Subgroups — C₂²×#15688974 #26374455

Order 12032.

5758 conjugacy classes of subgroups.

The subgroup lattice has 264 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	2	yes	maximal
3	C₂³×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688974	C₂	1	24	yes	maximal, direct product factor
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	C₄	1	32	yes	complemented
11	C₂²×Dic₄₇⋊C₄	C₂²	1	24	yes	—
12	C₂²×C₄×Dic₄₇	—	2	16	no	—
13	C₂²×C₄×Dic₄₇	C₄	1	16	yes	complemented
14	C₂²×C₄×Dic₄₇	C₂²	1	24	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
16	#15688974	C₂²	1	64	yes	direct product factor
17	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#95×C₄₇	C₂²	1	12	yes	—
21	C₂³×Dic₄₇	—	2	8	no	—
22	C₂³×Dic₄₇	—	2	8	no	—
23	C₂³×Dic₄₇	C₂×C₄	1	16	yes	—
24	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
25	C₂³×Dic₄₇	C₂×C₄	1	16	yes	—
26	C₂³×Dic₄₇	C₂×C₄	1	8	yes	—
27	C₂³×Dic₄₇	D₄	1	8	yes	—
28	C₂³×Dic₄₇	Q₈	1	8	yes	—
29	C₂³×Dic₄₇	C₂³	1	12	yes	—
30	C₂³×Dic₄₇	C₂³	1	6	yes	—
31	C₂³×Dic₄₇	C₂³	1	6	yes	—
32	C₂×Dic₄₇⋊C₄	C₂×C₄	1	192	yes	complemented
33	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
34	C₂×C₄×Dic₄₇	—	2	96	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	96	yes	complemented
36	C₂×C₄×Dic₄₇	C₂³	1	32	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
38	C₄₇⋊#95	C₂³	1	16	yes	—
39	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	—	2	8	no	—
41	C₂³×C₄×C₄₇	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
47	#95×C₄₇	C₂³	1	16	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
52	C₂⁴×C₄₇	#41	1	8	yes	—
53	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₄₇	—	2	32	no	—
55	C₂²×C₄×C₄₇	—	2	64	no	—
56	C₂²×C₄×C₄₇	—	2	24	no	—
57	C₂²×C₄×C₄₇	—	2	24	no	—
58	C₂²×C₄×C₄₇	—	2	24	no	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
63	C₂²×Dic₄₇	—	2	64	no	—
64	C₂²×Dic₄₇	—	2	24	no	—
65	C₂²×Dic₄₇	—	2	24	no	—
66	C₂²×Dic₄₇	C₄²	1	64	yes	complemented
67	C₂²×Dic₄₇	C₄⋊C₄	1	64	yes	complemented
68	C₂²×Dic₄₇	C₂²×C₄	1	48	yes	—
69	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
70	C₂²×Dic₄₇	C₂²×C₄	1	48	yes	—
71	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
72	C₂²×Dic₄₇	C₂×D₄	1	24	yes	—
73	C₂²×Dic₄₇	C₂×Q₈	1	24	yes	—
74	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
75	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
76	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
77	Dic₄₇⋊C₄	C₂²×C₄	1	128	yes	complemented
78	C₄×Dic₄₇	—	2	64	no	—
79	C₄×Dic₄₇	C₂²×C₄	1	64	yes	complemented
80	C₂³×C₄₇	C₂×C₄²	1	16	yes	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
82	C₂³×C₄₇	#117	1	16	yes	—
83	C₂³×C₄₇	C₄×D₄	1	32	yes	—
84	C₂³×C₄₇	C₄×Q₈	1	32	yes	—
85	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
87	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
88	C₂³×C₄₇	C₂²×Q₈	1	6	yes	—
89	C₂³×C₄₇	C₂×#41	1	12	yes	—
90	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	48	no	—
92	C₂×C₄×C₄₇	—	2	96	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
98	C₂×Dic₄₇	—	2	96	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	2	8	no	—
101	C₂×Dic₄₇	C₂×C₄²	1	96	yes	complemented
102	C₂×Dic₄₇	C₂×C₄⋊C₄	1	96	yes	complemented
103	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
104	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
105	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
106	C₂×Dic₄₇	C₂³×C₄	1	16	yes	—
107	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
108	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
109	C₂²×C₄×C₄⋊C₄	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	C₄×C₄⋊C₄	1	64	yes	complemented
111	C₂²×C₄₇	C₂²×C₄²	1	12	yes	—
112	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
113	C₂²×C₄₇	C₂×#117	1	12	yes	—
114	C₂²×C₄₇	C₂×C₄×D₄	1	24	yes	—
115	C₂²×C₄₇	C₂×C₄×Q₈	1	24	yes	—
116	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
118	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
120	C₂²×C₄₇	C₂²×#41	1	2	yes	—
121	C₄×C₄₇	—	2	8	no	—
122	C₄×C₄₇	—	2	16	no	—
123	Dic₄₇	—	2	16	no	—
124	Dic₄₇	C₂²×C₄²	1	16	yes	complemented
125	Dic₄₇	C₂²×C₄⋊C₄	1	16	yes	complemented
126	C₂³×C₄⋊C₄	—	47	2	no	—
127	C₂³×C₄²	—	47	1	no	—
128	C₂³×C₄²	—	47	2	no	—
129	C₂×C₄×C₄⋊C₄	—	47	24	no	—
130	C₂²×#95	—	47	1	no	—
131	C₂²×#95	D₄₇	1	1	yes	characteristic
