Subgroups — #26103107

Order 12032.

266 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
6	#15688899	—	2	1	no	—
7	#15688899	—	2	1	no	—
8	#15688844	—	2	1	no	—
9	#15688844	—	2	1	no	—
10	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#346	—	2	1	no	—
13	C₄₇×#346	—	2	1	no	—
14	C₂²×C₄₇⋊C₈	—	2	2	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	C₈	1	4	yes	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	—	2	2	no	—
24	C₂×C₁₆×C₄₇	—	2	2	no	—
25	C₂×C₁₆×C₄₇	C₈	1	4	yes	complemented
26	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
39	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
40	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×C₈	1	2	yes	—
43	C₂×C₄₇⋊C₈	—	2	4	no	—
44	C₂×C₄₇⋊C₈	—	4	4	no	—
45	C₁₆×C₄₇	—	2	4	no	—
46	C₄₇⋊C₁₆	—	2	4	no	—
47	C₂³×C₄₇	#97	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
57	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
58	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
59	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
60	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
61	C₄₇⋊C₈	—	4	4	no	—
62	#7137	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#322	1	1	yes	characteristic
64	C₂²×C₄₇	#322	1	1	yes	characteristic
65	C₂²×C₄₇	#336	1	1	yes	characteristic
66	C₂²×C₄₇	#347	1	4	yes	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	C₈²	1	1	yes	characteristic
70	C₄×C₄₇	#322	1	1	yes	characteristic
71	C₄×C₄₇	#336	1	1	yes	characteristic
72	C₄×C₄₇	#336	1	1	yes	characteristic
73	C₂×#346	—	47	1	no	—
74	C₂×#346	D₄₇	1	1	yes	characteristic
75	C₂×#322	—	47	1	no	—
76	C₂×C₄₇	#1273	1	1	yes	characteristic
77	C₂×C₄₇	#1313	1	2	yes	—
78	C₂×C₄₇	#1313	1	2	yes	—
79	C₂×C₄₇	#1375	1	2	yes	—
80	C₂²×C₁₆	—	47	2	no	—
81	C₂²×C₁₆	Dic₄₇	1	2	yes	—
82	C₂×C₄×C₈	—	47	2	no	—
83	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	#346	—	2	1	no	—
86	#346	—	94	1	no	—
87	#346	—	94	1	no	—
88	#322	—	94	1	no	—
89	#322	—	94	1	no	—
90	C₄₇	#7137	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	94	2	no	—
92	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
93	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	47	4	no	—
98	C₂×C₁₆	—	94	2	no	—
99	C₂×C₁₆	—	94	2	no	—
100	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	complemented
101	C₄×C₈	—	47	4	no	—
102	C₄×C₈	—	94	2	no	—
103	C₄×C₈	Dic₉₄	1	1	yes	characteristic
104	C₄×C₈	D₁₈₈	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₂²×C₄	#10625324	1	2	yes	—
108	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
109	C₂×C₈	—	94	4	no	—
110	C₂×C₈	—	188	4	no	—
111	C₂×C₈	C₈×D₄₇	1	2	yes	—
112	C₂×C₈	#10625324	1	2	yes	—
113	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
114	C₂×C₈	#10625329	1	2	yes	—
115	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
117	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
119	C₄²	C₈×D₄₇	1	2	yes	—
120	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
121	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₁₆	—	94	4	no	—
124	C₂³	#13093934	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
127	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
129	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
130	C₂×C₄	#13093934	1	1	yes	characteristic
131	C₂×C₄	#13093935	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
133	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
134	C₈	—	188	4	no	—
135	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
136	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
137	C₈	Dic₄₇⋊C₈	1	2	yes	—
138	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
139	C₂²	#15688844	1	1	yes	characteristic
140	C₂²	#15688845	1	1	yes	characteristic
141	C₂²	#15688877	1	1	yes	characteristic
142	C₂²	#15688900	1	4	yes	—
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
146	C₄	#15688845	1	1	yes	characteristic
147	C₄	#15688877	1	1	yes	characteristic
148	C₄	#15688878	1	1	yes	characteristic
149	C₂	C₄₇⋊#1273	1	1	yes	characteristic
150	C₂	#20241308	1	2	yes	—
151	C₂	#20241310	1	2	yes	—
152	C₂	#20241467	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₄₇
commutator	C₄×C₄₇	C₈²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle, prime radical	C₂³×C₄₇	#97
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#7137
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
17. C₂×C₄₇⋊C₁₆	C₈	semidirect
25. C₂×C₁₆×C₄₇	C₈	semidirect
90. C₄₇	#7137	semidirect
100. C₂×C₁₆	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×#15688899	2
C₂×#15688844	2
C₂×C₄₇×#346	2
#7137	47