Subgroups — C₂²×#15689734 #27123095

Order 12032.

7110 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	C₂×#15689734	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094019	C₂	1	4	yes	maximal
3	C₂²×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13094014	C₂	1	4	yes	maximal, complemented
5	C₂²×#13094000	C₂	1	2	yes	maximal
6	C₂²×Q₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#13093981	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	#15689734	C₂²	1	256	yes	direct product factor
11	C₂×#13094019	C₂²	1	96	yes	—
12	C₂×#13094016	C₂²	1	24	yes	complemented
13	C₂×#13094014	C₂²	1	96	yes	complemented
14	C₂×#13094000	C₂²	1	48	yes	—
15	C₂×Q₈×Dic₄₇	C₂²	1	48	yes	complemented
16	C₂×#13093981	C₂²	1	24	yes	—
17	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
19	C₂²×Dic₄₇⋊C₄	C₂²	1	2	yes	—
20	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
21	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
22	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
24	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
26	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₇	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
29	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₂⁴×D₄₇	C₂³	1	1	yes	characteristic
31	C₂³×Dic₄₇	C₂³	1	2	yes	—
32	C₂³×Dic₄₇	C₂³	1	4	yes	—
33	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×D₄₇	—	2	8	no	—
35	C₂²×C₄×D₄₇	C₂³	1	12	yes	—
36	#13094019	C₂³	1	256	yes	—
37	#13094016	C₂³	1	64	yes	complemented
38	#13094014	C₂³	1	256	yes	complemented
39	#13094000	C₂³	1	128	yes	—
40	Q₈×Dic₄₇	C₂³	1	128	yes	complemented
41	#13093981	C₂³	1	64	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	48	yes	—
44	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
45	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
46	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
47	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	24	yes	—
48	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
51	C₂²×Q₈×C₄₇	—	2	16	no	—
52	C₂²×Q₈×C₄₇	C₂³	1	24	yes	—
53	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₄₇	C₂³	1	64	yes	complemented
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	48	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
59	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂³×D₄₇	—	2	4	no	—
61	C₂³×D₄₇	—	2	8	no	—
62	C₂³×D₄₇	C₂⁴	1	6	yes	—
63	C₂×Q₈×C₄₇	—	2	96	no	—
64	C₂×Q₈×C₄₇	C₂⁴	1	32	yes	—
65	C₂²×C₄×C₄₇	—	2	16	no	—
66	C₂²×C₄×C₄₇	—	2	8	no	—
67	C₂²×C₄×C₄₇	#41	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	24	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	—	2	16	no	—
74	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
75	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
76	C₂²×Dic₄₇	C₂⁴	1	6	yes	—
77	C₂×C₄×D₄₇	—	2	48	no	—
78	C₂×C₄×D₄₇	C₂⁴	1	16	yes	—
79	C₄⋊C₄×C₄₇	C₂⁴	1	64	yes	—
80	C₄²×C₄₇	C₂⁴	1	16	yes	—
81	(C₂×D₄₇)⋊C₄	C₂⁴	1	32	yes	—
82	(C₂×D₄₇)⋊C₄	C₂⁴	1	64	yes	—
83	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
84	C₄⋊Dic₄₇	C₂⁴	1	64	yes	—
85	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
