Subgroups — #26114993

Order 12032.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244164	C₂	1	2	yes	maximal
2	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20241677	C₂	1	1	yes	maximal, characteristic
4	#20241285	C₂	1	2	yes	maximal
5	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	C₂×#13093960	—	2	2	no	—
8	#15689147	C₂²	1	2	yes	—
9	C₄×#10625329	C₂²	1	1	yes	characteristic
10	#15688888	—	2	2	no	—
11	#15688876	—	2	1	no	—
12	#15688876	—	2	1	no	—
13	#15688845	—	2	1	no	—
14	#15688844	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#334	C₄	1	2	yes	—
18	C₄₇×#334	C₄	1	2	yes	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	D₄	1	1	yes	characteristic
21	C₂×#10625329	D₄	1	1	yes	characteristic
22	#13093960	—	4	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	Q₈	1	2	yes	—
27	#13093935	—	2	1	no	—
28	C₄×C₄₇⋊C₈	—	2	1	no	—
29	C₄×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	4	no	—
45	C₂×C₈×C₄₇	—	4	1	no	—
46	C₂×C₈×C₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₄²×C₄₇	#41	1	1	yes	characteristic
56	#10625329	—	2	2	no	—
57	#10625329	—	2	2	no	—
58	#10625329	—	4	2	no	—
59	#10625329	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	2	1	no	—
61	C₂×C₄₇⋊C₈	—	2	1	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
74	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₄₇	#137	1	2	yes	—
76	C₈×C₄₇	—	2	1	no	—
77	C₈×C₄₇	—	2	1	no	—
78	C₈×C₄₇	—	2	1	no	—
79	C₈×C₄₇	—	4	4	no	—
80	C₈×C₄₇	#108	1	2	yes	complemented
81	C₄₇⋊C₈	—	4	2	no	—
82	C₄₇⋊C₈	—	4	1	no	—
83	C₄₇⋊C₈	—	4	1	no	—
84	C₄₇⋊C₈	—	8	2	no	—
85	#9626	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	#389	1	1	yes	characteristic
89	C₂²×C₄₇	#501	1	2	yes	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	1	no	—
93	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
94	C₄×C₄₇	#430	1	1	yes	characteristic
95	C₄×C₄₇	#440	1	2	yes	—
96	C₄×C₄₇	#469	1	1	yes	characteristic
97	C₄×C₄₇	#470	1	1	yes	characteristic
98	#1950	—	47	2	no	—
99	#1567	—	47	1	no	—
100	C₂×#334	D₄₇	1	1	yes	characteristic
101	#1451	—	47	1	no	—
102	#1301	—	47	2	no	—
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	#1944	1	1	yes	characteristic
105	C₂×C₄₇	#1950	1	1	yes	characteristic
106	C₂×C₄₇	#1953	1	1	yes	characteristic
107	#431	—	47	1	no	—
108	C₂×#108	—	94	2	no	—
109	#423	—	47	2	no	—
110	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
111	C₄×M₁₆	—	47	1	no	—
112	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
113	#341	—	94	2	no	—
114	#335	—	94	1	no	—
115	#335	—	94	1	no	—
116	#334	Dic₄₇	1	2	yes	—
117	#334	Dic₄₇	1	2	yes	—
118	#322	—	94	1	no	—
119	#322	—	94	1	no	—
120	C₄₇	#9626	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	47	1	no	—
122	C₂×M₁₆	—	47	1	no	—
123	C₂×M₁₆	—	94	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
126	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	#108	—	188	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	94	2	no	—
131	C₄⋊C₈	—	94	1	no	—
132	C₄⋊C₈	—	94	2	no	—
133	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
134	C₂²⋊C₈	—	47	2	no	—
135	#97	—	94	1	no	—
136	C₄×C₈	—	94	1	no	—
137	C₄×C₈	—	94	1	no	—
138	C₄×C₈	D₁₈₈	1	1	yes	characteristic
139	C₄×C₈	D₁₈₈	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
143	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
144	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
145	M₁₆	—	94	2	no	—
146	M₁₆	—	94	2	no	—
147	M₁₆	—	188	2	no	—
148	M₁₆	—	188	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	94	1	no	—
155	C₂×C₈	—	94	1	no	—
156	C₂×C₈	—	94	2	no	—
157	C₂×C₈	—	188	2	no	—
158	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
160	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
162	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
163	C₄²	#10625352	1	1	yes	characteristic
164	C₄²	#10625354	1	1	yes	characteristic
165	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
166	C₂³	#13094023	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
173	C₂×C₄	#13093999	1	1	yes	characteristic
174	C₂×C₄	#13094004	1	1	yes	characteristic
175	C₂×C₄	#13094015	1	1	yes	characteristic
176	C₂×C₄	#13094051	1	2	yes	—
177	C₂×C₄	#13094053	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	4	no	—
182	C₈	—	376	2	no	—
183	C₈	—	188	2	no	—
184	C₈	—	188	1	no	—
185	C₈	—	188	1	no	—
186	C₈	#13093960	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#15689022	1	1	yes	characteristic
190	C₂²	#15689530	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#13093960	1	1	yes	characteristic
195	C₄	#15689172	1	1	yes	characteristic
196	C₄	#15689224	1	2	yes	—
197	C₄	#15689377	1	1	yes	characteristic
198	C₄	#15689382	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#20244132	1	1	yes	characteristic
201	C₂	#20244164	1	1	yes	characteristic
202	C₂	#20244179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#9626
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
80. C₈×C₄₇	#108	semidirect
120. C₄₇	#9626	semidirect
186. C₈	#13093960	semidirect

Maximal subgroups

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

subgroup	index
#20244164	2
#20242147	2
#20241677	2
#20241285	2
C₂×C₄₇×#334	2
#9626	47