Subgroups — #26103194

Order 12032.

282 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×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₄×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
8	#15688876	—	2	1	no	—
9	#15688876	C₄	1	2	yes	complemented
10	C₂×#107×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
13	C₄₇×#340	—	2	1	no	—
14	C₂²×C₄₇⋊C₈	—	4	1	no	—
15	C₂×C₄₇⋊C₁₆	—	2	4	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₂×C₄²×C₄₇	—	4	1	no	—
23	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
24	#107×C₄₇	—	2	2	no	—
25	#107×C₄₇	C₈	1	4	yes	complemented
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	Q₈	1	1	yes	characteristic
30	#95×C₄₇	—	4	1	no	—
31	C₂²×C₄×C₄₇	—	4	2	no	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄⋊C₄×C₄₇	—	4	2	no	—
44	C₄²×C₄₇	—	8	1	no	—
45	C₄²×C₄₇	—	8	1	no	—
46	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
50	C₂×C₄₇⋊C₈	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	8	1	no	—
52	C₂×C₄₇⋊C₈	—	8	1	no	—
53	C₄₇⋊C₁₆	—	2	4	no	—
54	C₂³×C₄₇	#102	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	4	4	no	—
57	C₂×C₄×C₄₇	—	8	2	no	—
58	C₂×C₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#104	1	2	yes	—
65	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
67	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
71	C₈×C₄₇	—	2	1	no	—
72	C₈×C₄₇	—	2	1	no	—
73	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
74	C₈×C₄₇	#97	1	1	yes	characteristic
75	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
76	C₄₇⋊C₈	—	8	2	no	—
77	#7171	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
79	C₂²×C₄₇	#340	1	1	yes	characteristic
80	C₂²×C₄₇	#357	1	1	yes	characteristic
81	C₂²×C₄₇	#358	1	1	yes	characteristic
82	C₂²×C₄₇	#359	1	2	yes	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	8	4	no	—
86	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
87	C₄×C₄₇	#336	1	1	yes	characteristic
88	C₄×C₄₇	#339	1	1	yes	characteristic
89	C₄×C₄₇	#357	1	1	yes	characteristic
90	C₄×C₄₇	#358	1	1	yes	characteristic
91	C₄×C₄₇	#366	1	1	yes	characteristic
92	C₄×C₄₇	#367	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	47	1	no	—
94	C₄×#107	D₄₇	1	1	yes	characteristic
95	C₂×#335	—	47	1	no	—
96	C₂×C₄₇	#1279	1	1	yes	characteristic
97	C₂×C₄₇	#1331	1	2	yes	—
98	C₂×C₄₇	#1332	1	2	yes	—
99	C₂×C₄₇	#1380	1	1	yes	characteristic
100	C₂×C₄₇	#1382	1	1	yes	characteristic
101	C₂²×C₁₆	—	94	1	no	—
102	C₂×#107	Dic₄₇	1	2	yes	—
103	C₂×C₄⋊C₈	—	94	1	no	—
104	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₁₆	—	47	2	no	—
107	C₄×C₁₆	—	94	1	no	—
108	#340	—	2	1	no	—
109	#335	—	47	2	no	—
110	#335	—	94	1	no	—
111	C₄₇	#7171	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	188	1	no	—
113	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₂×C₁₆	—	94	4	no	—
119	C₂×C₁₆	—	94	2	no	—
120	#107	—	2	2	no	—
121	#107	C₄₇⋊C₈	1	4	yes	complemented
122	C₄⋊C₈	—	94	2	no	—
123	C₄⋊C₈	—	188	1	no	—
124	C₄×C₈	Dic₉₄	1	1	yes	characteristic
125	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
127	C₄×C₈	D₁₈₈	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#10625334	1	1	yes	characteristic
131	C₂²×C₄	#10625335	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	376	1	no	—
135	C₂×C₈	—	376	1	no	—
136	C₂×C₈	—	188	2	no	—
137	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
138	C₂×C₈	#10625329	1	2	yes	—
139	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
140	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
142	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
148	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
149	C₄²	#10625336	1	1	yes	characteristic
150	C₄²	#10625337	1	1	yes	characteristic
151	C₁₆	—	94	4	no	—
152	C₂³	#13093945	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
159	C₂×C₄	#13093944	1	1	yes	characteristic
160	C₂×C₄	#13093947	1	1	yes	characteristic
161	C₂×C₄	#13093948	1	1	yes	characteristic
162	C₂×C₄	#13093951	1	2	yes	—
163	C₂×C₄	#13093955	1	1	yes	characteristic
164	C₂×C₄	#13093957	1	1	yes	characteristic
165	C₂×C₄	#13093965	1	1	yes	characteristic
166	C₂×C₄	#13093967	1	1	yes	characteristic
167	C₂×C₄	#13093968	1	1	yes	characteristic
168	C₂×C₄	#13093970	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	376	2	no	—
172	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
173	C₈	#13093935	1	1	yes	characteristic
174	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
175	C₂²	D₄⋊(C₄₇⋊C₈)	1	2	yes	—
176	C₂²	#15688885	1	1	yes	characteristic
177	C₂²	#15688924	1	1	yes	characteristic
178	C₂²	#15688927	1	1	yes	characteristic
179	C₂²	#15688930	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
184	C₄	#15688878	1	1	yes	characteristic
185	C₄	#15688882	1	1	yes	characteristic
186	C₄	#15688923	1	1	yes	characteristic
187	C₄	#15688926	1	1	yes	characteristic
188	C₄	#15688948	1	1	yes	characteristic
189	C₄	#15688950	1	1	yes	characteristic
190	C₂	#20241229	1	1	yes	characteristic
191	C₂	#20241349	1	2	yes	—
192	C₂	#20241352	1	2	yes	—
193	C₂	#20241479	1	1	yes	characteristic
194	C₂	#20241483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	#7171
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×C₄₇⋊C₁₆	C₄	semidirect
9. #15688876	C₄	semidirect
25. #107×C₄₇	C₈	semidirect
111. C₄₇	#7171	semidirect
121. #107	C₄₇⋊C₈	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
C₄×#107×C₄₇	2
#7171	47