Subgroups — #26236088

Order 12032.

770 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×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#20246347	C₂	1	2	yes	maximal
3	#20244257	C₂	1	2	yes	maximal
4	#20244021	C₂	1	4	yes	maximal
5	#20243390	C₂	1	2	yes	maximal
6	#20243288	C₂	1	2	yes	maximal
7	#20243255	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₄×Dic₁₈₈	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#13093949	C₂²	1	2	yes	—
15	C₂×#13093947	C₂²	1	4	yes	—
16	C₂×#13093934	C₂²	1	2	yes	—
17	#15689028	C₂²	1	4	yes	—
18	#15689004	—	2	4	no	—
19	#15688988	—	2	2	no	—
20	C₄⋊C₄×Dic₄₇	—	2	2	no	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	4	yes	—
24	#15688877	C₂²	1	2	yes	—
25	C₂×#106×C₄₇	C₂²	1	2	yes	—
26	C₂×#106×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#384	C₂²	1	2	yes	—
29	C₂²×Dic₉₄	C₂³	1	2	yes	—
30	C₂×Dic₁₈₈	—	2	2	no	—
31	C₂×Dic₁₈₈	—	2	2	no	—
32	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₉₄	—	2	4	no	—
35	C₄×Dic₉₄	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×Dic₄₇⋊C₄	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
42	#13093958	C₂×C₄	1	4	yes	—
43	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
44	#13093949	C₂×C₄	1	8	yes	—
45	#13093947	—	4	4	no	—
46	#13093934	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	—	2	4	no	—
49	C₄₇⋊#95	—	2	4	no	—
50	C₄₇⋊#95	D₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#106×C₄₇	—	2	4	no	—
58	#106×C₄₇	—	4	2	no	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
67	C₂×Dic₉₄	—	2	2	no	—
68	C₂×Dic₉₄	—	2	2	no	—
69	C₂×Dic₉₄	—	2	2	no	—
70	C₂×Dic₉₄	—	2	2	no	—
71	C₂×Dic₉₄	—	2	2	no	—
72	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	#41	1	2	yes	—
78	C₂×C₈×C₄₇	#41	1	2	yes	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	4	4	no	—
82	C₄⋊C₄×C₄₇	—	8	2	no	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	#41	1	2	yes	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	2	no	—
87	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
88	Dic₄₇⋊C₄	—	2	8	no	—
89	C₄×Dic₄₇	—	2	8	no	—
90	C₄×Dic₄₇	—	2	2	no	—
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	—	8	4	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	Dic₁₈₈	—	2	8	no	—
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	#137	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	4	2	no	—
120	Dic₉₄	—	2	4	no	—
121	Dic₉₄	—	2	4	no	—
122	Dic₉₄	—	4	2	no	—
123	Dic₉₄	—	4	2	no	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	#117	1	4	yes	—
126	C₄₇⋊C₈	—	4	2	no	—
127	#22814	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₇	#547	1	1	yes	characteristic
130	C₂²×C₄₇	#550	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#137	1	2	yes	—
132	C₂²×C₄₇	#577	1	2	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	8	4	no	—
135	C₄×C₄₇	#441	1	4	yes	—
136	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₄₇	#520	1	1	yes	characteristic
138	C₄×C₄₇	#555	1	1	yes	characteristic
139	C₄×C₄₇	#556	1	1	yes	characteristic
140	Dic₄₇	—	4	4	no	—
141	Dic₄₇	—	4	4	no	—
142	Dic₄₇	—	4	4	no	—
143	C₂×C₄×Q₁₆	—	47	1	no	—
144	#2355	—	47	2	no	—
145	#1971	—	47	2	no	—
146	#1927	—	47	4	no	—
147	#1844	D₄₇	1	1	yes	characteristic
148	#1803	—	47	2	no	—
149	#1776	—	47	2	no	—
150	#1770	—	47	1	no	—
151	C₂×C₄₇	#2354	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
153	C₂×C₄₇	#2952	1	1	yes	characteristic
154	C₂×C₄₇	#3356	1	2	yes	—
155	C₂×C₄₇	#3400	1	2	yes	—
156	C₂²×Q₁₆	—	47	1	no	—
157	C₂×C₄×Q₈	—	47	2	no	—
158	C₄×Q₁₆	—	47	8	no	—
159	C₂×#107	—	47	1	no	—
160	C₂×#106	C₂×D₄₇	1	2	yes	—
