Subgroups — #26376050

Order 12032.

2290 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246494	C₂	1	4	yes	maximal, complemented
2	#20246448	C₂	1	4	yes	maximal, complemented
3	#20246419	C₂	1	8	yes	maximal
4	#20245886	C₂	1	4	yes	maximal
5	#20245866	C₂	1	4	yes	maximal, complemented
6	C₄×Q₈×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	C₄×#13093983	C₂	1	2	yes	maximal
8	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
10	C₂×#13093983	C₂²	1	4	yes	—
11	#15689030	C₂²	1	8	yes	—
12	#15689022	C₂²	1	8	yes	—
13	#15689016	C₂²	1	16	yes	—
14	#15689009	C₂²	1	16	yes	—
15	#15689009	C₂²	1	16	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	8	yes	—
17	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	8	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
23	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
24	#15688969	C₂²	1	4	yes	—
25	#15688969	C₂²	1	4	yes	—
26	C₄²×Dic₄₇	C₂²	1	4	yes	—
27	C₄²×Dic₄₇	C₂²	1	4	yes	—
28	C₄₇×#533	C₄	1	16	yes	complemented
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
30	C₂×#125×C₄₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
33	C₂×#117×C₄₇	C₂²	1	2	yes	—
34	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	Q₈×Dic₄₇	—	2	32	no	—
36	#13093983	—	2	16	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
46	C₄₇⋊#95	C₂³	1	4	yes	—
47	C₄₇⋊#95	C₂³	1	4	yes	—
48	C₄₇⋊#95	C₂³	1	8	yes	—
49	C₄₇⋊#95	C₂³	1	16	yes	—
50	C₄₇⋊#95	C₂³	1	16	yes	—
51	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
55	#125×C₄₇	C₂×C₄	1	16	yes	—
56	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
57	C₄×Q₈×C₄₇	C₂×C₄	1	32	yes	—
58	#117×C₄₇	C₂×C₄	1	16	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	16	no	—
70	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
81	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
83	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
84	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
85	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
90	C₂²⋊Dic₄₇	—	2	16	no	—
91	C₄⋊Dic₄₇	—	2	8	no	—
92	C₄⋊Dic₄₇	—	2	32	no	—
93	C₄⋊Dic₄₇	#41	1	16	yes	—
94	C₄×Dic₄₇	—	2	16	no	—
95	C₄×Dic₄₇	—	2	8	no	—
96	C₄×Dic₄₇	—	2	8	no	—
97	C₄×Dic₄₇	—	2	16	no	—
98	C₄×Dic₄₇	—	2	32	no	—
99	C₂³×C₄₇	—	2	4	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
103	Q₈×C₄₇	—	2	32	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	16	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	16	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
115	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
116	C₂×Dic₄₇	—	2	8	no	—
117	C₂×Dic₄₇	—	2	8	no	—
118	C₂×Dic₄₇	—	2	8	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	C₂×#41	1	16	yes	—
122	#34154	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	4	no	—
126	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
127	C₂²×C₄₇	C₂²×#41	1	4	yes	—
128	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	2	16	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	C₄×#41	1	8	yes	—
133	C₄×C₄₇	#519	1	4	yes	—
134	Dic₄₇	—	2	16	no	—
135	Dic₄₇	—	4	8	no	—
136	C₂×#533	D₄₇	1	1	yes	characteristic
137	#2371	—	47	4	no	—
138	#2365	—	47	4	no	—
139	#2362	—	47	8	no	—
140	#2298	—	47	4	no	—
141	#2295	—	47	4	no	—
142	C₄²×Q₈	—	47	4	no	—
143	C₄×#117	—	47	2	no	—
144	C₂×C₄₇	—	4	2	no	—
145	C₂×C₄₇	C₂×C₄×#41	1	2	yes	—
146	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
147	C₂×C₄₇	#3482	1	4	yes	—
148	#533	Dic₄₇	1	16	yes	complemented
149	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
150	C₂×#125	C₂×D₄₇	1	2	yes	—
151	C₂×C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
152	C₂×C₄×Q₈	—	47	8	no	—
153	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
154	C₂×#117	—	47	4	no	—
155	C₂×#117	C₂×D₄₇	1	2	yes	—
156	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
157	#391	—	47	8	no	—
158	#389	—	47	8	no	—
159	#388	—	47	16	no	—
160	#387	—	47	16	no	—
161	#387	—	47	16	no	—
