Subgroups — #26115507

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	#20244311	C₂	1	2	yes	maximal
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688868	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689528	—	2	2	no	—
8	#15689528	C₄	1	4	yes	complemented
9	C₂×#13093958	—	2	1	no	—
10	C₂×#13093958	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13093947	—	2	4	no	—
13	#15688876	—	2	2	no	—
14	#15688868	—	2	4	no	—
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	—	2	1	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	—	2	2	no	—
25	#13094024	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	#13093958	—	2	4	no	—
29	#13093958	—	2	4	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	#13093947	—	4	8	no	—
33	C₂²×C₈×C₄₇	—	2	1	no	—
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	1	no	—
40	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
41	C₄×C₈×C₄₇	—	4	2	no	—
42	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	#95×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂²×Dic₄₇	—	4	2	no	—
50	C₂×Dic₉₄	—	2	4	no	—
51	C₂×Dic₉₄	—	4	4	no	—
52	C₂×Dic₉₄	—	4	4	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×C₈×C₄₇	—	2	1	no	—
55	C₂×C₈×C₄₇	—	2	1	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	4	1	no	—
58	C₂×C₈×C₄₇	—	4	1	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
62	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₄⋊C₄×C₄₇	D₈	1	4	yes	—
65	C₄²×C₄₇	—	4	1	no	—
66	C₄²×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	#41	1	1	yes	characteristic
71	C₄⋊Dic₄₇	—	2	4	no	—
72	C₄⋊Dic₄₇	—	2	4	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	4	no	—
77	C₂³×C₄₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#135	1	2	yes	—
92	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	4	no	—
95	C₂×Dic₄₇	—	8	4	no	—
96	Dic₉₄	—	4	8	no	—
97	Dic₉₄	—	8	2	no	—
98	Dic₉₄	—	8	2	no	—
99	C₈×C₄₇	—	2	2	no	—
100	C₈×C₄₇	—	4	2	no	—
101	C₈×C₄₇	#103	1	4	yes	complemented
102	C₄₇⋊C₈	—	8	2	no	—
103	#9731	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	#390	1	1	yes	characteristic
105	C₂²×C₄₇	#461	1	1	yes	characteristic
106	C₂²×C₄₇	#462	1	1	yes	characteristic
107	C₂²×C₄₇	#466	1	2	yes	—
108	C₂²×C₄₇	#473	1	2	yes	—
109	C₄×C₄₇	—	2	1	no	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	4	4	no	—
112	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
113	C₄×C₄₇	#419	1	1	yes	characteristic
114	C₄×C₄₇	C₄×D₈	1	2	yes	—
115	C₄×C₄₇	#461	1	1	yes	characteristic
116	C₄×C₄₇	#462	1	1	yes	characteristic
117	C₄×C₄₇	#493	1	1	yes	characteristic
118	C₄×C₄₇	#495	1	1	yes	characteristic
119	Dic₄₇	—	8	4	no	—
120	C₂×#500	—	47	1	no	—
121	#1981	—	47	2	no	—
122	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
123	C₂×#335	—	47	1	no	—
124	C₂×#332	—	47	2	no	—
125	C₂×C₄₇	#1673	1	2	yes	—
126	C₂×C₄₇	#1674	1	2	yes	—
127	C₂×C₄₇	#1970	1	1	yes	characteristic
128	C₂×C₄₇	#1979	1	1	yes	characteristic
129	C₂×C₄₇	#1981	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	—	47	2	no	—
131	#500	—	47	4	no	—
132	#500	—	94	2	no	—
133	C₂×#107	—	94	1	no	—
134	C₂×#107	—	94	1	no	—
135	C₂×C₄⋊C₈	—	47	2	no	—
136	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
137	C₂×#103	—	94	4	no	—
138	C₂×C₄×C₈	—	2	1	no	—
139	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#335	—	94	2	no	—
143	#332	—	94	4	no	—
144	C₄₇	#9731	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	94	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	94	2	no	—
148	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₄⋊Q₈	—	47	4	no	—
151	C₄⋊Q₈	—	94	2	no	—
152	C₄⋊Q₈	—	94	2	no	—
153	C₄⋊Q₈	—	94	2	no	—
154	C₂×C₄⋊C₄	—	94	2	no	—
155	C₂×C₄⋊C₄	—	94	2	no	—
156	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
159	#107	—	94	4	no	—
160	#107	—	94	4	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	94	2	no	—
163	C₄⋊C₈	—	94	2	no	—
164	C₄⋊C₈	D₁₈₈	1	2	yes	—
165	#103	—	188	8	no	—
166	C₄×C₈	—	4	2	no	—
167	C₄×C₈	C₄×D₄₇	1	2	yes	—
168	C₄×C₈	D₁₈₈	1	2	yes	—
169	#95	—	2	1	no	—
170	C₂×Q₈	—	94	4	no	—
171	C₂×Q₈	—	188	4	no	—
172	C₂×Q₈	—	188	4	no	—
173	C₂×Q₈	—	188	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	188	2	no	—
176	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
177	C₂²×C₄	#10625350	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	188	2	no	—
185	C₂×C₈	—	188	4	no	—
186	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
187	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
188	C₂×C₈	#10625335	1	2	yes	—
189	C₂×C₈	#10625337	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	94	4	no	—
192	C₄⋊C₄	—	94	4	no	—
193	C₄⋊C₄	—	188	2	no	—
194	C₄⋊C₄	—	188	2	no	—
195	C₄⋊C₄	D₃₇₆	1	4	yes	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
200	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
201	C₄²	#10625350	1	1	yes	characteristic
202	C₂³	#13093998	1	1	yes	characteristic
203	Q₈	—	376	2	no	—
204	Q₈	—	376	2	no	—
205	Q₈	—	188	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	376	4	no	—
211	C₂×C₄	—	188	4	no	—
212	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
214	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
215	C₂×C₄	#13093998	1	1	yes	characteristic
216	C₂×C₄	#13094012	1	1	yes	characteristic
217	C₂×C₄	#13094021	1	1	yes	characteristic
218	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
219	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
220	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
221	C₂×C₄	#13094046	1	2	yes	—
222	C₂×C₄	#13094054	1	1	yes	characteristic
223	C₂×C₄	#13094054	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	376	2	no	—
227	C₈	#13093947	1	4	yes	complemented
228	C₂²	#15689025	1	1	yes	characteristic
229	C₂²	#15689328	1	1	yes	characteristic
230	C₂²	#15689333	1	1	yes	characteristic
231	C₂²	#15689359	1	2	yes	—
232	C₂²	#15689398	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	376	4	no	—
237	C₄	C₂×#13093947	1	1	yes	characteristic
238	C₄	#15689131	1	1	yes	characteristic
239	C₄	C₄×D₃₇₆	1	2	yes	—
240	C₄	#15689328	1	1	yes	characteristic
241	C₄	#15689333	1	1	yes	characteristic
242	C₄	#15689500	1	1	yes	characteristic
243	C₄	#15689507	1	1	yes	characteristic
244	C₂	#20242777	1	2	yes	—
245	C₂	#20242782	1	2	yes	—
246	C₂	#20244254	1	1	yes	characteristic
247	C₂	#20244303	1	1	yes	characteristic
248	C₂	#20244311	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#9731
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. #15689528	C₄	semidirect
101. C₈×C₄₇	#103	semidirect
144. C₄₇	#9731	semidirect
227. C₈	#13093947	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
#20244311	2
C₂×#15688876	2
C₂×#15688868	2
C₈×C₄⋊C₄×C₄₇	2
#9731	47