132	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	24	yes	complemented
133	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
134	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
136	C₂×C₄₇	C₂²×C₄×D₄	1	2	yes	—
137	C₂×C₄₇	C₂²×C₄×Q₈	1	2	yes	—
138	C₂⁴×C₄	—	47	1	no	—
139	C₂⁴×C₄	—	47	1	no	—
140	C₂⁴×C₄	—	47	2	no	—
141	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
142	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	—	47	32	no	—
144	C₂²×C₄⋊C₄	—	47	24	no	—
145	C₂²×C₄²	—	47	16	no	—
146	C₂²×C₄²	—	47	12	no	—
147	C₂²×C₄²	—	47	24	no	—
148	C₂²×C₄²	—	94	16	no	—
149	C₄×C₄⋊C₄	—	47	64	no	—
150	C₂×#95	—	47	12	no	—
151	C₂×#95	C₂×D₄₇	1	12	yes	—
152	C₄₇	C₂²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
153	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	47	16	no	—
158	C₂³×C₄	—	47	8	no	—
159	C₂³×C₄	—	47	8	no	—
160	C₂³×C₄	—	47	8	no	—
161	C₂³×C₄	—	47	8	no	—
162	C₂³×C₄	—	47	6	no	—
163	C₂³×C₄	—	47	6	no	—
164	C₂³×C₄	—	47	16	no	—
165	C₂³×C₄	—	47	12	no	—
166	C₂³×C₄	—	94	8	no	—
167	C₂³×C₄	—	94	8	no	—
168	C₂³×C₄	C₄×D₄₇	1	16	yes	—
169	C₂³×C₄	C₄×D₄₇	1	8	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
171	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
172	C₂×C₄⋊C₄	—	47	192	no	—
173	C₂×C₄⋊C₄	—	47	32	no	—
174	C₂×C₄²	—	47	96	no	—
175	C₂×C₄²	—	47	16	no	—
176	C₂×C₄²	—	47	32	no	—
177	C₂×C₄²	—	94	96	no	—
178	#95	—	47	16	no	—
179	#95	C₂²×D₄₇	1	16	yes	—
180	C₂⁴	C₂×C₄×D₄₇	1	4	yes	—
181	C₂⁴	C₂×C₄×D₄₇	1	8	yes	—
182	C₂⁴	D₄×D₄₇	1	4	yes	—
183	C₂⁴	#10625352	1	8	yes	—
184	C₂⁴	Q₈×D₄₇	1	4	yes	—
185	C₂⁴	C₂³×D₄₇	1	3	yes	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	2	64	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	24	no	—
190	C₂²×C₄	—	2	24	no	—
191	C₂²×C₄	—	47	64	no	—
192	C₂²×C₄	—	47	64	no	—
193	C₂²×C₄	—	47	24	no	—
194	C₂²×C₄	—	47	24	no	—
195	C₂²×C₄	—	47	48	no	—
196	C₂²×C₄	—	47	24	no	—
197	C₂²×C₄	—	47	24	no	—
198	C₂²×C₄	—	47	4	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	48	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	94	64	no	—
203	C₂²×C₄	—	94	24	no	—
204	C₂²×C₄	—	94	24	no	—
205	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	48	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
208	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
209	C₄⋊C₄	—	47	128	no	—
210	C₄²	—	47	64	no	—
211	C₄²	—	94	64	no	—
212	C₂³	C₄²×D₄₇	1	16	yes	—
213	C₂³	C₄⋊C₄×D₄₇	1	16	yes	—
214	C₂³	#13093980	1	16	yes	—
215	C₂³	(C₂²⋊D₄₇)⋊C₄	1	32	yes	—
216	C₂³	#13093989	1	32	yes	—
217	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
218	C₂³	C₂²×C₄×D₄₇	1	12	yes	—
219	C₂³	C₂×D₄×D₄₇	1	6	yes	—
220	C₂³	C₂×Q₈×D₄₇	1	6	yes	—
221	C₂³	C₂×#10625352	1	12	yes	—
222	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
223	C₂×C₄	—	2	48	no	—
224	C₂×C₄	—	2	96	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	47	96	no	—
229	C₂×C₄	—	47	96	no	—
230	C₂×C₄	—	47	8	no	—
231	C₂×C₄	—	47	8	no	—
232	C₂×C₄	—	47	16	no	—
233	C₂×C₄	—	47	8	no	—
234	C₂×C₄	—	47	8	no	—
235	C₂×C₄	—	47	16	no	—
236	C₂×C₄	—	94	96	no	—
237	C₂×C₄	—	94	8	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
240	C₂×C₄	C₂²×C₄×D₄₇	1	16	yes	—
241	C₂²	#15688974	1	64	yes	direct product factor
242	C₂²	C₂×C₄²×D₄₇	1	12	yes	—
243	C₂²	C₂×C₄⋊C₄×D₄₇	1	12	yes	—
244	C₂²	C₂×#13093980	1	12	yes	—
245	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	24	yes	—
246	C₂²	C₂×#13093989	1	24	yes	—
247	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
248	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
249	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
250	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
251	C₂²	C₂²×#10625352	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	2	16	no	—
254	C₄	—	47	16	no	—
255	C₄	—	47	16	no	—
256	C₄	—	94	16	no	—
257	C₂	C₂×#15688974	1	24	yes	direct product factor
258	C₂	C₂²×C₄²×D₄₇	1	1	yes	characteristic
259	C₂	C₂²×C₄⋊C₄×D₄₇	1	1	yes	characteristic
260	C₂	C₂²×#13093980	1	1	yes	characteristic
261	C₂	C₂²×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
262	C₂	C₂²×#13093989	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
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×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁵ ◁ C₂²×#95

