Subgroups — #26134402

Order 12032.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688945	C₂	1	2	yes	maximal
2	#20245144	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
4	#20241486	C₂	1	2	yes	maximal
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093960	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₂²	1	1	yes	characteristic
9	C₄×#10625329	—	2	1	no	—
10	#15688945	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₁₆)	C₄	1	4	yes	complemented
12	#15688888	—	2	1	no	—
13	C₂×#106×C₄₇	—	2	1	no	—
14	C₂×#106×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	—	2	1	no	—
17	C₂×#10625329	—	2	1	no	—
18	C₂×#10625329	—	2	1	no	—
19	#13093964	—	2	2	no	—
20	#13093964	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	#13093960	—	2	1	no	—
23	#13093960	—	2	1	no	—
24	#13093960	D₄	1	2	yes	—
25	#13093960	Q₈	1	2	yes	—
26	#13093935	—	4	1	no	—
27	C₄×C₄₇⋊C₈	—	4	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	#106×C₄₇	—	2	2	no	—
34	#106×C₄₇	—	2	2	no	—
35	#106×C₄₇	—	4	2	no	—
36	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	C₂²×C₄×C₄₇	—	4	2	no	—
39	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	4	2	no	—
52	C₄⋊C₄×C₄₇	—	8	4	no	—
53	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
55	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
56	#10625329	—	2	2	no	—
57	#10625329	—	2	2	no	—
58	#10625329	—	4	1	no	—
59	#10625329	—	4	1	no	—
60	C₂×C₄₇⋊C₈	—	4	1	no	—
61	C₂×C₄₇⋊C₈	—	4	1	no	—
62	C₄₇⋊C₁₆	—	2	4	no	—
63	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₄₇	—	2	4	no	—
65	C₂×C₄×C₄₇	—	8	4	no	—
66	C₂×C₄×C₄₇	—	8	2	no	—
67	C₂×C₄×C₄₇	#106	1	2	yes	—
68	C₂×C₄×C₄₇	#106	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
77	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₄₇	#125	1	1	yes	characteristic
79	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₄₇⋊C₈	—	4	2	no	—
81	C₄₇⋊C₈	—	4	2	no	—
82	#13183	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
85	C₂²×C₄₇	#460	1	1	yes	characteristic
86	C₂²×C₄₇	#475	1	1	yes	characteristic
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	16	2	no	—
89	C₄×C₄₇	#365	1	4	yes	complemented
90	C₄×C₄₇	#389	1	1	yes	characteristic
91	C₄×C₄₇	#428	1	1	yes	characteristic
92	C₄×C₄₇	#461	1	1	yes	characteristic
93	C₄×C₄₇	#476	1	1	yes	characteristic
94	C₂×#365	—	47	2	no	—
95	#2150	—	47	1	no	—
96	#1844	D₄₇	1	1	yes	characteristic
97	C₄×#108	—	47	1	no	—
98	#1383	—	47	2	no	—
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	#1948	1	1	yes	characteristic
102	C₂×C₄₇	C₂×#365	1	2	yes	—
103	C₂×M₃₂	—	47	2	no	—
104	C₂×#108	—	47	1	no	—
105	C₂×#108	—	47	1	no	—
106	C₂×#106	—	2	1	no	—
107	C₂×#106	C₂×D₄₇	1	2	yes	—
108	C₄×M₁₆	—	94	1	no	—
109	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#365	—	94	4	no	—
112	C₄⋊C₁₆	—	47	4	no	—
113	#341	—	94	1	no	—
114	C₄₇	#13183	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	94	1	no	—
116	C₂×M₁₆	—	94	1	no	—
117	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
122	M₃₂	—	94	2	no	—
123	M₃₂	—	94	2	no	—
124	C₂×C₁₆	—	47	4	no	—
125	#108	—	47	2	no	—
126	#108	—	47	2	no	—
127	#108	—	94	1	no	—
128	#108	—	94	1	no	—
129	#106	—	2	2	no	—
130	#106	—	2	2	no	—
131	#106	—	4	2	no	—
132	#97	—	188	1	no	—
133	C₄×C₈	—	188	1	no	—
134	C₄×C₈	Dic₉₄	1	2	yes	—
135	C₄×C₈	C₄×D₄₇	1	2	yes	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	#10625335	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
139	M₁₆	—	94	2	no	—
140	M₁₆	—	94	2	no	—
141	M₁₆	—	188	1	no	—
142	M₁₆	—	188	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	188	1	no	—
146	C₂×C₈	—	188	1	no	—
147	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
148	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
150	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
151	C₂×C₈	#10625352	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
153	C₂×C₈	#10625354	1	1	yes	characteristic
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	8	4	no	—
157	C₄²	Dic₄₇⋊C₄	1	2	yes	—
158	C₄²	#10625336	1	1	yes	characteristic
159	C₄²	#10625336	1	1	yes	characteristic
160	C₁₆	—	94	4	no	—
161	C₂³	C₂×#10625335	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	#13093955	1	2	yes	—
166	C₂×C₄	#13093955	1	2	yes	—
167	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#13093995	1	1	yes	characteristic
169	C₂×C₄	#13094000	1	1	yes	characteristic
170	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
171	C₂×C₄	#13094050	1	1	yes	characteristic
172	C₂×C₄	#13094055	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	188	2	no	—
175	C₈	—	188	2	no	—
176	C₈	#13093986	1	1	yes	characteristic
177	C₈	#13093989	1	1	yes	characteristic
178	C₈	#13094015	1	1	yes	characteristic
179	C₈	#13094023	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#13093955	1	1	yes	characteristic
182	C₂²	#15689320	1	1	yes	characteristic
183	C₂²	#15689405	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	16	2	no	—
186	C₄	#15688945	1	4	yes	complemented
187	C₄	#15689019	1	1	yes	characteristic
188	C₄	#15689163	1	1	yes	characteristic
189	C₄	#15689325	1	1	yes	characteristic
190	C₄	#15689410	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20244149	1	1	yes	characteristic
194	C₂	C₂×#15688945	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13093955
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂²×C₄₇	C₂×#106
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#13183
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₄⋊(C₄₇⋊C₁₆)	C₄	semidirect
89. C₄×C₄₇	#365	semidirect
114. C₄₇	#13183	semidirect
186. C₄	#15688945	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688945	2
#20245144	2
C₄×#13093960	2
#20241486	2
C₄₇×#1844	2
#13183	47