Subgroups — #26225149

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252323	C₂	1	2	yes	maximal
2	#20244370	C₂	1	2	yes	maximal
3	#20244142	C₂	1	4	yes	maximal
4	#20243563	C₂	1	1	yes	maximal, characteristic
5	#20243269	C₂	1	4	yes	maximal
6	#20243076	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#3162	C₂	1	1	yes	maximal, characteristic
8	#15689254	—	2	4	no	—
9	Q₈×C₄₇⋊C₈	—	2	4	no	—
10	C₂×#13093960	C₂²	1	4	yes	—
11	#15689150	C₂²	1	4	yes	—
12	#15689147	C₂²	1	2	yes	—
13	#15689079	C₂²	1	4	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	#15688969	C₂²	1	1	yes	characteristic
18	#15688882	C₂²	1	8	yes	—
19	C₄₇×#533	C₂²	1	2	yes	—
20	C₄₇×#501	C₄	1	8	yes	complemented
21	C₄₇×#427	C₂²	1	4	yes	—
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
27	#13093960	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	#13093935	—	2	4	no	—
31	#13093934	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
34	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
35	C₈×Dic₄₇	Q₈	1	8	yes	complemented
36	C₄₇⋊#95	—	2	2	no	—
37	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
39	#125×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²⋊Q₈×C₄₇	—	2	4	no	—
41	C₄×Q₈×C₄₇	—	2	4	no	—
42	#117×C₄₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#107×C₄₇	C₂×C₄	1	8	yes	—
45	#106×C₄₇	C₂×C₄	1	8	yes	—
46	#97×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×Q₈×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	#41	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
53	C₂×C₈×C₄₇	#41	1	4	yes	—
54	C₄⋊C₄×C₄₇	—	2	8	no	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
57	C₂²⋊C₄×C₄₇	—	2	4	no	—
58	C₄²×C₄₇	—	2	4	no	—
59	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₄₇	C₂×D₄	1	2	yes	—
61	C₄×Dic₄₇	—	2	2	no	—
62	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
63	#10625329	—	2	8	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	4	no	—
66	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
70	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
71	Q₈×C₄₇	—	4	8	no	—
72	C₂×C₄×C₄₇	—	2	8	no	—
73	C₂×C₄×C₄₇	—	4	2	no	—
74	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
79	C₂×Dic₄₇	—	2	4	no	—
80	C₂×Dic₄₇	—	4	4	no	—
81	C₈×C₄₇	C₄×Q₈	1	8	yes	—
82	C₄₇⋊C₈	—	2	2	no	—
83	C₄₇⋊C₈	—	4	4	no	—
84	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
85	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
86	#21735	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂²×C₄₇	#533	1	2	yes	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	4	8	no	—
92	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
93	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₄₇	#532	1	1	yes	characteristic
95	Dic₄₇	—	4	4	no	—
96	#3162	D₄₇	1	1	yes	characteristic
97	#2960	—	47	2	no	—
98	#1995	—	47	2	no	—
99	#1946	—	47	4	no	—
100	#1849	—	47	1	no	—
101	#1772	—	47	4	no	—
102	#1726	—	47	1	no	—
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄₇	#2955	1	2	yes	—
106	#533	C₂×D₄₇	1	2	yes	—
107	#501	Dic₄₇	1	8	yes	complemented
108	#446	—	94	4	no	—
109	C₈×Q₈	—	94	4	no	—
110	C₂×#108	—	47	4	no	—
111	#427	C₂×D₄₇	1	4	yes	—
112	#424	—	47	4	no	—
113	#423	—	47	2	no	—
114	#405	—	47	4	no	—
115	C₂×#97	—	47	2	no	—
116	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₈	—	47	2	no	—
118	C₂×C₄×C₈	—	47	1	no	—
119	#376	—	47	1	no	—
120	#339	—	47	8	no	—
121	C₄₇	#21735	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	47	4	no	—
123	C₂²×C₈	—	47	1	no	—
124	C₂²×C₈	—	47	1	no	—
125	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
126	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
127	#125	C₂×Dic₄₇	1	4	yes	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	#117	C₂²×D₄₇	1	4	yes	—
131	C₂×C₄²	—	47	2	no	—
132	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
133	#108	—	94	8	no	—
134	#107	C₂×Dic₄₇	1	8	yes	—
135	#106	C₂×Dic₄₇	1	8	yes	—
136	C₄⋊C₈	—	94	8	no	—
137	C₄⋊C₈	—	94	4	no	—
138	#97	—	94	4	no	—
139	#97	—	94	4	no	—
140	#97	C₂×Dic₄₇	1	4	yes	—
141	C₄×C₈	—	47	8	no	—
142	C₄×C₈	—	47	2	no	—
143	C₄×C₈	—	47	2	no	—
144	C₄×C₈	—	94	4	no	—
145	#95	—	94	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	—	94	2	no	—
148	C₂²×C₄	#10625352	1	2	yes	—
149	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
150	M₁₆	—	94	8	no	—
151	C₂×C₈	—	47	1	no	—
152	C₂×C₈	—	47	1	no	—
153	C₂×C₈	—	47	4	no	—
154	C₂×C₈	—	47	2	no	—
155	C₂×C₈	—	94	2	no	—
156	C₂×C₈	—	94	4	no	—
157	C₂×C₈	Q₈×D₄₇	1	4	yes	—
158	C₂×C₈	#10625354	1	4	yes	—
159	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	47	4	no	—
166	C₄²	—	94	2	no	—
167	C₄²	D₄×D₄₇	1	2	yes	—
168	C₄²	C₂²×Dic₄₇	1	2	yes	—
169	C₂³	C₂×#10625352	1	1	yes	characteristic
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	188	4	no	—
175	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
176	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
179	C₂×C₄	C₂×#10625354	1	2	yes	—
180	C₈	—	47	2	no	—
181	C₈	—	47	2	no	—
182	C₈	—	94	2	no	—
183	C₈	—	188	4	no	—
184	C₈	Q₈×Dic₄₇	1	8	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
187	C₂²	#15689678	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	8	no	—
190	C₄	—	188	4	no	—
191	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
192	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
193	C₄	#15689674	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
196	C₂	#20252291	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#3162	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3162	D₄₇
hypocenter	C₄₇	#21735
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

Complements

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

N	H ≅ G/N	type
20. C₄₇×#501	C₄	semidirect
35. C₈×Dic₄₇	Q₈	semidirect
107. #501	Dic₄₇	semidirect
121. C₄₇	#21735	semidirect

Maximal subgroups

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

subgroup	index
#20252323	2
#20244370	2
#20244142	2
#20243563	2
#20243269	2
#20243076	2
C₄₇×#3162	2
#21735	47