Subgroups — #26127922

Order 12032.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 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	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20241461	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
6	#15689147	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15688876	—	2	1	no	—
11	#15688876	C₂²	1	2	yes	—
12	C₄₇×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#335	—	2	1	no	—
15	C₄₇×#335	C₂²	1	2	yes	—
16	C₂×#10625329	—	4	1	no	—
17	C₂×C₄₇⋊C₁₆	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
22	C₂×M₁₆×C₄₇	—	4	1	no	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	—	2	2	no	—
27	C₄⋊C₈×C₄₇	—	4	1	no	—
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₄₇	—	8	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₇	D₈	1	1	yes	characteristic
45	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	#10625329	—	8	2	no	—
48	C₂×C₄₇⋊C₈	—	4	2	no	—
49	C₄₇⋊C₁₆	—	2	2	no	—
50	C₄₇⋊C₁₆	M₁₆	1	4	yes	complemented
51	C₂³×C₄₇	#107	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
58	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
59	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₄₇	—	2	1	no	—
64	C₈×C₄₇	—	2	1	no	—
65	C₈×C₄₇	—	8	2	no	—
66	C₈×C₄₇	#108	1	2	yes	—
67	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
68	C₄₇⋊C₈	—	8	2	no	—
69	#12023	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	—	2	1	no	—
71	C₂²×C₄₇	—	2	1	no	—
72	C₂²×C₄₇	#367	1	2	yes	—
73	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	—	2	1	no	—
76	C₄×C₄₇	#367	1	2	yes	—
77	C₄×C₄₇	#368	1	2	yes	—
78	C₄×C₄₇	#423	1	1	yes	characteristic
79	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
80	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
81	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
82	C₂×C₄×C₁₆	—	47	1	no	—
83	#1567	—	47	1	no	—
84	#1567	D₄₇	1	1	yes	characteristic
85	#1372	—	47	4	no	—
86	C₂×C₄₇	—	2	2	no	—
87	C₂×C₄₇	#1567	1	1	yes	characteristic
88	C₂×C₄₇	C₂×#367	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#368	1	1	yes	characteristic
90	C₂²×C₁₆	—	47	2	no	—
91	#423	—	2	1	no	—
92	#423	—	94	1	no	—
93	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
94	C₄×C₁₆	—	47	1	no	—
95	C₄×C₁₆	—	47	1	no	—
96	C₄×C₁₆	—	47	2	no	—
97	#335	—	2	1	no	—
98	#335	—	47	2	no	—
99	#335	—	94	1	no	—
100	#335	C₂×D₄₇	1	2	yes	—
101	C₄₇	#12023	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	188	1	no	—
104	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
105	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₂×C₁₆	—	47	2	no	—
108	C₂×C₁₆	—	47	2	no	—
109	C₂×C₁₆	—	94	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	94	2	no	—
113	C₄⋊C₈	—	188	1	no	—
114	C₄×C₈	Dic₉₄	1	1	yes	characteristic
115	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	#10625334	1	1	yes	characteristic
120	C₂²×C₄	#10625337	1	1	yes	characteristic
121	M₁₆	—	8	2	no	—
122	M₁₆	—	376	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	188	2	no	—
127	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
131	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
132	C₄²	#10625334	1	1	yes	characteristic
133	C₄²	#10625337	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
135	C₁₆	—	47	4	no	—
136	C₁₆	—	94	2	no	—
137	C₂³	#13093957	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#13093957	1	1	yes	characteristic
142	C₂×C₄	#13093957	1	1	yes	characteristic
143	C₂×C₄	#13093957	1	1	yes	characteristic
144	C₂×C₄	#13093967	1	2	yes	—
145	C₂×C₄	#13093968	1	2	yes	—
146	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
148	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	376	2	no	—
153	C₈	#13093959	1	2	yes	—
154	C₈	M₁₆×D₄₇	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#15688950	1	2	yes	—
158	C₂²	C₂×#13093957	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#15688950	1	2	yes	—
162	C₄	#15688952	1	2	yes	—
163	C₄	#15689146	1	1	yes	characteristic
164	C₄	C₂×#13093959	1	1	yes	characteristic
165	C₄	C₂×#13093967	1	1	yes	characteristic
166	C₄	C₂×#13093968	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#20242146	1	1	yes	characteristic
169	C₂	C₂×#15688950	1	1	yes	characteristic
170	C₂	C₂×#15688952	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1567	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	#107
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1567	D₄₇
hypocenter	C₄₇	#12023
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #20241461	C₂	semidirect
50. C₄₇⋊C₁₆	M₁₆	semidirect
101. C₄₇	#12023	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
#20242147	2
#20241461	2
C₄₇×#1567	2
#12023	47