Subgroups — #26214968

Order 12032.

1354 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	#20243166	C₂	1	8	yes	maximal, complemented
3	#20243150	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	16	yes	complemented
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×#13093935	C₂²	1	8	yes	complemented
10	#15688878	C₂²	1	8	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
14	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
15	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
16	C₂×#95×C₄₇	C₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	4	8	no	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
19	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
20	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
21	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
22	#13093935	—	4	8	no	—
23	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	4	4	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₄₇	—	4	4	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
36	#95×C₄₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₄₇	—	4	2	no	—
38	C₂⁴×C₄₇	—	4	2	no	—
39	C₂⁴×C₄₇	C₄²	1	4	yes	—
40	C₂⁴×C₄₇	M₁₆	1	8	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	8	no	—
46	C₂²×C₄×C₄₇	—	4	16	no	—
47	C₂²×C₄×C₄₇	—	4	4	no	—
48	C₂²×C₄×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
52	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²⋊C₄×C₄₇	—	2	16	no	—
61	C₄²×C₄₇	—	2	8	no	—
62	C₄²×C₄₇	—	4	4	no	—
63	C₂×C₄₇⋊C₈	—	2	16	no	—
64	C₂×C₄₇⋊C₈	—	4	16	no	—
65	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	—	2	4	no	—
68	C₂³×C₄₇	—	2	2	no	—
69	C₂³×C₄₇	—	2	4	no	—
70	C₂³×C₄₇	—	4	4	no	—
71	C₂³×C₄₇	—	4	4	no	—
72	C₂³×C₄₇	—	4	16	no	—
73	C₂³×C₄₇	#97	1	16	yes	complemented
74	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
75	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
76	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
77	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
78	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	4	8	no	—
88	C₂×C₄×C₄₇	—	4	4	no	—
89	C₂×C₄×C₄₇	—	4	16	no	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
94	C₄₇⋊C₈	—	2	16	no	—
95	#20526	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	—	2	8	no	—
99	C₂²×C₄₇	—	2	16	no	—
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	4	16	no	—
103	C₂²×C₄₇	C₂×#97	1	8	yes	—
104	C₂²×C₄₇	C₂×#97	1	4	yes	—
105	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
106	C₂²×C₄₇	#518	1	2	yes	—
107	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
108	C₂²×C₄₇	#568	1	2	yes	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	4	8	no	—
112	C₄×C₄₇	#518	1	4	yes	—
113	C₂²×C₂²⋊C₈	—	47	2	no	—
114	C₂×C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
115	#1752	—	47	8	no	—
116	#1749	—	47	4	no	—
117	C₂×C₄₇	—	2	4	no	—
118	C₂×C₄₇	—	2	8	no	—
119	C₂×C₄₇	#2273	1	1	yes	characteristic
120	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
121	C₂×C₄₇	#2873	1	1	yes	characteristic
122	C₂×C₄₇	#2973	1	4	yes	—
123	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
124	C₂³×C₈	—	47	4	no	—
125	C₂²×C₂²⋊C₄	Dic₄₇	1	2	yes	—
126	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
127	C₂×C₂²⋊C₈	—	47	16	no	—
128	C₂×C₂²⋊C₈	—	47	4	no	—
129	C₂×C₂²⋊C₈	—	47	4	no	—
130	C₂×#97	—	47	8	no	—
131	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
132	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
133	C₂×#95	Dic₄₇	1	2	yes	—
134	#336	—	47	8	no	—
135	C₄₇	#20526	1	1	yes	characteristic, complemented, Sylow
136	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂³×C₄	—	4	4	no	—
138	C₂³×C₄	C₄×D₄₇	1	8	yes	—
139	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
140	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
141	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
142	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
143	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
144	C₂²×C₈	—	47	8	no	—
145	C₂²×C₈	—	47	16	no	—
146	C₂²×C₈	—	188	8	no	—
147	C₂×C₂²⋊C₄	—	2	8	no	—
148	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
149	C₂×C₄²	—	4	4	no	—
150	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
152	C₂²⋊C₈	—	47	16	no	—
153	C₂²⋊C₈	—	47	16	no	—
154	#97	—	188	8	no	—
155	#95	C₂×Dic₄₇	1	8	yes	—
156	C₂⁴	—	4	2	no	—
157	C₂⁴	—	4	2	no	—
158	C₂⁴	#10625329	1	8	yes	—
159	C₂⁴	C₄×Dic₄₇	1	4	yes	—
160	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
161	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	4	16	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	—	4	4	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
171	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
172	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
176	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
178	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
179	C₂×C₈	—	47	16	no	—
180	C₂×C₈	—	94	16	no	—
181	C₂×C₈	—	188	16	no	—
182	C₂²⋊C₄	—	2	16	no	—
183	C₄²	—	2	8	no	—
184	C₄²	—	4	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	4	no	—
190	C₂³	—	4	16	no	—
191	C₂³	—	4	4	no	—
192	C₂³	#13093935	1	16	yes	complemented
193	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
194	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
195	C₂³	C₂×#10625329	1	8	yes	—
196	C₂³	C₂×#10625329	1	4	yes	—
197	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	16	no	—
209	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
211	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
212	C₂×C₄	#13094069	1	4	yes	—
213	C₈	—	94	16	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	16	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	16	no	—
221	C₂²	C₂×#13093935	1	8	yes	—
222	C₂²	C₂×#13093935	1	4	yes	—
223	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
224	C₂²	#15689598	1	2	yes	—
225	C₂²	C₂²×#10625329	1	2	yes	—
226	C₂²	#15689890	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	8	no	—
230	C₄	#15689596	1	4	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	2	8	no	—
233	C₂	#20245741	1	1	yes	characteristic
234	C₂	C₂²×#13093935	1	1	yes	characteristic
235	C₂	#20251807	1	1	yes	characteristic
236	C₂	#20252417	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#20526
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20243166	C₂	semidirect
3. #20243150	C₂	semidirect
6. C₂×C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
9. C₂×#13093935	C₂²	semidirect
10. #15688878	C₂²	semidirect
18. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
21. C₂²⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
73. C₂³×C₄₇	#97	semidirect
135. C₄₇	#20526	semidirect
192. C₂³	#13093935	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊(C₄₇⋊C₈)	2
#20243166	2
#20243150	2
C₂×C₄×C₂²⋊C₄×C₄₇	2
#20526	47