Subgroups — #26225150

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253912	C₂	1	1	yes	maximal, characteristic
2	#20252325	C₂	1	2	yes	maximal
3	#20244364	C₂	1	2	yes	maximal
4	#20244139	C₂	1	2	yes	maximal
5	#20244134	C₂	1	2	yes	maximal
6	#20243275	C₂	1	2	yes	maximal
7	#20243270	C₂	1	2	yes	maximal
8	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	2	yes	—
11	#15689532	C₄	1	8	yes	complemented
12	#15689252	—	2	8	no	—
13	#15689161	C₂²	1	2	yes	—
14	#15689161	C₂²	1	2	yes	—
15	#15689151	C₂²	1	4	yes	—
16	#15689081	C₂²	1	4	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	1	yes	characteristic
19	C₂×#13093934	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688884	C₂²	1	4	yes	—
24	#15688884	C₂²	1	4	yes	—
25	C₂×#108×C₄₇	C₂²	1	2	yes	—
26	C₂×#108×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#423	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	#13094024	C₂×C₄	1	4	yes	—
32	#13094016	C₂×C₄	1	4	yes	—
33	#13094003	—	2	4	no	—
34	C₄×Dic₉₄	—	2	4	no	—
35	#13093983	C₂³	1	4	yes	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
39	#13093958	C₂×C₄	1	4	yes	—
40	#13093958	C₂×C₄	1	4	yes	—
41	#13093956	C₂×C₄	1	4	yes	—
42	#13093956	C₂×C₄	1	4	yes	—
43	Dic₄₇⋊C₈	—	2	8	no	—
44	#13093935	D₄	1	4	yes	—
45	#13093934	—	2	2	no	—
46	#13093934	—	2	4	no	—
47	#13093934	Q₈	1	4	yes	complemented
48	C₈×Dic₄₇	—	2	2	no	—
49	C₈×Dic₄₇	—	2	4	no	—
50	C₈×Dic₄₇	Q₈	1	4	yes	complemented
51	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	#108×C₄₇	—	2	4	no	—
56	#108×C₄₇	—	2	4	no	—
57	C₄⋊C₈×C₄₇	—	2	4	no	—
58	#97×C₄₇	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	1	no	—
62	C₂²×Dic₄₇	—	2	1	no	—
63	C₂×Dic₉₄	—	2	4	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₄₇	#41	1	2	yes	—
72	C₂×C₈×C₄₇	#41	1	2	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
78	Dic₄₇⋊C₄	—	2	8	no	—
79	C₄×Dic₄₇	—	2	4	no	—
80	C₄×Dic₄₇	—	2	1	no	—
81	C₄×Dic₄₇	—	2	1	no	—
82	C₄×Dic₄₇	—	4	4	no	—
83	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
84	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
91	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₂×Dic₄₇	—	2	4	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	8	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	Dic₉₄	—	8	4	no	—
106	C₈×C₄₇	—	4	4	no	—
107	C₈×C₄₇	C₄×Q₈	1	4	yes	—
108	C₈×C₄₇	C₄×Q₈	1	4	yes	—
109	C₄₇⋊C₈	—	2	2	no	—
110	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
111	#21736	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄₇	#533	1	1	yes	characteristic
115	C₂²×C₄₇	#533	1	1	yes	characteristic
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
118	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₄×C₄₇	#532	1	1	yes	characteristic
121	Dic₄₇	—	2	4	no	—
122	Dic₄₇	—	4	2	no	—
123	Dic₄₇	—	4	8	no	—
124	#3162	—	47	1	no	—
125	#2961	—	47	2	no	—
126	#1994	—	47	2	no	—
127	#1946	—	47	2	no	—
128	#1945	—	47	2	no	—
129	#1849	D₄₇	1	1	yes	characteristic
130	#1773	—	47	2	no	—
131	#1772	—	47	2	no	—
132	C₄×#97	—	47	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	1	no	—
135	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄₇	#2956	1	2	yes	—
137	#533	—	47	2	no	—
138	#501	—	47	8	no	—
139	#446	—	94	8	no	—
140	C₂×#108	C₂×D₄₇	1	2	yes	—
141	C₂×#108	C₂×D₄₇	1	2	yes	—
142	#427	—	47	2	no	—
143	#427	—	47	2	no	—
144	#424	—	47	4	no	—
