Subgroups — #26103147

Order 12032.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₄×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	—	2	1	no	—
8	#15688876	—	2	1	no	—
9	#15688876	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#335	—	2	1	no	—
13	C₄₇×#335	C₄	1	2	yes	—
14	C₂²×C₄₇⋊C₈	—	4	1	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	C₈	1	4	yes	complemented
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
20	C₂²×C₈×C₄₇	—	4	1	no	—
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	—	2	2	no	—
25	C₄⋊C₈×C₄₇	—	4	1	no	—
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	—	8	1	no	—
36	C₂×C₈×C₄₇	—	8	1	no	—
37	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
43	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
47	C₂×C₄₇⋊C₈	—	4	2	no	—
48	C₂×C₄₇⋊C₈	—	8	1	no	—
49	C₂×C₄₇⋊C₈	—	8	1	no	—
50	C₄₇⋊C₁₆	—	2	4	no	—
51	C₂³×C₄₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#108	1	2	yes	—
61	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
63	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
64	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	—	8	2	no	—
68	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
69	C₈×C₄₇	#97	1	1	yes	characteristic
70	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
71	C₄₇⋊C₈	—	8	2	no	—
72	#7154	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	#335	1	2	yes	—
74	C₂²×C₄₇	#340	1	1	yes	characteristic
75	C₂²×C₄₇	#366	1	1	yes	characteristic
76	C₂²×C₄₇	#367	1	1	yes	characteristic
77	C₂²×C₄₇	#368	1	2	yes	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	#334	1	2	yes	—
81	C₄×C₄₇	#336	1	1	yes	characteristic
82	C₄×C₄₇	#341	1	1	yes	characteristic
83	C₄×C₄₇	#357	1	1	yes	characteristic
84	C₄×C₄₇	#357	1	1	yes	characteristic
85	C₄×C₄₇	#358	1	1	yes	characteristic
86	C₄×C₄₇	#358	1	1	yes	characteristic
87	C₂×C₄×C₁₆	—	47	1	no	—
88	C₂×#335	—	47	1	no	—
89	C₂×#335	D₄₇	1	1	yes	characteristic
90	C₂×C₄₇	#1277	1	1	yes	characteristic
91	C₂×C₄₇	#1371	1	2	yes	—
92	C₂×C₄₇	#1372	1	2	yes	—
93	C₂×C₄₇	#1380	1	1	yes	characteristic
94	C₂×C₄₇	#1381	1	1	yes	characteristic
95	C₂²×C₁₆	—	47	2	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₄⋊C₈	—	94	1	no	—
98	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₄×C₁₆	—	94	1	no	—
100	C₄×C₁₆	—	94	1	no	—
101	#335	—	2	1	no	—
102	#335	—	47	2	no	—
103	#335	—	94	1	no	—
104	#335	Dic₄₇	1	2	yes	—
105	C₄₇	#7154	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	—	188	1	no	—
108	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
110	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂×C₁₆	—	47	4	no	—
112	C₂×C₁₆	—	94	2	no	—
113	C₂×C₁₆	—	94	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	94	2	no	—
117	C₄⋊C₈	—	188	1	no	—
118	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
119	C₄×C₈	D₁₈₈	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#10625334	1	1	yes	characteristic
124	C₂²×C₄	#10625337	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	376	1	no	—
130	C₂×C₈	—	376	1	no	—
131	C₂×C₈	—	188	2	no	—
132	C₂×C₈	C₈×D₄₇	1	2	yes	—
133	C₂×C₈	#10625324	1	2	yes	—
134	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
135	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
137	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
138	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
139	C₄²	#10625335	1	1	yes	characteristic
140	C₄²	#10625336	1	1	yes	characteristic
141	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
142	C₁₆	—	94	4	no	—
143	C₂³	#13093957	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
146	C₂×C₄	#13093944	1	1	yes	characteristic
147	C₂×C₄	#13093945	1	1	yes	characteristic
148	C₂×C₄	#13093947	1	1	yes	characteristic
149	C₂×C₄	#13093948	1	1	yes	characteristic
150	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
151	C₂×C₄	#13093955	1	1	yes	characteristic
152	C₂×C₄	#13093959	1	2	yes	—
153	C₂×C₄	#13093965	1	1	yes	characteristic
154	C₂×C₄	#13093967	1	1	yes	characteristic
155	C₂×C₄	#13093968	1	1	yes	characteristic
156	C₂×C₄	#13093970	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	376	2	no	—
161	C₈	C₈×Dic₄₇	1	1	yes	characteristic
162	C₈	#13093934	1	1	yes	characteristic
163	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
164	C₂²	#15688875	1	2	yes	—
165	C₂²	#15688885	1	1	yes	characteristic
166	C₂²	#15688948	1	1	yes	characteristic
167	C₂²	#15688950	1	1	yes	characteristic
168	C₂²	#15688952	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#15688873	1	2	yes	—
172	C₄	#15688877	1	1	yes	characteristic
173	C₄	#15688887	1	1	yes	characteristic
174	C₄	#15688923	1	1	yes	characteristic
175	C₄	#15688924	1	1	yes	characteristic
176	C₄	#15688926	1	1	yes	characteristic
177	C₄	#15688927	1	1	yes	characteristic
178	C₂	#20241222	1	1	yes	characteristic
179	C₂	#20241459	1	2	yes	—
180	C₂	#20241461	1	2	yes	—
181	C₂	#20241479	1	1	yes	characteristic
182	C₂	#20241481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	#107
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#7154
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₂×C₄₇⋊C₁₆	C₈	semidirect
105. C₄₇	#7154	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
C₂×#15688876	2
C₂×C₄₇×#335	2
#7154	47