Subgroups — C₂²×#15688987 #26374551

Order 12032.

4286 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₄₇⋊C₄	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688987	C₂	1	24	yes	maximal, direct product factor
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×Dic₄₇⋊C₄	—	2	16	no	—
11	C₂²×Dic₄₇⋊C₄	C₂²	1	24	yes	—
12	C₂²×C₄×Dic₄₇	C₄	1	16	yes	complemented
13	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
14	#15688987	C₂²	1	64	yes	direct product factor
15	C₂×C₄₇⋊#95	C₂²	1	24	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
17	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#95×C₄₇	C₂²	1	12	yes	—
20	C₂³×Dic₄₇	—	2	8	no	—
21	C₂³×Dic₄₇	—	2	8	no	—
22	C₂³×Dic₄₇	—	2	8	no	—
23	C₂³×Dic₄₇	—	2	8	no	—
24	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
25	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
26	C₂³×Dic₄₇	D₄	1	8	yes	—
27	C₂³×Dic₄₇	Q₈	1	8	yes	—
28	C₂³×Dic₄₇	C₂³	1	12	yes	—
29	C₂³×Dic₄₇	C₂³	1	6	yes	—
30	C₂³×Dic₄₇	C₂³	1	6	yes	—
31	C₂×Dic₄₇⋊C₄	—	2	96	no	—
32	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	96	yes	complemented
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	32	yes	—
37	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	—	2	8	no	—
39	C₂³×C₄×C₄₇	—	2	8	no	—
40	C₂³×C₄×C₄₇	—	2	8	no	—
41	C₂³×C₄×C₄₇	—	2	8	no	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
45	#95×C₄₇	C₂³	1	16	yes	—
46	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
48	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂⁴×C₄₇	#41	1	8	yes	—
50	C₂⁴×C₄₇	#41	1	8	yes	—
51	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
52	C₂²×C₄×C₄₇	—	2	32	no	—
53	C₂²×C₄×C₄₇	—	2	24	no	—
54	C₂²×C₄×C₄₇	—	2	24	no	—
55	C₂²×C₄×C₄₇	—	2	24	no	—
56	C₂²×C₄×C₄₇	—	2	24	no	—
57	C₂²×C₄×C₄₇	—	4	32	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
61	C₂²×Dic₄₇	—	2	32	no	—
62	C₂²×Dic₄₇	—	2	24	no	—
63	C₂²×Dic₄₇	—	2	24	no	—
64	C₂²×Dic₄₇	—	2	24	no	—
65	C₂²×Dic₄₇	—	2	24	no	—
66	C₂²×Dic₄₇	—	4	32	no	—
67	C₂²×Dic₄₇	C₄⋊C₄	1	64	yes	—
68	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
69	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
70	C₂²×Dic₄₇	C₂×D₄	1	24	yes	—
71	C₂²×Dic₄₇	C₂×Q₈	1	24	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
73	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
74	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
75	Dic₄₇⋊C₄	—	2	64	no	—
76	C₄×Dic₄₇	C₂²×C₄	1	64	yes	complemented
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
78	C₂³×C₄₇	#117	1	16	yes	—
79	C₂³×C₄₇	#117	1	16	yes	—
80	C₂³×C₄₇	#124	1	32	yes	—
81	C₂³×C₄₇	#125	1	32	yes	—
82	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
83	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
84	C₂³×C₄₇	C₂²×Q₈	1	6	yes	—
85	C₂³×C₄₇	C₂×#41	1	12	yes	—
86	C₂³×C₄₇	C₂×#41	1	12	yes	—
87	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	48	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	4	48	no	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
95	C₂×Dic₄₇	—	2	48	no	—
96	C₂×Dic₄₇	—	2	8	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	C₂×C₄⋊C₄	1	96	yes	—
102	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
103	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
104	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
105	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
106	C₂²×#382	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	#382	1	64	yes	complemented
108	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
109	C₂²×C₄₇	C₂×#117	1	12	yes	—
110	C₂²×C₄₇	C₂×#117	1	12	yes	—
111	C₂²×C₄₇	C₂×#124	1	24	yes	—
112	C₂²×C₄₇	C₂×#125	1	24	yes	—
113	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	C₂²×#41	1	2	yes	—
117	C₂²×C₄₇	C₂²×#41	1	2	yes	—
118	C₄×C₄₇	—	2	8	no	—
119	C₄×C₄₇	—	4	8	no	—
120	Dic₄₇	—	2	8	no	—
121	Dic₄₇	—	4	8	no	—
122	Dic₄₇	C₂²×C₄⋊C₄	1	16	yes	—
123	C₂³×C₄⋊C₄	—	47	2	no	—
124	C₂³×C₄²	—	47	1	no	—
125	C₂×#382	—	47	24	no	—
126	C₂²×#95	—	47	2	no	—
127	C₂²×#95	—	47	1	no	—
128	C₂²×#95	D₄₇	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#382	1	24	yes	complemented
130	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
132	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