145	#423	C₂×D₄₇	1	2	yes	—
146	#405	—	47	4	no	—
147	C₂×#97	—	47	1	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×#97	—	47	1	no	—
150	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
151	C₂×C₄×C₈	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₄³	—	47	1	no	—
154	#339	—	47	4	no	—
155	#339	—	47	4	no	—
156	C₄₇	#21736	1	1	yes	characteristic, complemented, Sylow
157	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
158	C₂²×C₈	—	47	1	no	—
159	C₂²×C₈	—	47	1	no	—
160	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	47	4	no	—
163	#125	—	47	4	no	—
164	C₂²⋊Q₈	—	94	4	no	—
165	C₄×Q₈	—	94	4	no	—
166	#117	—	47	4	no	—
167	C₂×C₄²	—	47	1	no	—
168	C₂×C₄²	—	47	1	no	—
169	C₂×C₄²	—	94	2	no	—
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	#108	—	2	4	no	—
172	#108	—	2	4	no	—
173	#107	—	47	4	no	—
174	#107	—	47	4	no	—
175	#106	—	47	4	no	—
176	#106	—	47	4	no	—
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	—	94	8	no	—
179	#97	—	47	4	no	—
180	#97	—	47	4	no	—
181	#97	—	94	4	no	—
182	#97	—	94	2	no	—
183	#97	C₄×D₄₇	1	4	yes	—
184	C₄×C₈	—	47	4	no	—
185	C₄×C₈	—	94	4	no	—
186	C₄×C₈	—	94	2	no	—
187	C₂×Q₈	—	94	4	no	—
188	C₂²×C₄	—	94	1	no	—
189	C₂²×C₄	—	94	1	no	—
190	C₂²×C₄	#10625354	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	47	2	no	—
196	C₂×C₈	—	47	2	no	—
197	C₂×C₈	—	47	2	no	—
198	C₂×C₈	—	47	2	no	—
199	C₂×C₈	—	94	1	no	—
200	C₂×C₈	—	94	1	no	—
201	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
203	C₂×C₈	#10625352	1	2	yes	—
204	C₂×C₈	#10625352	1	2	yes	—
205	C₂×C₈	Q₈×D₄₇	1	2	yes	—
206	C₂×C₈	Q₈×D₄₇	1	2	yes	—
207	C₄⋊C₄	—	47	8	no	—
208	C₄⋊C₄	—	94	8	no	—
209	C₄⋊C₄	—	94	4	no	—
210	C₂²⋊C₄	—	94	4	no	—
211	C₄²	—	47	2	no	—
212	C₄²	—	47	2	no	—
213	C₄²	—	94	4	no	—
214	C₄²	—	94	1	no	—
215	C₄²	—	94	1	no	—
216	C₄²	—	188	4	no	—
217	C₄²	C₂×C₄×D₄₇	1	2	yes	—
218	C₄²	D₄×D₄₇	1	2	yes	—
219	C₂³	C₂×#10625354	1	1	yes	characteristic
220	Q₈	—	376	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	94	4	no	—
223	C₂×C₄	—	94	2	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	—	94	2	no	—
226	C₂×C₄	—	188	2	no	—
227	C₂×C₄	#13093986	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
233	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
234	C₈	—	4	4	no	—
235	C₈	—	47	4	no	—
236	C₈	—	94	2	no	—
237	C₈	#13093989	1	4	yes	—
238	C₈	#13093989	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×#13093986	1	1	yes	characteristic
241	C₂²	#15689678	1	1	yes	characteristic
242	C₂²	#15689678	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	94	4	no	—
245	C₄	—	188	2	no	—
246	C₄	—	188	8	no	—
247	C₄	C₂×#13093989	1	1	yes	characteristic
248	C₄	C₂×#13093989	1	1	yes	characteristic
249	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
250	C₄	#15689671	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#20245955	1	1	yes	characteristic
254	C₂	#20252297	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#21736
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
11. #15689532	C₄	semidirect
47. #13093934	Q₈	semidirect
50. C₈×Dic₄₇	Q₈	semidirect
156. C₄₇	#21736	semidirect

Maximal subgroups

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

subgroup	index
#20253912	2
#20252325	2
#20244364	2
#20244139	2
#20244134	2
#20243275	2
#20243270	2
#97×Dic₄₇	2
C₄₇×#1849	2
#21736	47