Subgroups — #26376415

Order 12032.

2250 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246353	C₂	1	8	yes	maximal, complemented
2	#20246345	C₂	1	8	yes	maximal, complemented
3	#20245991	C₂	1	4	yes	maximal
4	#20245974	C₂	1	2	yes	maximal
5	#20245795	C₂	1	2	yes	maximal
6	#20245786	C₂	1	4	yes	maximal
7	#20245745	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	4	yes	—
10	#15689030	C₂²	1	16	yes	—
11	#15689009	C₂²	1	32	yes	—
12	#15688993	C₂²	1	4	yes	—
13	#15688991	C₂²	1	8	yes	—
14	#15688989	C₂²	1	8	yes	—
15	#15688989	C₂²	1	4	yes	—
16	#15688986	C₂²	1	8	yes	—
17	#15688983	C₂²	1	4	yes	—
18	#15688983	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
23	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
24	#15688969	C₂²	1	8	yes	—
25	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₄₇	C₄	1	16	yes	complemented
27	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
29	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
33	#13093983	—	2	16	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
35	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
37	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
40	C₄₇⋊#95	C₂³	1	8	yes	—
41	C₄₇⋊#95	C₂³	1	16	yes	—
42	C₄₇⋊#95	C₂³	1	16	yes	—
43	C₄₇⋊#95	C₂³	1	8	yes	—
44	C₄₇⋊#95	C₂³	1	16	yes	—
45	C₂²×Q₈×C₄₇	—	2	8	no	—
46	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
47	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	—
54	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
65	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂×Q₈×C₄₇	—	2	32	no	—
67	C₂×Q₈×C₄₇	—	2	8	no	—
68	C₂×Q₈×C₄₇	—	2	8	no	—
69	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	4	4	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	#41	1	8	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
88	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
93	C₂²⋊Dic₄₇	—	2	16	no	—
94	C₄⋊Dic₄₇	—	2	16	no	—
95	C₄⋊Dic₄₇	—	2	16	no	—
96	C₄⋊Dic₄₇	—	4	8	no	—
97	C₄×Dic₄₇	—	2	16	no	—
98	C₄×Dic₄₇	—	4	16	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂³×C₄₇	C₂×#41	1	4	yes	—
103	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
104	Q₈×C₄₇	—	2	32	no	—
105	C₂×C₄×C₄₇	—	2	16	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	#117	1	16	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
122	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
123	C₂×Dic₄₇	—	2	16	no	—
124	C₂×Dic₄₇	—	2	16	no	—
125	C₂×Dic₄₇	—	2	16	no	—
126	#34187	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	C₂×#117	1	4	yes	—
131	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×#41	1	2	yes	—
133	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
134	C₂²×C₄₇	#585	1	2	yes	—
135	C₄×C₄₇	—	2	8	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	—	2	8	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	C₂×#117	1	8	yes	—
140	C₄×C₄₇	#519	1	4	yes	—
141	Dic₄₇	—	4	16	no	—
142	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
143	#2356	—	47	8	no	—
144	#2355	—	47	8	no	—
145	#2313	—	47	4	no	—
146	#2310	—	47	2	no	—
147	#2285	—	47	2	no	—
148	#2283	—	47	4	no	—
149	#2274	—	47	2	no	—
150	C₂×C₄₇	—	2	4	no	—
151	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
153	C₂×C₄₇	#3428	1	1	yes	characteristic
154	C₂×C₄₇	#3507	1	4	yes	—
155	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	Dic₄₇	1	16	yes	complemented
157	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
158	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
159	C₂×#117	—	47	4	no	—
160	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
161	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
162	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	#391	—	47	16	no	—