133	C₂×C₄₇	C₂²×#124	1	2	yes	—
134	C₂×C₄₇	C₂²×#125	1	2	yes	—
135	C₂⁴×C₄	—	47	1	no	—
136	C₂⁴×C₄	—	47	1	no	—
137	C₂⁴×C₄	—	47	2	no	—
138	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
139	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	—	47	24	no	—
141	C₂²×C₄⋊C₄	—	94	16	no	—
142	C₂²×C₄²	—	47	16	no	—
143	C₂²×C₄²	—	47	12	no	—
144	#382	—	47	64	no	—
145	C₂×#95	—	47	24	no	—
146	C₂×#95	—	47	12	no	—
147	C₂×#95	C₂×D₄₇	1	12	yes	—
148	C₄₇	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
149	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	47	8	no	—
155	C₂³×C₄	—	47	8	no	—
156	C₂³×C₄	—	47	8	no	—
157	C₂³×C₄	—	47	6	no	—
158	C₂³×C₄	—	47	8	no	—
159	C₂³×C₄	—	47	6	no	—
160	C₂³×C₄	—	47	12	no	—
161	C₂³×C₄	—	94	8	no	—
162	C₂³×C₄	—	94	8	no	—
163	C₂³×C₄	—	94	8	no	—
164	C₂³×C₄	—	94	8	no	—
165	C₂³×C₄	C₄×D₄₇	1	8	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
168	C₂×C₄⋊C₄	—	47	32	no	—
169	C₂×C₄⋊C₄	—	94	96	no	—
170	C₂×C₄²	—	47	96	no	—
171	C₂×C₄²	—	47	16	no	—
172	#95	—	47	32	no	—
173	#95	—	47	16	no	—
174	#95	C₂²×D₄₇	1	16	yes	—
175	C₂⁴	C₂×C₄×D₄₇	1	4	yes	—
176	C₂⁴	#10625350	1	8	yes	—
177	C₂⁴	D₄×D₄₇	1	4	yes	—
178	C₂⁴	#10625352	1	8	yes	—
179	C₂⁴	Q₈×D₄₇	1	4	yes	—
180	C₂⁴	C₂³×D₄₇	1	3	yes	—
181	C₂²×C₄	—	2	32	no	—
182	C₂²×C₄	—	2	24	no	—
183	C₂²×C₄	—	2	24	no	—
184	C₂²×C₄	—	2	24	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	4	32	no	—
187	C₂²×C₄	—	47	64	no	—
188	C₂²×C₄	—	47	24	no	—
189	C₂²×C₄	—	47	24	no	—
190	C₂²×C₄	—	47	24	no	—
191	C₂²×C₄	—	47	4	no	—
192	C₂²×C₄	—	47	24	no	—
193	C₂²×C₄	—	47	4	no	—
194	C₂²×C₄	—	47	8	no	—
195	C₂²×C₄	—	94	32	no	—
196	C₂²×C₄	—	94	24	no	—
197	C₂²×C₄	—	94	24	no	—
198	C₂²×C₄	—	94	24	no	—
199	C₂²×C₄	—	94	24	no	—
200	C₂²×C₄	—	188	32	no	—
201	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
203	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
204	C₄⋊C₄	—	94	64	no	—
205	C₄²	—	47	64	no	—
206	C₂³	C₄⋊C₄×D₄₇	1	16	yes	—
207	C₂³	#13093980	1	16	yes	—
208	C₂³	#13093981	1	16	yes	—
209	C₂³	#13094010	1	32	yes	—
210	C₂³	#13094014	1	32	yes	—
211	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
212	C₂³	C₂×D₄×D₄₇	1	6	yes	—
213	C₂³	C₂×Q₈×D₄₇	1	6	yes	—
214	C₂³	C₂×#10625352	1	12	yes	—
215	C₂³	C₂×#10625350	1	12	yes	—
216	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
217	C₂×C₄	—	2	48	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	48	no	—
223	C₂×C₄	—	47	96	no	—
224	C₂×C₄	—	47	8	no	—
225	C₂×C₄	—	47	8	no	—
226	C₂×C₄	—	47	8	no	—
227	C₂×C₄	—	47	8	no	—
228	C₂×C₄	—	94	48	no	—
229	C₂×C₄	—	94	8	no	—
230	C₂×C₄	—	94	8	no	—
231	C₂×C₄	—	94	8	no	—
232	C₂×C₄	—	94	8	no	—
233	C₂×C₄	—	188	48	no	—
234	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
235	C₂²	#15688987	1	64	yes	direct product factor
236	C₂²	C₂×C₄⋊C₄×D₄₇	1	12	yes	—
237	C₂²	C₂×#13093980	1	12	yes	—
238	C₂²	C₂×#13093981	1	12	yes	—
239	C₂²	C₂×#13094010	1	24	yes	—
240	C₂²	C₂×#13094014	1	24	yes	—
241	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
242	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
243	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
244	C₂²	C₂²×#10625352	1	2	yes	—
245	C₂²	C₂²×#10625350	1	2	yes	—
246	C₄	—	2	8	no	—
247	C₄	—	4	8	no	—
248	C₄	—	47	16	no	—
249	C₄	—	94	8	no	—
250	C₄	—	188	8	no	—
251	C₂	C₂×#15688987	1	24	yes	direct product factor
252	C₂	C₂²×C₄⋊C₄×D₄₇	1	1	yes	characteristic
253	C₂	C₂²×#13093980	1	1	yes	characteristic
254	C₂	C₂²×#13093981	1	1	yes	characteristic
255	C₂	C₂²×#13094010	1	2	yes	—
256	C₂	C₂²×#13094014	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×#382
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
3. C₂×#15688987	C₂	direct (N × H)
12. C₂²×C₄×Dic₄₇	C₄	semidirect
14. #15688987	C₂²	direct (N × H)
33. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
76. C₄×Dic₄₇	C₂²×C₄	semidirect
107. C₂²×C₄₇	#382	semidirect
129. C₂×C₄₇	C₂×#382	semidirect
148. C₄₇	C₂²×#382	semidirect
235. C₂²	#15688987	direct (N × H)
251. C₂	C₂×#15688987	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×Dic₄₇⋊C₄	2
C₂³×C₄×Dic₄₇	2
C₂×#15688987	2
C₂²×C₄₇⋊#95	2
C₂²×#95×C₄₇	2
C₂²×#382	47

Subgroups — C22×#15688987 #26374551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688987 #26374551