Subgroups — C₂²×#15688990 #26374555

Order 12032.

3678 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688990	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄₇⋊#95	C₂	1	2	yes	maximal
4	C₂²×C₄₇⋊#95	C₂	1	2	yes	maximal
5	C₂²×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂²×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₂²	1	2	yes	—
8	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₄	1	16	yes	complemented
11	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
12	#15688990	C₂²	1	64	yes	direct product factor
13	C₂×C₄₇⋊#95	C₂²	1	24	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	24	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
16	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
17	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#95×C₄₇	C₂²	1	12	yes	—
19	C₂³×Dic₄₇	—	2	8	no	—
20	C₂³×Dic₄₇	—	2	4	no	—
21	C₂³×Dic₄₇	—	2	4	no	—
22	C₂³×Dic₄₇	—	2	8	no	—
23	C₂³×Dic₄₇	—	2	8	no	—
24	C₂³×Dic₄₇	—	2	8	no	—
25	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
26	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
27	C₂³×Dic₄₇	C₂³	1	12	yes	—
28	C₂³×Dic₄₇	C₂³	1	6	yes	—
29	C₂³×Dic₄₇	C₂³	1	6	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	96	yes	complemented
31	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
32	C₄₇⋊#95	C₂³	1	16	yes	—
33	C₄₇⋊#95	C₂³	1	32	yes	—
34	C₄₇⋊#95	C₂³	1	32	yes	—
35	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	—	2	8	no	—
37	C₂³×C₄×C₄₇	—	2	8	no	—
38	C₂³×C₄×C₄₇	—	2	8	no	—
39	C₂³×C₄×C₄₇	—	2	8	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
43	#95×C₄₇	C₂³	1	16	yes	—
44	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₄₇	#41	1	8	yes	—
46	C₂⁴×C₄₇	#41	1	8	yes	—
47	C₂⁴×C₄₇	#41	1	4	yes	—
48	C₂⁴×C₄₇	#41	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
50	C₂²×C₄×C₄₇	—	2	32	no	—
51	C₂²×C₄×C₄₇	—	2	24	no	—
52	C₂²×C₄×C₄₇	—	2	24	no	—
53	C₂²×C₄×C₄₇	—	2	24	no	—
54	C₂²×C₄×C₄₇	—	2	24	no	—
55	C₂²×C₄×C₄₇	—	4	32	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
59	C₂²×Dic₄₇	—	2	32	no	—
60	C₂²×Dic₄₇	—	2	32	no	—
61	C₂²×Dic₄₇	—	2	12	no	—
62	C₂²×Dic₄₇	—	2	24	no	—
63	C₂²×Dic₄₇	—	2	12	no	—
64	C₂²×Dic₄₇	—	2	24	no	—
65	C₂²×Dic₄₇	—	2	24	no	—
66	C₂²×Dic₄₇	—	2	24	no	—
67	C₂²×Dic₄₇	—	4	32	no	—
68	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
69	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
70	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
71	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
73	C₄×Dic₄₇	C₂²×C₄	1	64	yes	complemented
74	C₂³×C₄₇	#117	1	16	yes	—
75	C₂³×C₄₇	#117	1	16	yes	—
76	C₂³×C₄₇	#117	1	16	yes	—
77	C₂³×C₄₇	#126	1	32	yes	—
78	C₂³×C₄₇	#126	1	32	yes	—
79	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
80	C₂³×C₄₇	C₂×#41	1	12	yes	—
81	C₂³×C₄₇	C₂×#41	1	6	yes	—
82	C₂³×C₄₇	C₂×#41	1	12	yes	—
83	C₂³×C₄₇	C₂×#41	1	6	yes	—