161	C₂×#106	C₂×D₄₇	1	2	yes	—
162	C₂×C₄⋊C₈	—	47	1	no	—
163	C₂×#103	—	47	2	no	—
164	C₂×#103	—	47	4	no	—
165	C₂×#97	—	47	2	no	—
166	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
167	#391	—	47	4	no	—
168	#387	—	94	4	no	—
169	#384	C₂×D₄₇	1	2	yes	—
170	#382	—	94	2	no	—
171	C₄×C₄⋊C₄	—	47	1	no	—
172	C₄×C₄⋊C₄	—	47	4	no	—
173	C₄×C₄⋊C₄	—	94	2	no	—
174	#340	—	47	4	no	—
175	#336	—	47	2	no	—
176	C₄₇	#22814	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	—	47	2	no	—
178	C₂×Q₁₆	—	47	4	no	—
179	C₂×Q₁₆	—	94	2	no	—
180	C₂×Q₁₆	—	94	2	no	—
181	C₂²×C₈	—	47	2	no	—
182	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
184	C₄×Q₈	—	47	8	no	—
185	C₄×Q₈	—	94	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	47	2	no	—
189	C₂×C₄⋊C₄	—	47	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
192	C₂×C₄²	—	47	2	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	—	94	4	no	—
195	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
196	#107	—	47	4	no	—
197	#106	—	2	4	no	—
198	#106	—	4	2	no	—
199	C₄⋊C₈	—	94	2	no	—
200	#103	—	47	8	no	—
201	#103	—	188	4	no	—
202	#97	—	94	4	no	—
203	C₄×C₈	C₄×D₄₇	1	4	yes	—
204	#95	—	47	2	no	—
205	#95	—	94	4	no	—
206	#95	—	94	4	no	—
207	#95	—	94	4	no	—
208	C₂×Q₈	—	47	4	no	—
209	C₂×Q₈	—	94	2	no	—
210	C₂×Q₈	—	94	2	no	—
211	C₂×Q₈	—	94	2	no	—
212	C₂×Q₈	—	94	2	no	—
213	C₂×Q₈	—	94	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	47	4	no	—
216	C₂²×C₄	—	94	2	no	—
217	C₂²×C₄	—	94	2	no	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	Q₁₆	—	94	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
229	C₂×C₈	#10625352	1	2	yes	—
230	C₂×C₈	#10625354	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₄⋊C₄	—	47	4	no	—
236	C₄⋊C₄	—	94	8	no	—
237	C₄⋊C₄	—	94	4	no	—
238	C₄⋊C₄	—	94	2	no	—
239	C₄²	—	94	8	no	—
240	C₄²	—	94	2	no	—
241	C₄²	—	94	2	no	—
242	C₄²	—	376	4	no	—
243	C₄²	C₂×C₄×D₄₇	1	2	yes	—
244	C₄²	#10625354	1	2	yes	—
245	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
246	Q₈	—	94	4	no	—
247	Q₈	—	94	4	no	—
248	Q₈	—	188	2	no	—
249	Q₈	—	188	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	—	94	4	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	2	no	—
261	C₂×C₄	—	188	2	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	#13093986	1	2	yes	—
264	C₂×C₄	#13093986	1	2	yes	—
265	C₂×C₄	#13094053	1	4	yes	—
266	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
268	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
270	C₂×C₄	#13094069	1	1	yes	characteristic
271	C₂×C₄	#13094072	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	188	2	no	—
274	C₈	#13093982	1	4	yes	—
275	C₂²	C₂×#13093986	1	1	yes	characteristic
276	C₂²	#15689776	1	1	yes	characteristic
277	C₂²	#15689793	1	1	yes	characteristic
278	C₂²	C₂×#13094053	1	2	yes	—
279	C₂²	#15689929	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	8	4	no	—
282	C₄	—	188	4	no	—
283	C₄	—	188	4	no	—
284	C₄	—	188	4	no	—
285	C₄	#15689229	1	4	yes	—
286	C₄	C₂×#13093982	1	1	yes	characteristic
287	C₄	#15689604	1	1	yes	characteristic
288	C₄	#15689839	1	1	yes	characteristic
289	C₄	#15689847	1	1	yes	characteristic
290	C₂	#20246339	1	1	yes	characteristic
291	C₂	C₂×#15689229	1	1	yes	characteristic
292	C₂	#20252271	1	1	yes	characteristic
293	C₂	#20256002	1	2	yes	—
294	C₂	#20256446	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#22814
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₁₈₈	C₄	semidirect
176. C₄₇	#22814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
#20246347	2
#20244257	2
#20244021	2
#20243390	2
#20243288	2
#20243255	2
C₄₇×#1844	2
#22814	47