Subgroups — #26422087

Order 12032.

1626 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251766	C₂	1	3	yes	maximal
2	#20248754	C₂	1	6	yes	maximal
3	#20247996	C₂	1	12	yes	maximal
4	#20246351	C₂	1	6	yes	maximal
5	C₄×#13094024	C₂	1	3	yes	maximal
6	C₄²×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094024	C₂²	1	3	yes	—
8	C₂×C₄×Dic₉₄	C₂²	1	6	yes	—
9	C₄₇⋊#401	C₂²	1	8	yes	—
10	#15689061	C₂²	1	12	yes	—
11	#15689042	C₂²	1	4	yes	—
12	#15689029	C₂²	1	12	yes	—
13	#15689029	C₂²	1	12	yes	—
14	#15689029	C₂²	1	6	yes	—
15	#15689021	C₂²	1	12	yes	—
16	#15689004	C₂²	1	24	yes	—
17	#15688993	C₂²	1	1	yes	characteristic
18	#15688990	C₂²	1	12	yes	—
19	#15688989	C₂²	1	6	yes	—
20	#15688989	C₂²	1	6	yes	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	12	yes	—
22	#15688968	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₄₇	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₂²	1	6	yes	—
26	C₄³×C₄₇	C₂²	1	3	yes	—
27	C₂²×Dic₉₄	C₂³	1	6	yes	—
28	#13094024	—	2	12	no	—
29	C₄×Dic₉₄	—	2	24	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
32	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
34	C₄₇⋊#95	C₂³	1	12	yes	—
35	C₄₇⋊#95	C₂³	1	24	yes	—
36	C₄₇⋊#95	C₂³	1	24	yes	—
37	C₄₇⋊#95	C₂³	1	12	yes	—
38	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₄₇	—	2	12	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#95×C₄₇	C₂³	1	1	yes	characteristic
48	#95×C₄₇	C₂³	1	3	yes	—
49	#95×C₄₇	C₂³	1	2	yes	—
50	C₂×Q₈×C₄₇	—	2	6	no	—
51	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
52	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
55	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
56	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
57	C₂×Dic₉₄	—	2	12	no	—
58	C₂×Dic₉₄	—	2	12	no	—
59	C₂×Dic₉₄	—	2	12	no	—
60	C₄⋊C₄×C₄₇	—	2	12	no	—
61	C₄⋊C₄×C₄₇	—	2	6	no	—
62	C₄²×C₄₇	—	2	12	no	—
63	C₄²×C₄₇	—	2	12	no	—
64	C₄²×C₄₇	—	2	6	no	—
65	C₄²×C₄₇	#41	1	12	yes	—
66	C₄²×C₄₇	#41	1	4	yes	—
67	C₄⋊Dic₄₇	—	2	12	no	—
68	C₄⋊Dic₄₇	—	2	24	no	—
69	Dic₄₇⋊C₄	—	2	24	no	—
70	C₄×Dic₄₇	—	2	24	no	—
71	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
72	Q₈×C₄₇	—	2	8	no	—
73	C₂×C₄×C₄₇	—	2	6	no	—
74	C₂×C₄×C₄₇	—	2	6	no	—
75	C₂×C₄×C₄₇	—	2	6	no	—
76	C₂×C₄×C₄₇	—	2	6	no	—
77	C₂×C₄×C₄₇	—	2	6	no	—
78	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
79	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
80	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
81	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
82	C₂×C₄×C₄₇	2₋¹⁺⁴	1	6	yes	—
83	C₂×Dic₄₇	—	2	12	no	—
84	C₂×Dic₄₇	—	2	12	no	—
85	C₂×Dic₄₇	—	2	12	no	—
86	C₂×Dic₄₇	—	2	12	no	—
87	Dic₉₄	—	4	24	no	—
88	#36608	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	C₂²×#41	1	3	yes	—
90	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	3	yes	—
92	C₄×C₄₇	—	2	6	no	—
93	C₄×C₄₇	—	2	6	no	—
94	C₄×C₄₇	—	4	6	no	—
95	C₄×C₄₇	#556	1	12	yes	—
96	Dic₄₇	—	4	4	no	—
97	Dic₄₇	—	4	12	no	—
98	#2862	—	47	3	no	—
99	#2581	—	47	6	no	—
100	#2514	—	47	12	no	—
101	#2356	—	47	6	no	—
102	C₄×C₄⋊Q₈	—	47	3	no	—
103	C₄²×Q₈	D₄₇	1	1	yes	characteristic
104	C₂×C₄₇	#3444	1	1	yes	characteristic
105	C₂×C₄₇	C₂×#556	1	3	yes	—