84	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	48	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	4	48	no	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
92	C₂×Dic₄₇	—	2	48	no	—
93	C₂×Dic₄₇	—	2	48	no	—
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	2	8	no	—
96	C₂×Dic₄₇	—	2	4	no	—
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	8	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	4	48	no	—
101	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
102	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
103	C₂²×#383	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	#383	1	64	yes	complemented
105	C₂²×C₄₇	C₂×#117	1	12	yes	—
106	C₂²×C₄₇	C₂×#117	1	12	yes	—
107	C₂²×C₄₇	C₂×#117	1	12	yes	—
108	C₂²×C₄₇	C₂×#126	1	24	yes	—
109	C₂²×C₄₇	C₂×#126	1	24	yes	—
110	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
111	C₂²×C₄₇	C₂²×#41	1	2	yes	—
112	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
113	C₂²×C₄₇	C₂²×#41	1	2	yes	—
114	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
115	C₄×C₄₇	—	2	8	no	—
116	C₄×C₄₇	—	4	8	no	—
117	Dic₄₇	—	2	8	no	—
118	Dic₄₇	—	2	8	no	—
119	Dic₄₇	—	4	8	no	—
120	C₂³×C₄²	—	47	1	no	—
121	C₂×#383	—	47	24	no	—
122	C₂²×#95	—	47	2	no	—
123	C₂²×#95	—	47	2	no	—
124	C₂²×#95	—	47	1	no	—
125	C₂²×#95	D₄₇	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#383	1	24	yes	complemented
127	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
128	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
129	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
130	C₂×C₄₇	C₂²×#126	1	2	yes	—
131	C₂×C₄₇	C₂²×#126	1	2	yes	—
132	C₂⁴×C₄	—	47	1	no	—
133	C₂⁴×C₄	—	47	1	no	—
134	C₂⁴×C₄	—	47	2	no	—
135	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
136	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
137	C₂²×C₄²	—	47	16	no	—
138	C₂²×C₄²	—	47	12	no	—
139	#383	—	47	64	no	—
140	C₂×#95	—	47	24	no	—
141	C₂×#95	—	47	24	no	—
142	C₂×#95	—	47	12	no	—
143	C₂×#95	C₂×D₄₇	1	12	yes	—
144	C₄₇	C₂²×#383	1	1	yes	characteristic, complemented, Sylow
145	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
146	C₂³×C₄	—	2	8	no	—
147	C₂³×C₄	—	2	8	no	—
148	C₂³×C₄	—	2	8	no	—
149	C₂³×C₄	—	2	8	no	—
150	C₂³×C₄	—	47	8	no	—
151	C₂³×C₄	—	47	8	no	—
152	C₂³×C₄	—	47	6	no	—
153	C₂³×C₄	—	47	6	no	—
154	C₂³×C₄	—	47	12	no	—
155	C₂³×C₄	—	94	8	no	—
156	C₂³×C₄	—	94	4	no	—
157	C₂³×C₄	—	94	4	no	—
158	C₂³×C₄	—	94	8	no	—
159	C₂³×C₄	—	94	8	no	—
160	C₂³×C₄	—	94	8	no	—
161	C₂³×C₄	C₄×D₄₇	1	8	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
164	C₂×C₄²	—	47	96	no	—
165	C₂×C₄²	—	47	16	no	—
166	#95	—	47	32	no	—
167	#95	—	47	32	no	—
168	#95	—	47	16	no	—
169	#95	C₂²×D₄₇	1	16	yes	—
170	C₂⁴	C₂×C₄×D₄₇	1	4	yes	—
171	C₂⁴	#10625350	1	8	yes	—
172	C₂⁴	#10625352	1	8	yes	—
173	C₂⁴	#10625352	1	4	yes	—
174	C₂⁴	#10625354	1	4	yes	—
175	C₂⁴	C₂³×D₄₇	1	3	yes	—
176	C₂²×C₄	—	2	32	no	—
177	C₂²×C₄	—	2	24	no	—
178	C₂²×C₄	—	2	24	no	—
179	C₂²×C₄	—	2	24	no	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	4	32	no	—