86	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
87	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
88	C₄×Dic₄₇	C₂⁴	1	16	yes	—
89	C₄×Dic₄₇	C₂⁴	1	64	yes	—
90	C₂³×C₄₇	C₂×#41	1	4	yes	—
91	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
92	C₂³×C₄₇	C₂⁵	1	3	yes	—
93	C₂²×D₄₇	—	2	12	no	—
94	C₂²×D₄₇	—	2	24	no	—
95	C₂²×D₄₇	—	4	16	no	—
96	C₂²×D₄₇	C₂⁵	1	4	yes	—
97	Q₈×C₄₇	—	2	64	no	—
98	C₂×C₄×C₄₇	—	2	48	no	—
99	C₂×C₄×C₄₇	—	2	24	no	—
100	C₂×C₄×C₄₇	C₂×#41	1	24	yes	—
101	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
102	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
103	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
104	C₂×Dic₄₇	—	2	12	no	—
105	C₂×Dic₄₇	—	2	24	no	—
106	C₂×Dic₄₇	—	2	48	no	—
107	C₂×Dic₄₇	C₂⁵	1	8	yes	—
108	C₂×Dic₄₇	C₂⁵	1	16	yes	—
109	C₂×Dic₄₇	C₂⁵	1	4	yes	—
110	C₄×D₄₇	—	2	32	no	—
111	C₂²×#541	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	#541	1	16	yes	complemented
113	C₂²×C₄₇	C₂²×#41	1	6	yes	—
114	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	12	yes	—
115	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
116	C₂×D₄₇	—	2	4	no	—
117	C₂×D₄₇	—	2	8	no	—
118	C₂×D₄₇	—	4	24	no	—
119	C₄×C₄₇	—	2	16	no	—
120	C₄×C₄₇	—	2	8	no	—
121	C₄×C₄₇	C₂²×#41	1	8	yes	—
122	Dic₄₇	—	2	4	no	—
123	Dic₄₇	—	2	8	no	—
124	Dic₄₇	—	2	16	no	—
125	C₂×#541	—	47	48	no	—
126	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
127	C₂²×#126	—	47	4	no	—
128	C₂²×#125	—	47	1	no	—
129	C₂²×#125	—	47	4	no	—
130	C₂²×C₂²⋊Q₈	—	47	2	no	—
131	C₂²×C₄×Q₈	—	47	2	no	—
132	C₂²×#117	—	47	1	no	—
133	C₂×C₄₇	C₂×#541	1	12	yes	complemented
134	C₂×C₄₇	C₂³×#41	1	1	yes	characteristic
135	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	2	yes	—
136	D₄₇	—	4	4	no	—
137	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
138	C₂⁴×C₄	—	47	1	no	—
139	#541	—	47	256	no	—
140	C₂×C₄⋊Q₈	C₂×D₄₇	1	24	yes	complemented
141	C₂×#126	—	47	96	no	—
142	C₂×#125	—	47	96	no	—
143	C₂×#125	—	47	24	no	—
144	C₂×C₂²⋊Q₈	—	47	48	no	—
145	C₂×C₄×Q₈	—	47	48	no	—
146	C₂×#117	—	47	24	no	—
147	C₂²×C₄⋊C₄	—	47	2	no	—
148	C₂²×C₄⋊C₄	—	47	2	no	—
149	C₂²×C₄⋊C₄	—	47	4	no	—
150	C₂²×C₄⋊C₄	—	47	4	no	—
151	C₂²×C₄⋊C₄	—	47	4	no	—
152	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
153	C₂²×C₂²⋊C₄	—	47	2	no	—
154	C₂²×C₂²⋊C₄	—	47	4	no	—
155	C₂²×C₄²	—	47	1	no	—
156	C₂²×C₄²	—	47	4	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	C₄₇	C₂²×#541	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	—	47	1	no	—
160	C₂²×Q₈	—	2	16	no	—
161	C₂²×Q₈	C₂²×D₄₇	1	24	yes	—
162	C₂³×C₄	—	47	4	no	—
163	C₂³×C₄	—	47	2	no	—
164	C₂³×C₄	—	47	1	no	—
165	C₂³×C₄	—	47	12	no	—
166	C₂³×C₄	—	94	8	no	—
167	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
168	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊Q₈	C₂²×D₄₇	1	64	yes	complemented
171	#126	—	47	256	no	—
172	#125	—	47	256	no	—
173	#125	—	47	64	no	—
174	C₂²⋊Q₈	—	47	128	no	—
175	C₄×Q₈	—	47	128	no	—
176	#117	—	47	64	no	—
177	C₂×C₄⋊C₄	—	47	24	no	—
178	C₂×C₄⋊C₄	—	47	24	no	—
179	C₂×C₄⋊C₄	—	47	48	no	—
180	C₂×C₄⋊C₄	—	47	48	no	—
181	C₂×C₄⋊C₄	—	47	48	no	—
182	C₂×C₄⋊C₄	C₂²×D₄₇	1	48	yes	—
183	C₂×C₂²⋊C₄	—	47	24	no	—
