₈	C₂²×D₄₇	1	4	yes	—
219	C₄×C₈	C₂²×D₄₇	1	4	yes	—
220	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
221	C₂×Q₈	—	94	8	no	—
222	C₂×Q₈	—	94	8	no	—
223	C₂×Q₈	—	188	16	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	94	8	no	—
226	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
227	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
228	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
229	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
230	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
231	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	M₁₆	C₂×Dic₉₄	1	32	yes	complemented
236	C₂×C₈	C₂×Dic₉₄	1	16	yes	complemented
237	C₂×C₈	C₂³×D₄₇	1	8	yes	—
238	C₄⋊C₄	—	47	16	no	—
239	C₄⋊C₄	—	94	16	no	—
240	C₄⋊C₄	—	94	16	no	—
241	C₄⋊C₄	—	94	8	no	—
242	C₄⋊C₄	—	94	8	no	—
243	C₄⋊C₄	—	94	8	no	—
244	C₂²⋊C₄	—	94	8	no	—
245	C₂²⋊C₄	—	94	8	no	—
246	C₄²	—	94	8	no	—
247	C₄²	—	94	8	no	—
248	C₄²	C₂×D₁₈₈	1	4	yes	—
249	C₄²	C₂×D₁₈₈	1	4	yes	—
250	C₄²	C₂×D₁₈₈	1	2	yes	—
251	C₄²	C₂×D₁₈₈	1	2	yes	—
252	C₄²	C₂³×D₄₇	1	1	yes	characteristic
253	C₄²	C₂³×D₄₇	1	1	yes	characteristic
254	C₄²	C₂³×D₄₇	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#13094024	1	4	yes	—
257	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
258	C₂³	C₂²×D₁₈₈	1	2	yes	—
259	Q₈	—	188	16	no	—
260	Q₈	—	188	16	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	94	16	no	—
266	C₂×C₄	—	188	16	no	—
267	C₂×C₄	#13094024	1	4	yes	—
268	C₂×C₄	#13094024	1	8	yes	—
269	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
270	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
271	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
272	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
273	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
275	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
276	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
277	C₈	C₂²×Dic₉₄	1	16	yes	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	C₂×#13094024	1	2	yes	—
282	C₂²	C₂×#13094024	1	4	yes	—
283	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	188	16	no	—
287	C₄	C₂×#13094024	1	4	yes	—
288	C₄	C₂×#13094024	1	2	yes	—
289	C₄	#15689923	1	4	yes	—
290	C₄	C₂³×Dic₉₄	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂²×#13094024	1	1	yes	characteristic
294	C₂	C₂×#15689923	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#60663
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20253940	C₂	semidirect
3. #20253934	C₂	semidirect
4. #20253912	C₂	semidirect
5. #20253902	C₂	semidirect
19. #15689532	C₂²	semidirect
20. #15689528	C₂²	semidirect
21. #15689525	C₂²	semidirect
22. #15689521	C₂²	semidirect
60. C₂×M₁₆×C₄₇	Q₈	semidirect
87. M₁₆×C₄₇	C₂×Q₈	semidirect
88. C₂×C₈×C₄₇	C₂×Q₈	semidirect
178. C₄₇	#60663	semidirect
182. C₂×M₁₆	Dic₉₄	semidirect
235. M₁₆	C₂×Dic₉₄	semidirect
236. C₂×C₈	C₂×Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689680	2
#20253940	2
#20253934	2
#20253912	2
#20253902	2
#20251978	2
C₂×C₄×M₁₆×C₄₇	2
#60663	47