Subgroups — #26226464

Order 12032.

628 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	#20244003	C₂	1	4	yes	maximal
2	#20243954	C₂	1	4	yes	maximal, complemented
3	#20243542	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243508	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243372	C₂	1	2	yes	maximal
6	#20243241	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2298	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	4	yes	—
13	#15688878	C₂²	1	4	yes	—
14	#15688878	C₂²	1	4	yes	—
15	#15688878	C₂²	1	4	yes	—
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#388	C₄	1	4	yes	—
18	C₄₇×#387	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#376	C₂²	1	1	yes	characteristic
22	C₄³×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
29	#13093935	—	4	4	no	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	#117×C₄₇	—	2	2	no	—
32	#117×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#95×C₄₇	C₂×C₄	1	4	yes	—
43	#95×C₄₇	C₂×C₄	1	4	yes	—
44	#95×C₄₇	C₂×C₄	1	4	yes	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	4	no	—
58	C₂²⋊C₄×C₄₇	—	2	4	no	—
59	C₂²⋊C₄×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	—	2	1	no	—
61	C₄²×C₄₇	—	2	1	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	4	4	no	—
67	C₄²×C₄₇	#41	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂³×C₄₇	—	2	1	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	4	1	no	—
91	C₂×C₄×C₄₇	—	4	1	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	#117	1	4	yes	—
94	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
97	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
98	C₄₇⋊C₈	—	4	4	no	—
99	C₄₇⋊C₈	—	4	4	no	—
100	#21927	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	—	4	1	no	—
102	C₂²×C₄₇	—	4	2	no	—
103	C₂²×C₄₇	—	4	4	no	—
104	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
105	C₂²×C₄₇	#520	1	2	yes	—
106	C₂²×C₄₇	#568	1	4	yes	—
107	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	4	1	no	—
109	C₄×C₄₇	—	4	2	no	—
110	C₄×C₄₇	—	4	4	no	—
111	C₄×C₄₇	—	4	4	no	—
112	C₄×C₄₇	#540	1	1	yes	characteristic
113	C₄×C₄₇	#541	1	1	yes	characteristic
114	C₄×C₄₇	#542	1	2	yes	—
115	#2298	D₄₇	1	1	yes	characteristic
116	#1924	—	47	4	no	—
117	#1917	—	47	4	no	—
118	#1843	—	47	1	no	—
119	#1832	—	47	1	no	—
120	#1798	—	47	2	no	—
121	#1767	—	47	2	no	—
122	C₂×C₄₇	—	8	1	no	—
123	C₂×C₄₇	#2314	1	1	yes	characteristic
124	C₂×C₄₇	#2923	1	2	yes	—
125	C₂×C₄₇	#2995	1	4	yes	—
126	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	47	2	no	—
128	C₂×C₄⋊C₈	—	47	2	no	—
129	C₂×C₄⋊C₈	—	47	4	no	—
130	C₂×C₂²⋊C₈	—	47	4	no	—
131	C₂×#97	—	47	4	no	—
132	#388	Dic₄₇	1	4	yes	—
133	#387	Dic₄₇	1	4	yes	—
134	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
135	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
136	#376	C₂×D₄₇	1	1	yes	characteristic
137	C₄³	C₂×D₄₇	1	1	yes	characteristic
138	#336	—	47	4	no	—
139	#336	—	47	4	no	—
140	#336	—	47	4	no	—
141	C₄₇	#21927	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂²×C₈	—	47	4	no	—
144	C₂²×C₈	—	47	4	no	—
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
151	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
156	C₄⋊C₈	—	94	4	no	—
157	C₄⋊C₈	—	94	4	no	—
158	C₄⋊C₈	—	188	4	no	—
159	C₂²⋊C₈	—	188	4	no	—
160	#97	—	188	4	no	—
161	#95	C₂×Dic₄₇	1	4	yes	—
162	#95	C₂×Dic₄₇	1	4	yes	—
163	#95	C₂×Dic₄₇	1	4	yes	—
164	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
170	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	C₂×C₈	—	94	4	no	—
176	C₂×C₈	—	94	4	no	—
177	C₂×C₈	—	94	4	no	—
178	C₂×C₈	—	94	4	no	—
179	C₂×C₈	—	94	4	no	—
180	C₂×C₈	—	94	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	4	4	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	4	no	—
192	C₄²	#10625350	1	2	yes	—
193	C₄²	#10625350	1	2	yes	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	#13093983	1	4	yes	—
213	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
214	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
215	C₂×C₄	#13094070	1	2	yes	—
216	C₂×C₄	#13094071	1	2	yes	—
217	C₈	—	188	4	no	—
218	C₈	—	188	4	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	—	4	1	no	—
222	C₂²	C₂×#13093983	1	1	yes	characteristic
223	C₂²	#15689606	1	2	yes	—
224	C₂²	#15689890	1	4	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	4	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	1	no	—
230	C₄	#15689728	1	1	yes	characteristic
231	C₄	#15689735	1	1	yes	characteristic
232	C₄	#15689744	1	2	yes	—
233	C₂	—	8	1	no	—
234	C₂	#20245997	1	1	yes	characteristic
235	C₂	#20252039	1	2	yes	—
236	C₂	#20252558	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2298	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2298	D₄₇
hypocenter	C₄₇	#21927
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
2. #20243954	C₂	semidirect
3. #20243542	C₂	semidirect
4. #20243508	C₂	semidirect
6. #20243241	C₂	semidirect
141. C₄₇	#21927	semidirect

Maximal subgroups

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

subgroup	index
#20244003	2
#20243954	2
#20243542	2
#20243508	2
#20243372	2
#20243241	2
C₄₇×#2298	2
#21927	47