165	#387	—	47	32	no	—
166	#384	—	47	4	no	—
167	#383	—	47	8	no	—
168	#382	—	47	8	no	—
169	#382	—	47	4	no	—
170	#381	—	47	8	no	—
171	#380	—	47	4	no	—
172	#380	—	47	4	no	—
173	C₄×C₄⋊C₄	—	47	16	no	—
174	C₄×C₄⋊C₄	—	47	8	no	—
175	C₄×C₄⋊C₄	—	47	4	no	—
176	C₄×C₄⋊C₄	—	47	4	no	—
177	C₄×C₂²⋊C₄	—	47	8	no	—
178	#376	—	47	8	no	—
179	C₄₇	#34187	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	8	no	—
181	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
182	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
189	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
190	#117	—	94	16	no	—
191	C₂×C₄⋊C₄	—	47	8	no	—
192	C₂×C₄⋊C₄	—	47	8	no	—
193	C₂×C₄⋊C₄	—	47	8	no	—
194	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
196	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
197	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
198	C₂×C₂²⋊C₄	—	47	8	no	—
199	C₂×C₄²	—	47	8	no	—
200	C₂×C₄²	—	47	16	no	—
201	C₂×C₄²	C₂×Dic₄₇	1	16	yes	—
202	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
206	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
207	#95	—	47	8	no	—
208	#95	—	47	16	no	—
209	#95	—	47	16	no	—
210	#95	—	47	8	no	—
211	#95	—	47	16	no	—
212	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	32	no	—
214	C₂×Q₈	—	2	8	no	—
215	C₂×Q₈	—	2	8	no	—
216	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	47	16	no	—
222	C₂²×C₄	#10625350	1	8	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
231	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
233	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	C₄⋊C₄	—	94	16	no	—
235	C₄⋊C₄	—	94	16	no	—
236	C₄⋊C₄	—	188	8	no	—
237	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
238	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
239	C₂²⋊C₄	—	94	16	no	—
240	C₄²	—	94	16	no	—
241	C₄²	—	188	16	no	—
242	C₄²	C₂²×Dic₄₇	1	16	yes	—
243	C₄²	C₂²×Dic₄₇	1	8	yes	—
244	C₄²	C₂²×Dic₄₇	1	16	yes	—
245	C₄²	C₂²×Dic₄₇	1	8	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂³×Dic₄₇	1	2	yes	—
249	C₂³	C₂×#10625350	1	4	yes	—
250	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
251	Q₈	—	2	32	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	94	16	no	—
261	C₂×C₄	—	94	16	no	—
262	C₂×C₄	—	94	16	no	—
263	C₂×C₄	#13093983	1	16	yes	—
264	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
266	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
267	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
268	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
269	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
270	C₂×C₄	C₂×#10625350	1	4	yes	—
271	C₂×C₄	C₂×#10625350	1	4	yes	—
272	C₂×C₄	#13094072	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×#13093983	1	4	yes	—
277	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
278	C₂²	C₂²×#10625350	1	2	yes	—
279	C₂²	C₂×#13094072	1	2	yes	—
280	C₂²	#15689959	1	2	yes	—
281	C₄	—	2	8	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	8	no	—
284	C₄	—	2	4	no	—
285	C₄	—	188	16	no	—
286	C₄	C₂×#13093983	1	8	yes	—
287	C₄	#15689601	1	4	yes	—
288	C₂	—	2	4	no	—
289	C₂	C₂²×#13093983	1	1	yes	characteristic
290	C₂	C₂×#15689601	1	1	yes	characteristic
291	C₂	#20256591	1	1	yes	characteristic
292	C₂	#20257253	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	#34187
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20246353	C₂	semidirect
2. #20246345	C₂	semidirect
26. C₂×C₄×Q₈×C₄₇	C₄	semidirect
156. C₂×C₄×Q₈	Dic₄₇	semidirect
179. C₄₇	#34187	semidirect

Maximal subgroups

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

subgroup	index
#20246353	2
#20246345	2
#20245991	2
#20245974	2
#20245795	2
#20245786	2
#20245745	2
C₂²×C₄×Q₈×C₄₇	2
#34187	47