182	C₂²×C₄	—	47	24	no	—
183	C₂²×C₄	—	47	24	no	—
184	C₂²×C₄	—	47	4	no	—
185	C₂²×C₄	—	47	4	no	—
186	C₂²×C₄	—	47	8	no	—
187	C₂²×C₄	—	94	32	no	—
188	C₂²×C₄	—	94	32	no	—
189	C₂²×C₄	—	94	12	no	—
190	C₂²×C₄	—	94	24	no	—
191	C₂²×C₄	—	94	12	no	—
192	C₂²×C₄	—	94	24	no	—
193	C₂²×C₄	—	94	24	no	—
194	C₂²×C₄	—	94	24	no	—
195	C₂²×C₄	—	188	32	no	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
197	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
199	C₄²	—	47	64	no	—
200	C₂³	#13093980	1	16	yes	—
201	C₂³	#13093981	1	16	yes	—
202	C₂³	#13093982	1	16	yes	—
203	C₂³	#13094017	1	32	yes	—
204	C₂³	#13094019	1	32	yes	—
205	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
206	C₂³	C₂×#10625352	1	12	yes	—
207	C₂³	C₂×#10625352	1	6	yes	—
208	C₂³	C₂×#10625350	1	12	yes	—
209	C₂³	C₂×#10625354	1	6	yes	—
210	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
211	C₂×C₄	—	2	48	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	48	no	—
217	C₂×C₄	—	47	8	no	—
218	C₂×C₄	—	47	8	no	—
219	C₂×C₄	—	94	48	no	—
220	C₂×C₄	—	94	48	no	—
221	C₂×C₄	—	94	4	no	—
222	C₂×C₄	—	94	8	no	—
223	C₂×C₄	—	94	4	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	—	94	8	no	—
226	C₂×C₄	—	94	8	no	—
227	C₂×C₄	—	188	48	no	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
229	C₂²	#15688990	1	64	yes	direct product factor
230	C₂²	C₂×#13093980	1	12	yes	—
231	C₂²	C₂×#13093981	1	12	yes	—
232	C₂²	C₂×#13093982	1	12	yes	—
233	C₂²	C₂×#13094017	1	24	yes	—
234	C₂²	C₂×#13094019	1	24	yes	—
235	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
236	C₂²	C₂²×#10625352	1	2	yes	—
237	C₂²	C₂²×#10625352	1	1	yes	characteristic
238	C₂²	C₂²×#10625350	1	2	yes	—
239	C₂²	C₂²×#10625354	1	1	yes	characteristic
240	C₄	—	2	8	no	—
241	C₄	—	4	8	no	—
242	C₄	—	94	8	no	—
243	C₄	—	94	8	no	—
244	C₄	—	188	8	no	—
245	C₂	C₂×#15688990	1	24	yes	direct product factor
246	C₂	C₂²×#13093980	1	1	yes	characteristic
247	C₂	C₂²×#13093981	1	1	yes	characteristic
248	C₂	C₂²×#13093982	1	1	yes	characteristic
249	C₂	C₂²×#13094017	1	2	yes	—
250	C₂	C₂²×#13094019	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×#95×C₄₇	C₂
socle, prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#95	D₄₇
hypocenter	C₄₇	C₂²×#383
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

Complements

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

N	H ≅ G/N	type
2. C₂×#15688990	C₂	direct (N × H)
10. C₂²×C₄×Dic₄₇	C₄	semidirect
12. #15688990	C₂²	direct (N × H)
30. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
73. C₄×Dic₄₇	C₂²×C₄	semidirect
104. C₂²×C₄₇	#383	semidirect
126. C₂×C₄₇	C₂×#383	semidirect
144. C₄₇	C₂²×#383	semidirect
229. C₂²	#15688990	direct (N × H)
245. C₂	C₂×#15688990	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×Dic₄₇	2
C₂×#15688990	2
C₂²×C₄₇⋊#95	2
C₂²×#95×C₄₇	2
C₂²×#383	47

Subgroups — C22×#15688990 #26374555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688990 #26374555