Subgroups — #26103471

Order 12032.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 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	#20243560	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂×#13093960	—	2	1	no	—
6	C₂×#13093960	C₄	1	2	yes	—
7	#15689147	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
10	C₄₇×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#335	—	2	1	no	—
13	C₄₇×#334	C₄	1	2	yes	—
14	C₂×#10625329	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	4	1	no	—
17	C₂×C₄₇⋊C₁₆	—	4	1	no	—
18	#13093960	—	4	1	no	—
19	#13093960	—	4	1	no	—
20	#13093960	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	—	2	2	no	—
27	C₂²⋊C₈×C₄₇	—	4	1	no	—
28	C₄×C₈×C₄₇	—	4	1	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
47	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₇	D₈	1	1	yes	characteristic
50	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
51	#10625329	—	4	2	no	—
52	#10625329	—	4	2	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₄₇⋊C₁₆	—	4	2	no	—
55	C₂³×C₄₇	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	—	8	2	no	—
71	C₈×C₄₇	#102	1	1	yes	characteristic
72	C₈×C₄₇	#103	1	1	yes	characteristic
73	C₈×C₄₇	#106	1	1	yes	characteristic
74	C₈×C₄₇	#107	1	1	yes	characteristic
75	C₄₇⋊C₈	—	8	2	no	—
76	#7276	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	#332	1	1	yes	characteristic
80	C₂²×C₄₇	#333	1	1	yes	characteristic
81	C₂²×C₄₇	#340	1	1	yes	characteristic
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	#331	1	1	yes	characteristic
86	C₄×C₄₇	#332	1	1	yes	characteristic
87	C₄×C₄₇	#340	1	1	yes	characteristic
88	C₄×C₄₇	#343	1	1	yes	characteristic
89	C₄×C₄₇	#368	1	2	yes	—
90	C₂×C₄⋊C₁₆	—	47	1	no	—
91	#1848	—	47	1	no	—
92	#1564	D₄₇	1	1	yes	characteristic
93	C₂×C₄₇	—	2	1	no	—
94	C₂×C₄₇	—	2	1	no	—
95	C₂×C₄₇	#1296	1	1	yes	characteristic
96	C₂×C₄₇	#1381	1	1	yes	characteristic
97	C₂×C₄₇	#1389	1	1	yes	characteristic
98	C₂²×C₁₆	—	94	1	no	—
99	#431	—	2	1	no	—
100	C₂×#108	—	47	2	no	—
101	C₂×#108	—	94	1	no	—
102	#423	—	94	1	no	—
103	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
104	C₄⋊C₁₆	—	47	2	no	—
105	C₄⋊C₁₆	—	94	1	no	—
106	#335	—	2	1	no	—
107	#334	Dic₄₇	1	2	yes	—
108	C₄₇	#7276	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	94	2	no	—
110	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
111	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂×C₁₆	—	94	2	no	—
114	C₂×C₁₆	—	188	1	no	—
115	C₂×C₁₆	—	188	1	no	—
116	#108	—	188	2	no	—
117	#108	—	188	1	no	—
118	#108	—	188	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	188	1	no	—
121	C₄⋊C₈	—	188	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	Dic₉₄	1	1	yes	characteristic
125	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
128	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	#10625335	1	1	yes	characteristic
130	C₂²×C₄	#10625337	1	1	yes	characteristic
131	M₁₆	—	188	2	no	—
132	M₁₆	—	188	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	188	2	no	—
137	C₂×C₈	#10625325	1	1	yes	characteristic
138	C₂×C₈	#10625325	1	1	yes	characteristic
139	C₂×C₈	D₃₇₆	1	1	yes	characteristic
140	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
141	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
142	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
145	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
146	C₄²	#10625334	1	1	yes	characteristic
147	C₄²	#10625336	1	1	yes	characteristic
148	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
149	C₁₆	—	188	2	no	—
150	C₂³	#13093947	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
154	C₂×C₄	#13093940	1	1	yes	characteristic
155	C₂×C₄	#13093942	1	1	yes	characteristic
156	C₂×C₄	#13093944	1	1	yes	characteristic
157	C₂×C₄	#13093945	1	1	yes	characteristic
158	C₂×C₄	#13093946	1	1	yes	characteristic
159	C₂×C₄	#13093948	1	1	yes	characteristic
160	C₂×C₄	#13093949	1	1	yes	characteristic
161	C₂×C₄	#13093955	1	1	yes	characteristic
162	C₂×C₄	#13093957	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	376	2	no	—
167	C₈	#13093946	1	1	yes	characteristic
168	C₈	#13093949	1	1	yes	characteristic
169	C₈	#13093956	1	1	yes	characteristic
170	C₈	#13093958	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#15688868	1	1	yes	characteristic
174	C₂²	#15688871	1	1	yes	characteristic
175	C₂²	#15688885	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#15688866	1	1	yes	characteristic
180	C₄	#15688870	1	1	yes	characteristic
181	C₄	#15688886	1	1	yes	characteristic
182	C₄	#15688891	1	1	yes	characteristic
183	C₄	#15688952	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#20241272	1	1	yes	characteristic
187	C₂	#20241481	1	1	yes	characteristic
188	C₂	#20241504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093944
commutator	C₂×C₈×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#103
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#7276
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
108. C₄₇	#7276	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
#20243560	2
C₄₇×#1564	2
#7276	47