162	C₄×C₄⋊C₄	—	47	8	no	—
163	C₄×C₄⋊C₄	—	47	16	no	—
164	C₄×C₄⋊C₄	—	47	4	no	—
165	C₄×C₄⋊C₄	—	47	2	no	—
166	C₄×C₄⋊C₄	—	47	2	no	—
167	C₄×C₄⋊C₄	—	47	8	no	—
168	C₄×C₂²⋊C₄	—	47	4	no	—
169	C₄×C₂²⋊C₄	—	47	4	no	—
170	#376	—	47	4	no	—
171	#376	—	47	4	no	—
172	C₄³	—	47	4	no	—
173	C₄³	—	47	4	no	—
174	C₄₇	#34154	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
176	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
179	#125	C₂×Dic₄₇	1	16	yes	—
180	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
181	C₄×Q₈	—	94	32	no	—
182	C₄×Q₈	C₂×Dic₄₇	1	32	yes	—
183	#117	—	94	16	no	—
184	#117	C₂×Dic₄₇	1	16	yes	—
185	C₂×C₄⋊C₄	—	47	4	no	—
186	C₂×C₄⋊C₄	—	47	4	no	—
187	C₂×C₄⋊C₄	—	47	16	no	—
188	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
189	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
190	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
191	C₂×C₂²⋊C₄	—	47	8	no	—
192	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
193	C₂×C₄²	—	47	8	no	—
194	C₂×C₄²	—	47	8	no	—
195	C₂×C₄²	—	47	16	no	—
196	C₂×C₄²	—	47	4	no	—
197	C₂×C₄²	—	47	4	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
202	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
203	#95	—	47	4	no	—
204	#95	—	47	8	no	—
205	#95	—	47	4	no	—
206	#95	—	47	16	no	—
207	#95	—	47	16	no	—
208	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
209	C₂×Q₈	—	2	16	no	—
210	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	47	8	no	—
214	C₂²×C₄	—	47	8	no	—
215	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
216	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
217	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
219	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
220	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
222	C₄⋊C₄	—	47	16	no	—
223	C₄⋊C₄	—	94	32	no	—
224	C₄⋊C₄	—	94	8	no	—
225	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
226	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
227	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
228	C₂²⋊C₄	—	94	16	no	—
229	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
230	C₄²	—	94	16	no	—
231	C₄²	—	94	32	no	—
232	C₄²	—	94	16	no	—
233	C₄²	—	94	8	no	—
234	C₄²	—	94	8	no	—
235	C₄²	C₂²×Dic₄₇	1	8	yes	—
236	C₄²	C₂²×Dic₄₇	1	4	yes	—
237	C₄²	C₂²×Dic₄₇	1	4	yes	—
238	C₄²	C₂²×Dic₄₇	1	16	yes	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₂³×Dic₄₇	1	2	yes	—
242	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
243	Q₈	—	2	32	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	47	16	no	—
250	C₂×C₄	—	94	8	no	—
251	C₂×C₄	—	94	8	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	94	8	no	—
254	C₂×C₄	—	94	8	no	—
255	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
256	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
258	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
259	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
260	C₂×C₄	#41×D₄₇	1	8	yes	—
261	C₂×C₄	#13094072	1	4	yes	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
266	C₂²	C₂×#41×D₄₇	1	4	yes	—
267	C₂²	C₂×#13094072	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	16	no	—
270	C₄	—	2	2	no	—
271	C₄	—	94	16	no	—
272	C₄	—	188	8	no	—
273	C₄	#41×Dic₄₇	1	8	yes	—
274	C₄	#15689601	1	4	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₂×#41×Dic₄₇	1	2	yes	—
277	C₂	C₂×#15689601	1	1	yes	characteristic
278	C₂	#20257006	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#34154
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20246494	C₂	semidirect
2. #20246448	C₂	semidirect
5. #20245866	C₂	semidirect
6. C₄×Q₈×Dic₄₇	C₂	semidirect
28. C₄₇×#533	C₄	semidirect
148. #533	Dic₄₇	semidirect
174. C₄₇	#34154	semidirect

Maximal subgroups

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

subgroup	index
#20246494	2
#20246448	2
#20246419	2
#20245886	2
#20245866	2
C₄×Q₈×Dic₄₇	2
C₄×#13093983	2
C₂×C₄₇×#533	2
#34154	47