Subgroups — #26132667

Order 12032.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 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	#20241481	C₂	1	4	yes	maximal
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093960	—	2	1	no	—
7	C₂×#13093960	C₂²	1	2	yes	—
8	#15689147	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
11	C₂×#108×C₄₇	—	2	1	no	—
12	C₂×#108×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#10625329	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
19	#13093960	—	2	4	no	—
20	#13093960	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	#108×C₄₇	—	2	4	no	—
27	#108×C₄₇	—	4	2	no	—
28	C₄⋊C₈×C₄₇	—	4	2	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	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₄₇	—	4	2	no	—
36	M₁₆×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
42	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	2	yes	—
46	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	D₈	1	2	yes	—
49	#10625329	—	4	2	no	—
50	#10625329	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₄₇⋊C₁₆	—	2	4	no	—
53	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	#107	1	2	yes	—
56	C₂×C₄×C₄₇	#107	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₄₇	—	8	2	no	—
64	C₈×C₄₇	#106	1	2	yes	—
65	C₈×C₄₇	#106	1	2	yes	—
66	C₈×C₄₇	#117	1	2	yes	—
67	C₈×C₄₇	#125	1	2	yes	—
68	C₄₇⋊C₈	—	8	2	no	—
69	#12891	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
72	C₂²×C₄₇	#487	1	1	yes	characteristic
73	C₂²×C₄₇	#487	1	1	yes	characteristic
74	C₄×C₄₇	—	2	2	no	—
75	C₄×C₄₇	#368	1	4	yes	—
76	C₄×C₄₇	#382	1	1	yes	characteristic
77	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
78	C₄×C₄₇	#486	1	1	yes	characteristic
79	C₄×C₄₇	#486	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	47	1	no	—
81	#1848	—	47	1	no	—
82	#1848	D₄₇	1	1	yes	characteristic
83	#1381	—	47	4	no	—
84	C₂×C₄₇	—	2	1	no	—
85	C₂×C₄₇	—	2	1	no	—
86	C₂×C₄₇	#1834	1	1	yes	characteristic
87	C₂×C₄₇	C₂×#368	1	2	yes	—
88	C₂²×C₁₆	—	47	2	no	—
89	C₂×#108	—	2	1	no	—
90	C₂×#108	—	47	2	no	—
91	C₂×#108	—	94	1	no	—
92	C₂×#108	C₂×D₄₇	1	2	yes	—
93	#423	—	2	1	no	—
94	#423	—	94	1	no	—
95	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
96	C₄×C₁₆	—	47	2	no	—
97	C₄×C₁₆	—	47	2	no	—
98	C₄₇	#12891	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	94	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂×C₁₆	—	47	4	no	—
105	C₂×C₁₆	—	94	2	no	—
106	C₂×C₁₆	—	94	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	94	4	no	—
110	#108	—	188	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	188	2	no	—
113	C₄×C₈	Dic₉₄	1	1	yes	characteristic
114	C₄×C₈	Dic₉₄	1	1	yes	characteristic
115	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
116	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
117	C₂²×C₄	#10625337	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	188	2	no	—
122	M₁₆	—	188	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	188	2	no	—
125	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#10625335	1	2	yes	—
128	C₂×C₈	#10625336	1	2	yes	—
129	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
131	C₂×C₈	#10625352	1	2	yes	—
132	C₂×C₈	#10625354	1	2	yes	—
133	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
134	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
135	C₄²	#10625334	1	2	yes	—
136	C₁₆	—	94	4	no	—
137	C₂³	C₂×#10625337	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#13093957	1	2	yes	—
140	C₂×C₄	#13093957	1	2	yes	—
141	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#13094011	1	1	yes	characteristic
143	C₂×C₄	#13094013	1	1	yes	characteristic
144	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
145	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
146	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	376	2	no	—
149	C₈	#13093955	1	2	yes	—
150	C₈	#13093955	1	2	yes	—
151	C₈	#13093982	1	2	yes	—
152	C₈	#13094015	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#13093957	1	1	yes	characteristic
155	C₂²	#15689473	1	1	yes	characteristic
156	C₂²	#15689475	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#15688952	1	4	yes	—
159	C₄	#15688988	1	1	yes	characteristic
160	C₄	C₂×#13093955	1	1	yes	characteristic
161	C₄	#15689469	1	1	yes	characteristic
162	C₄	#15689471	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#20243515	1	1	yes	characteristic
166	C₂	C₂×#15688952	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#12891
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
98. C₄₇	#12891	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
#20241481	2
C₄₇×#1848	2
#12891	47