Subgroups — #26126098

Order 12032.

326 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241371	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689680	C₄	1	1	yes	characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	#15689253	—	2	1	no	—
8	#15689253	C₂²	1	2	yes	—
9	C₈×Dic₉₄	—	2	1	no	—
10	C₈×Dic₉₄	C₄	1	2	yes	—
11	#15689149	—	2	1	no	—
12	#15689147	C₄	1	1	yes	characteristic
13	#15689078	—	2	1	no	—
14	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂×M₃₂×C₄₇	—	2	1	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#346	C₂²	1	2	yes	—
19	C₄×C₁₆×C₄₇	—	2	1	no	—
20	C₂×#10625329	—	2	1	no	—
21	#13094024	C₈	1	1	yes	characteristic
22	#13094024	C₈	1	1	yes	characteristic
23	#13094016	—	2	1	no	—
24	#13094003	—	2	1	no	—
25	#13094003	C₈	1	2	yes	—
26	C₄×Dic₉₄	—	2	1	no	—
27	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
28	#13093983	—	2	1	no	—
29	C₂×C₄₇⋊C₁₆	—	2	2	no	—
30	C₂×C₄₇⋊C₁₆	—	4	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
33	Dic₄₇⋊C₈	—	2	2	no	—
34	Dic₄₇⋊C₈	—	2	2	no	—
35	#13093934	—	2	2	no	—
36	C₈×Dic₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₄₇	—	2	4	no	—
41	C₂×C₁₆×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
49	C₂×Dic₉₄	—	2	1	no	—
50	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
55	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
61	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²⋊Dic₄₇	—	2	2	no	—
63	C₄⋊Dic₄₇	—	2	1	no	—
64	C₄⋊Dic₄₇	—	2	1	no	—
65	C₄⋊Dic₄₇	—	2	1	no	—
66	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
67	Dic₄₇⋊C₄	—	2	2	no	—
68	Dic₄₇⋊C₄	—	2	2	no	—
69	C₄×Dic₄₇	—	2	2	no	—
70	#10625329	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₁₆×C₄₇	—	2	4	no	—
73	C₄₇⋊C₁₆	—	4	2	no	—
74	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	4	1	no	—
77	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#103	1	2	yes	—
81	C₂×C₄×C₄₇	#104	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	2	2	no	—
87	Dic₉₄	—	4	2	no	—
88	Dic₉₄	—	4	2	no	—
89	C₈×C₄₇	—	2	1	no	—
90	C₈×C₄₇	—	2	1	no	—
91	C₈×C₄₇	#103	1	2	yes	—
92	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₄₇⋊C₈	—	4	2	no	—
95	#11610	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	—	2	1	no	—
98	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	—	2	1	no	—
103	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
104	C₄×C₄₇	#350	1	2	yes	—
105	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
106	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
107	Dic₄₇	—	4	2	no	—
108	Dic₄₇	—	4	2	no	—
109	#2960	—	47	1	no	—
110	C₂×C₄⋊C₁₆	—	47	1	no	—
111	C₄×M₃₂	D₄₇	1	1	yes	characteristic
112	#1338	—	47	4	no	—
113	C₂×C₄₇	—	2	2	no	—
114	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₄₇	#2115	1	1	yes	characteristic
116	C₂×C₄₇	C₂×#350	1	1	yes	characteristic
117	#533	—	47	1	no	—
118	C₂×M₃₂	—	2	1	no	—
119	C₂²×C₁₆	—	94	1	no	—
120	#446	—	47	2	no	—
121	#446	—	94	1	no	—
122	C₈×Q₈	—	47	2	no	—
123	C₈×Q₈	—	94	1	no	—
124	#424	—	94	1	no	—
125	#423	—	47	1	no	—
126	#405	—	94	1	no	—
127	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₄⋊C₁₆	—	47	2	no	—
129	C₄⋊C₁₆	—	94	1	no	—
130	#346	C₂×D₄₇	1	2	yes	—
131	C₄×C₁₆	—	2	1	no	—
132	C₄₇	#11610	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	94	1	no	—
134	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₄⋊Q₈	—	47	1	no	—
137	C₄⋊Q₈	—	47	1	no	—
138	#125	—	94	1	no	—
139	C₂²⋊Q₈	—	47	2	no	—
140	C₂²⋊Q₈	—	94	1	no	—
141	C₄×Q₈	—	47	2	no	—
142	C₄×Q₈	—	94	1	no	—
143	#117	—	94	1	no	—
144	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
145	M₃₂	—	2	4	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	94	2	no	—
148	C₂×C₁₆	—	188	2	no	—
149	C₄⋊C₈	—	47	2	no	—
150	C₄⋊C₈	—	94	2	no	—
151	C₄⋊C₈	—	94	2	no	—
152	C₄⋊C₈	—	94	1	no	—
153	#97	—	94	2	no	—
154	C₄×C₈	—	94	2	no	—
155	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
156	C₄×C₈	D₁₈₈	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×Q₈	—	47	2	no	—
160	C₂×Q₈	—	94	1	no	—
161	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
162	C₂²×C₄	#10625324	1	1	yes	characteristic
163	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
164	M₁₆	—	188	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	94	2	no	—
167	C₂×C₈	#10625325	1	2	yes	—
168	C₂×C₈	Dic₁₈₈	1	2	yes	—
169	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
170	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
173	C₄⋊C₄	—	47	2	no	—
174	C₄⋊C₄	—	94	2	no	—
175	C₄⋊C₄	—	94	1	no	—
176	C₄⋊C₄	—	94	1	no	—
177	C₄⋊C₄	—	94	1	no	—
178	C₄⋊C₄	—	94	2	no	—
179	C₂²⋊C₄	—	94	2	no	—
180	C₄²	—	94	2	no	—
181	C₄²	C₈×D₄₇	1	1	yes	characteristic
182	C₄²	#10625324	1	1	yes	characteristic
183	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	188	2	no	—
187	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
188	Q₈	—	188	2	no	—
189	Q₈	—	188	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	94	2	no	—
193	C₂×C₄	—	94	2	no	—
194	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
195	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
196	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
197	C₂×C₄	#13093949	1	2	yes	—
198	C₂×C₄	#13093952	1	2	yes	—
199	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
201	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	188	2	no	—
205	C₈	#13093949	1	2	yes	—
206	C₈	C₂×#10625325	1	1	yes	characteristic
207	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#15688855	1	2	yes	—
211	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	188	2	no	—
216	C₄	—	188	2	no	—
217	C₄	#15688855	1	2	yes	—
218	C₄	#15688908	1	2	yes	—
219	C₄	C₂×#13093949	1	1	yes	characteristic
220	C₄	C₂×#13093952	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#15688855	1	1	yes	characteristic
223	C₂	#20245008	1	1	yes	characteristic
224	C₂	C₂×#15688908	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#11610
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20241371	C₂	semidirect
132. C₄₇	#11610	semidirect

Maximal subgroups

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

subgroup	index
#20252322	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241371	2
C₄×M₃₂×C₄₇	2
#11610	47