Subgroups — #26226717

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252026	C₂	1	2	yes	maximal
2	#20244164	C₂	1	8	yes	maximal, complemented
3	#20243430	C₂	1	2	yes	maximal
4	#20243426	C₂	1	2	yes	maximal
5	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
6	#15689184	—	2	8	no	—
7	C₂×#13093960	C₂²	1	4	yes	—
8	C₂×#13093960	C₂²	1	4	yes	—
9	#15689147	C₂²	1	4	yes	—
10	#15689087	C₂²	1	4	yes	—
11	C₄×#10625329	C₂²	1	4	yes	—
12	#15688888	C₂²	1	4	yes	—
13	#15688888	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#424	C₄	1	8	yes	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
19	C₂×#10625329	C₂³	1	8	yes	—
20	#13093960	—	4	4	no	—
21	#13093960	—	4	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	#13093935	—	2	4	no	—
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	—	4	2	no	—
28	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₄₇	—	4	2	no	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
33	#117×C₄₇	C₂×C₄	1	4	yes	—
34	#117×C₄₇	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
38	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	—
39	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₄₇	—	4	4	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	4	4	no	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
53	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
54	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
56	C₄²×C₄₇	C₂×D₄	1	4	yes	—
57	#10625329	—	2	8	no	—
58	#10625329	—	2	8	no	—
59	C₂×C₄₇⋊C₈	—	2	8	no	—
60	C₂³×C₄₇	—	2	2	no	—
61	C₂³×C₄₇	—	4	2	no	—
62	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
74	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
75	C₈×C₄₇	—	2	4	no	—
76	C₈×C₄₇	—	2	4	no	—
77	C₄₇⋊C₈	—	4	8	no	—
78	#21971	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	—	4	4	no	—
83	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄₇	#544	1	2	yes	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	2	8	no	—
87	C₄×C₄₇	#518	1	1	yes	characteristic
88	C₄×C₄₇	#519	1	1	yes	characteristic
89	C₄×C₄₇	#537	1	2	yes	—
90	#2921	—	47	2	no	—
91	C₂×#424	D₄₇	1	1	yes	characteristic
92	#1950	—	47	8	no	—
93	#1811	—	47	2	no	—
94	#1810	—	47	2	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	—	4	2	no	—
97	C₂×C₄₇	#2316	1	1	yes	characteristic
98	C₂×C₄₇	#2916	1	2	yes	—
99	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
100	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
102	#433	—	94	8	no	—
103	C₂×#108	—	47	4	no	—
104	C₂×#108	—	47	4	no	—
105	#424	Dic₄₇	1	8	yes	—
106	#423	—	47	4	no	—
107	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
108	#407	—	47	4	no	—
109	C₄×M₁₆	—	47	4	no	—
110	#341	—	47	4	no	—
111	#341	—	47	4	no	—
112	C₄₇	#21971	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×M₁₆	—	4	2	no	—
115	C₂×M₁₆	—	47	8	no	—
116	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
117	C₂²×C₈	—	4	2	no	—
118	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
119	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
120	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
121	#117	D₁₈₈	1	4	yes	—
122	#117	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
124	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
125	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
126	#108	—	188	4	no	—
127	#108	—	188	4	no	—
128	C₄⋊C₈	—	94	8	no	—
129	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	—
130	C₂²⋊C₈	—	94	8	no	—
131	#97	—	94	4	no	—
132	C₄×C₈	—	94	4	no	—
133	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
138	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
140	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
141	M₁₆	—	4	4	no	—
142	M₁₆	—	94	8	no	—
143	M₁₆	—	94	8	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	94	8	no	—
148	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
149	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
150	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
151	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
152	C₄²	C₂×D₁₈₈	1	4	yes	—
153	C₄²	C₂²×Dic₄₇	1	4	yes	—
154	C₂³	—	2	2	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
157	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
165	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
167	C₂×C₄	#13094069	1	2	yes	—
168	C₂×C₄	#13094072	1	2	yes	—
169	C₈	—	2	4	no	—
170	C₈	—	2	4	no	—
171	C₈	—	188	8	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
177	C₂²	#15689754	1	2	yes	—
178	C₄	—	2	8	no	—
179	C₄	—	2	2	no	—
180	C₄	#15689598	1	1	yes	characteristic
181	C₄	#15689601	1	1	yes	characteristic
182	C₄	#15689704	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	4	2	no	—
185	C₂	#20246006	1	1	yes	characteristic
186	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#424	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#21971
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
2. #20244164	C₂	semidirect
112. C₄₇	#21971	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20252026	2
#20244164	2
#20243430	2
#20243426	2
C₂×C₄₇×#424	2
#21971	47