Complements

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

N	H ≅ G/N	type
4. C₂×#15688974	C₂	direct (N × H)
10. C₂²×Dic₄₇⋊C₄	C₄	semidirect
13. C₂²×C₄×Dic₄₇	C₄	semidirect
16. #15688974	C₂²	direct (N × H)
32. C₂×Dic₄₇⋊C₄	C₂×C₄	semidirect
35. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
66. C₂²×Dic₄₇	C₄²	semidirect
67. C₂²×Dic₄₇	C₄⋊C₄	semidirect
77. Dic₄₇⋊C₄	C₂²×C₄	semidirect
79. C₄×Dic₄₇	C₂²×C₄	semidirect
101. C₂×Dic₄₇	C₂×C₄²	semidirect
102. C₂×Dic₄₇	C₂×C₄⋊C₄	semidirect
110. C₂²×C₄₇	C₄×C₄⋊C₄	semidirect
124. Dic₄₇	C₂²×C₄²	semidirect
125. Dic₄₇	C₂²×C₄⋊C₄	semidirect
132. C₂×C₄₇	C₂×C₄×C₄⋊C₄	semidirect
152. C₄₇	C₂²×C₄×C₄⋊C₄	semidirect
241. C₂²	#15688974	direct (N × H)
257. C₂	C₂×#15688974	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×#15688974	2
C₂²×C₄₇⋊#95	2
C₂²×#95×C₄₇	2
C₂²×C₄×C₄⋊C₄	47

Subgroups — C22×#15688974 #26374455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688974 #26374455