106	C₂×C₄₇	#3509	1	3	yes	—
107	C₂×C₄⋊Q₈	—	47	3	no	—
108	C₂×C₄×Q₈	—	47	6	no	—
109	C₂×C₄×Q₈	C₂×D₄₇	1	3	yes	—
110	#401	—	47	8	no	—
111	#400	—	47	12	no	—
112	#395	—	47	4	no	—
113	#391	—	47	6	no	—
114	#391	—	47	12	no	—
115	#391	—	47	12	no	—
116	#389	—	47	12	no	—
117	#387	—	47	24	no	—
118	#384	—	47	1	no	—
119	#383	—	47	12	no	—
120	#382	—	47	6	no	—
121	#382	—	47	6	no	—
122	C₄×C₄⋊C₄	—	47	12	no	—
123	C₄×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
124	C₄×C₄⋊C₄	C₂×D₄₇	1	6	yes	—
125	#376	—	47	4	no	—
126	C₄³	C₂×D₄₇	1	3	yes	—
127	C₄₇	#36608	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	47	6	no	—
129	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	94	12	no	—
131	C₄×Q₈	—	2	12	no	—
132	C₄×Q₈	—	94	24	no	—
133	C₂×C₄⋊C₄	—	47	12	no	—
134	C₂×C₄⋊C₄	—	47	6	no	—
135	C₂×C₄⋊C₄	—	47	12	no	—
136	C₂×C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
137	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
138	C₂×C₄²	—	47	12	no	—
139	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
140	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
141	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
142	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
143	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
144	#95	—	47	12	no	—
145	#95	—	47	24	no	—
146	#95	—	47	24	no	—
147	#95	—	47	12	no	—
148	#95	C₂²×D₄₇	1	1	yes	characteristic
149	#95	C₂²×D₄₇	1	3	yes	—
150	#95	C₂²×D₄₇	1	2	yes	—
151	C₂×Q₈	—	2	6	no	—
152	C₂×Q₈	—	94	12	no	—
153	C₂×Q₈	—	94	12	no	—
154	C₂×Q₈	—	94	12	no	—
155	C₂²×C₄	—	47	12	no	—
156	C₂²×C₄	—	47	4	no	—
157	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
158	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
159	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
160	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	—	2	6	no	—
163	C₄⋊C₄	—	94	24	no	—
164	C₄⋊C₄	—	94	12	no	—
165	C₄⋊C₄	—	94	24	no	—
166	C₄²	—	2	12	no	—
167	C₄²	—	2	12	no	—
168	C₄²	—	2	6	no	—
169	C₄²	—	94	24	no	—
170	C₄²	#10625350	1	12	yes	—
171	C₄²	#10625354	1	4	yes	—
172	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	188	24	no	—
175	C₂×C₄	—	2	6	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	6	no	—
180	C₂×C₄	—	94	12	no	—
181	C₂×C₄	—	94	12	no	—
182	C₂×C₄	—	94	12	no	—
183	C₂×C₄	—	94	12	no	—
184	C₂×C₄	C₂×#10625350	1	6	yes	—
185	C₂×C₄	C₂×#10625350	1	6	yes	—
186	C₂×C₄	C₂×#10625350	1	6	yes	—
187	C₂×C₄	C₂×#10625354	1	6	yes	—
188	C₂×C₄	#13094071	1	6	yes	—
189	C₂²	C₂²×#10625350	1	3	yes	—
190	C₂²	C₂²×#10625354	1	1	yes	characteristic
191	C₂²	C₂×#13094071	1	3	yes	—
192	C₄	—	2	6	no	—
193	C₄	—	2	6	no	—
194	C₄	—	4	6	no	—
195	C₄	—	188	12	no	—
196	C₄	—	188	4	no	—
197	C₄	#15689846	1	12	yes	—
198	C₂	#20256684	1	1	yes	characteristic
199	C₂	C₂×#15689846	1	3	yes	—
200	C₂	#20257274	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator, socle, prime radical, cosocle	C₂³×C₄₇	C₂⁵
Fitting	C₄²×Q₈×C₄₇	C₂
hypercenter	C₄²×Q₈	D₄₇
hypocenter	C₄₇	#36608
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
127. C₄₇	#36608	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20248754	2
#20247996	2
#20246351	2
C₄×#13094024	2
C₄²×Q₈×C₄₇	2
#36608	47