Subgroups — #26114830

Order 12032.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244106	C₂	1	2	yes	maximal
2	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	#20241228	C₂	1	2	yes	maximal
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093956	C₄	1	1	yes	characteristic
7	C₂×#13093956	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₄₇⋊C₈	—	2	2	no	—
10	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15688886	C₄	1	2	yes	—
12	#15688877	—	2	2	no	—
13	#15688876	—	2	2	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#322	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	—	2	2	no	—
22	#13093956	—	2	2	no	—
23	#13093956	C₈	1	4	yes	complemented
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	Dic₄₇⋊C₈	—	4	4	no	—
27	C₄₇⋊#95	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	1	no	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₄₇	—	2	1	no	—
35	C₄×C₈×C₄₇	—	2	1	no	—
36	C₄×C₈×C₄₇	—	2	1	no	—
37	C₄×C₈×C₄₇	D₄	1	2	yes	—
38	C₄×C₈×C₄₇	Q₈	1	2	yes	—
39	C₄×C₈×C₄₇	Q₈	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂²×Dic₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	—	2	1	no	—
48	C₂×C₈×C₄₇	—	2	1	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	4	2	no	—
53	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₄₇	#41	1	1	yes	characteristic
61	C₄⋊Dic₄₇	—	2	2	no	—
62	C₄⋊Dic₄₇	—	2	2	no	—
63	C₄⋊Dic₄₇	—	2	2	no	—
64	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
65	C₄×Dic₄₇	—	4	2	no	—
66	C₄×Dic₄₇	—	4	2	no	—
67	C₂×C₄₇⋊C₈	—	4	2	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
82	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₄₇	#137	1	2	yes	—
84	C₂×Dic₄₇	—	2	4	no	—
85	C₂×Dic₄₇	—	4	2	no	—
86	C₂×Dic₄₇	—	4	2	no	—
87	C₂×Dic₄₇	—	4	2	no	—
88	C₂×Dic₄₇	—	4	2	no	—
89	C₂×Dic₄₇	—	4	2	no	—
90	C₈×C₄₇	—	2	2	no	—
91	C₈×C₄₇	—	4	2	no	—
92	C₈×C₄₇	—	4	1	no	—
93	C₈×C₄₇	—	4	1	no	—
94	C₈×C₄₇	C₄⋊C₈	1	4	yes	complemented
95	C₄₇⋊C₈	—	8	2	no	—
96	#9596	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	#389	1	1	yes	characteristic
98	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
99	C₂²×C₄₇	#436	1	1	yes	characteristic
100	C₂²×C₄₇	#440	1	2	yes	—
101	C₂²×C₄₇	#444	1	2	yes	—
102	C₄×C₄₇	—	2	1	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₄₇	#424	1	1	yes	characteristic
106	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
107	C₄×C₄₇	#446	1	1	yes	characteristic
108	C₄×C₄₇	#469	1	1	yes	characteristic
109	C₄×C₄₇	#470	1	1	yes	characteristic
110	C₄×C₄₇	#501	1	2	yes	—
111	Dic₄₇	—	4	4	no	—
112	Dic₄₇	—	8	2	no	—
113	#1939	—	47	2	no	—
114	C₄×#106	—	47	1	no	—
115	C₂×#335	—	47	1	no	—
116	C₂×#322	D₄₇	1	1	yes	characteristic
117	#1279	—	47	2	no	—
118	C₂×C₄₇	SD₁₆⋊C₈	1	2	yes	—
119	C₂×C₄₇	#1592	1	2	yes	—
120	C₂×C₄₇	#1939	1	1	yes	characteristic
121	C₂×C₄₇	#1944	1	1	yes	characteristic
122	C₂×C₄₇	#1946	1	1	yes	characteristic
123	C₂×#106	—	47	1	no	—
124	C₂×#106	—	47	1	no	—
125	C₂×C₄⋊C₈	—	47	2	no	—
126	C₂×C₄⋊C₈	—	94	2	no	—
127	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	47	2	no	—
131	#340	—	47	2	no	—
132	#336	—	94	2	no	—
133	#335	—	94	2	no	—
134	#322	—	2	2	no	—
135	C₄₇	#9596	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	94	2	no	—
138	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
139	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
140	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
141	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	47	2	no	—
143	C₂×C₄⋊C₄	—	47	2	no	—
144	C₂×C₄²	—	94	2	no	—
145	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
146	#106	—	47	4	no	—
147	#106	—	94	2	no	—
148	C₄⋊C₈	—	94	2	no	—
149	C₄⋊C₈	—	94	2	no	—
150	C₄⋊C₈	—	188	4	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	Dic₉₄	1	2	yes	—
155	C₄×C₈	Dic₉₄	1	2	yes	—
156	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
157	#95	—	94	2	no	—
158	C₂²×C₄	—	94	2	no	—
159	C₂²×C₄	—	94	2	no	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
162	C₂²×C₄	#10625350	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	188	2	no	—
172	C₂×C₈	—	188	2	no	—
173	C₂×C₈	—	188	2	no	—
174	C₂×C₈	C₈×D₄₇	1	2	yes	—
175	C₂×C₈	#10625324	1	2	yes	—
176	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
177	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
178	C₄⋊C₄	—	47	4	no	—
179	C₄⋊C₄	—	94	2	no	—
180	C₄⋊C₄	—	94	2	no	—
181	C₄⋊C₄	—	94	2	no	—
182	C₄²	—	188	2	no	—
183	C₄²	—	188	2	no	—
184	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
185	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
186	C₄²	#10625350	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	94	4	no	—
191	C₂×C₄	—	188	2	no	—
192	C₂×C₄	—	188	2	no	—
193	C₂×C₄	—	188	2	no	—
194	C₂×C₄	—	188	2	no	—
195	C₂×C₄	—	188	2	no	—
196	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
198	C₂×C₄	#13093998	1	1	yes	characteristic
199	C₂×C₄	#13094003	1	1	yes	characteristic
200	C₂×C₄	#13094016	1	1	yes	characteristic
201	C₂×C₄	#13094024	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
203	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
204	C₂×C₄	#13094030	1	1	yes	characteristic
205	C₂×C₄	#13094030	1	1	yes	characteristic
206	C₂×C₄	#13094052	1	2	yes	—
207	C₂×C₄	#13094057	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	376	2	no	—
213	C₈	Dic₄₇⋊C₈	1	4	yes	complemented
214	C₂²	#15689021	1	1	yes	characteristic
215	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
216	C₂²	#15689202	1	1	yes	characteristic
217	C₂²	#15689225	1	2	yes	—
218	C₂²	#15689246	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	376	2	no	—
222	C₄	—	188	4	no	—
223	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
224	C₄	#15689149	1	1	yes	characteristic
225	C₄	C₈×Dic₉₄	1	1	yes	characteristic
226	C₄	#15689253	1	1	yes	characteristic
227	C₄	#15689376	1	1	yes	characteristic
228	C₄	#15689381	1	1	yes	characteristic
229	C₄	#15689532	1	2	yes	—
230	C₂	#20242202	1	2	yes	—
231	C₂	#20242292	1	2	yes	—
232	C₂	#20244106	1	1	yes	characteristic
233	C₂	#20244131	1	1	yes	characteristic
234	C₂	#20244141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9596
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
23. #13093956	C₈	semidirect
94. C₈×C₄₇	C₄⋊C₈	semidirect
135. C₄₇	#9596	semidirect
213. C₈	Dic₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20244106	2
C₄×#13093956	2
C₂×#15688876	2
#20241228	2
C₂×C₄₇×#322	2
#9596	47