Subgroups — #26233946

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246281	C₂	1	2	yes	maximal
2	#20244122	C₂	1	2	yes	maximal, complemented
3	#20243968	C₂	1	4	yes	maximal, complemented
4	#20243483	C₂	1	2	yes	maximal
5	#20243417	C₂	1	1	yes	maximal, characteristic
6	#107×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₈×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093958	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
12	#15689022	C₂²	1	4	yes	—
13	#15689016	—	2	4	no	—
14	#15688991	—	2	2	no	—
15	#15688983	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	—	2	2	no	—
18	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688886	C₂²	1	2	yes	—
20	#15688885	C₂²	1	4	yes	—
21	#15688885	C₂²	1	4	yes	—
22	#15688877	C₂²	1	2	yes	—
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₇	C₂²	1	2	yes	—
25	C₄₇×#427	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	#13093958	—	2	4	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
38	C₈×Dic₄₇	—	2	4	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	4	no	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₄₇⋊#95	D₄	1	1	yes	characteristic
43	C₄₇⋊#95	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	—	2	4	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	#117×C₄₇	—	2	4	no	—
50	#117×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	#107×C₄₇	C₂×C₄	1	4	yes	—
56	#107×C₄₇	C₂×C₄	1	4	yes	—
57	#106×C₄₇	C₂×C₄	1	8	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
69	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	#41	1	4	yes	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	—	2	8	no	—
81	C₂²⋊Dic₄₇	—	2	4	no	—
82	C₂²⋊Dic₄₇	—	8	2	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	4	8	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	C₄×Dic₄₇	—	8	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	C₄×D₄	1	2	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	4	4	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	#135	1	4	yes	—
121	C₈×C₄₇	—	2	2	no	—
122	C₈×C₄₇	#117	1	4	yes	—
123	C₄₇⋊C₈	—	4	2	no	—
124	#22614	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₇	#547	1	2	yes	—
132	C₂²×C₄₇	C₂×#135	1	2	yes	—
133	C₂²×C₄₇	#576	1	1	yes	characteristic
134	C₂²×C₄₇	#578	1	1	yes	characteristic
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	—	4	4	no	—
138	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₄₇	#519	1	1	yes	characteristic
140	C₄×C₄₇	#552	1	2	yes	—
141	Dic₄₇	—	2	4	no	—
142	Dic₄₇	—	4	2	no	—
143	Dic₄₇	—	8	4	no	—
144	C₂×#427	D₄₇	1	1	yes	characteristic
145	#2347	—	47	2	no	—
146	#1942	—	47	2	no	—
147	#1919	—	47	4	no	—
148	#1825	—	47	2	no	—
149	#1808	—	47	1	no	—
150	C₄×#107	—	47	2	no	—
151	C₈×C₂²⋊C₄	—	47	1	no	—
152	C₂×C₄₇	—	4	2	no	—
153	C₂×C₄₇	#2344	1	1	yes	characteristic
154	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
155	C₂×C₄₇	#2951	1	1	yes	characteristic
156	C₂×C₄₇	#3339	1	2	yes	—
157	C₂×C₄₇	#3342	1	2	yes	—
158	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#117	C₂×D₄₇	1	2	yes	—
160	#427	Dic₄₇	1	8	yes	complemented
161	C₂×#107	—	47	2	no	—
162	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#106	C₂×D₄₇	1	2	yes	—
165	C₂×C₂²⋊C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	#389	—	47	4	no	—
168	#388	—	94	4	no	—
169	#383	—	94	2	no	—
170	#380	—	47	2	no	—
171	C₄×C₄⋊C₄	—	47	4	no	—
172	C₄×C₄⋊C₄	—	94	2	no	—
173	C₄×C₂²⋊C₄	—	47	1	no	—
174	#340	—	47	2	no	—
175	#340	—	47	4	no	—
176	#340	—	47	4	no	—
177	#336	—	47	2	no	—
178	C₄₇	#22614	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	—	47	2	no	—
182	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
183	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	#117	—	2	4	no	—
186	#117	C₂×Dic₄₇	1	8	yes	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	94	4	no	—
189	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
190	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	47	2	no	—
193	C₂×C₂²⋊C₄	—	94	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	47	2	no	—
196	C₂×C₄²	—	94	4	no	—
197	#107	—	94	4	no	—
198	#107	C₂×Dic₄₇	1	4	yes	—
199	#107	C₂×Dic₄₇	1	4	yes	—
200	#106	C₂×Dic₄₇	1	8	yes	—
201	C₂²⋊C₈	—	94	2	no	—
202	C₄×C₈	—	94	4	no	—
203	#95	—	47	1	no	—
204	#95	—	47	1	no	—
205	#95	—	94	4	no	—
206	#95	—	94	4	no	—
207	#95	—	94	2	no	—
208	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	47	2	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	#10625352	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	#10625350	1	4	yes	—
226	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
227	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₄⋊C₄	—	94	4	no	—
232	C₄⋊C₄	—	188	8	no	—
233	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
234	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	94	4	no	—
237	C₂²⋊C₄	—	376	2	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	94	2	no	—
240	C₄²	—	94	2	no	—
241	C₄²	—	376	4	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	D₄×Dic₄₇	1	2	yes	—
247	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	47	4	no	—
258	C₂×C₄	—	94	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	—	94	2	no	—
262	C₂×C₄	—	94	2	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
267	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625350	1	2	yes	—
270	C₂×C₄	#13094072	1	2	yes	—
271	C₈	—	2	2	no	—
272	C₈	—	188	2	no	—
273	C₈	#13093983	1	4	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
280	C₂²	#15689774	1	2	yes	—
281	C₂²	#135×D₄₇	1	2	yes	—
282	C₂²	#15689925	1	1	yes	characteristic
283	C₂²	#15689937	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	94	4	no	—
288	C₄	—	376	4	no	—
289	C₄	—	188	2	no	—
290	C₄	C₂×#13093983	1	1	yes	characteristic
291	C₄	#15689601	1	1	yes	characteristic
292	C₄	#15689808	1	2	yes	—
293	C₂	—	4	2	no	—
294	C₂	#20246257	1	1	yes	characteristic
295	C₂	#135×Dic₄₇	1	1	yes	characteristic
296	C₂	#20252264	1	1	yes	characteristic
297	C₂	#20255790	1	2	yes	—
298	C₂	#20255816	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#22614
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #20244122	C₂	semidirect
3. #20243968	C₂	semidirect
6. #107×Dic₄₇	C₂	semidirect
25. C₄₇×#427	C₄	semidirect
160. #427	Dic₄₇	semidirect
178. C₄₇	#22614	semidirect

Maximal subgroups

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

subgroup	index
#20246281	2
#20244122	2
#20243968	2
#20243483	2
#20243417	2
#107×Dic₄₇	2
C₈×C₂²⋊Dic₄₇	2
C₂×C₄₇×#427	2
#22614	47