184	C₂×C₂²⋊C₄	—	47	48	no	—
185	C₂×C₄²	—	47	12	no	—
186	C₂×C₄²	—	47	48	no	—
187	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
188	C₂⁴	—	47	6	no	—
189	C₂⁴	—	94	4	no	—
190	C₂⁴	—	94	8	no	—
191	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
192	C₂×Q₈	—	2	96	no	—
193	C₂×Q₈	C₂³×D₄₇	1	32	yes	—
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	47	12	no	—
197	C₂²×C₄	—	47	24	no	—
198	C₂²×C₄	—	47	6	no	—
199	C₂²×C₄	—	47	16	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	—	94	16	no	—
202	C₂²×C₄	—	94	8	no	—
203	C₂²×C₄	—	94	48	no	—
204	C₂²×C₄	#10625352	1	8	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	24	yes	—
206	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
208	C₄⋊C₄	—	47	32	no	—
209	C₄⋊C₄	—	47	64	no	—
210	C₄⋊C₄	—	47	64	no	—
211	C₄⋊C₄	—	47	32	no	—
212	C₄⋊C₄	—	47	64	no	—
213	C₄⋊C₄	C₂³×D₄₇	1	64	yes	—
214	C₂²⋊C₄	—	47	32	no	—
215	C₂²⋊C₄	—	47	64	no	—
216	C₄²	—	47	16	no	—
217	C₄²	—	47	64	no	—
218	C₄²	C₂³×D₄₇	1	16	yes	—
219	C₂³	—	47	4	no	—
220	C₂³	—	94	12	no	—
221	C₂³	—	94	24	no	—
222	C₂³	—	188	16	no	—
223	C₂³	C₂×#10625352	1	4	yes	—
224	C₂³	#13094072	1	8	yes	—
225	C₂³	C₂⁴×D₄₇	1	3	yes	—
226	Q₈	—	2	64	no	—
227	C₂×C₄	—	2	48	no	—
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	47	8	no	—
230	C₂×C₄	—	47	16	no	—
231	C₂×C₄	—	47	4	no	—
232	C₂×C₄	—	94	12	no	—
233	C₂×C₄	—	94	24	no	—
234	C₂×C₄	—	94	48	no	—
235	C₂×C₄	—	94	32	no	—
236	C₂×C₄	C₂×#10625352	1	24	yes	—
237	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
238	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
239	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
240	C₂²	—	94	4	no	—
241	C₂²	—	94	8	no	—
242	C₂²	—	188	24	no	—
243	C₂²	#15689734	1	16	yes	direct product factor
244	C₂²	C₂²×#10625352	1	6	yes	—
245	C₂²	C₂×#13094072	1	12	yes	—
246	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
247	C₄	—	2	16	no	—
248	C₄	—	2	8	no	—
249	C₄	—	94	4	no	—
250	C₄	—	94	8	no	—
251	C₄	—	94	16	no	—
252	C₄	C₂²×#10625352	1	8	yes	—
253	C₂	—	188	4	no	—
254	C₂	C₂×#15689734	1	12	yes	direct product factor
255	C₂	C₂³×#10625352	1	1	yes	characteristic
256	C₂	C₂²×#13094072	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁴×D₄₇	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#541
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689734	C₂	direct (N × H)
3. C₂²×#13094016	C₂	semidirect
4. C₂²×#13094014	C₂	semidirect
6. C₂²×Q₈×Dic₄₇	C₂	semidirect
8. C₂²×C₄⋊Q₈×C₄₇	C₂	semidirect
10. #15689734	C₂²	direct (N × H)
12. C₂×#13094016	C₂²	semidirect
13. C₂×#13094014	C₂²	semidirect
15. C₂×Q₈×Dic₄₇	C₂²	semidirect
27. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
37. #13094016	C₂³	semidirect
38. #13094014	C₂³	semidirect
40. Q₈×Dic₄₇	C₂³	semidirect
56. C₄⋊Q₈×C₄₇	C₂³	semidirect
112. C₂²×C₄₇	#541	semidirect
126. C₂²×C₄⋊Q₈	D₄₇	semidirect
133. C₂×C₄₇	C₂×#541	semidirect
140. C₂×C₄⋊Q₈	C₂×D₄₇	semidirect
158. C₄₇	C₂²×#541	semidirect
170. C₄⋊Q₈	C₂²×D₄₇	semidirect
243. C₂²	#15689734	direct (N × H)
254. C₂	C₂×#15689734	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689734	2
C₂²×#13094019	2
C₂²×#13094016	2
C₂²×#13094014	2
C₂²×#13094000	2
C₂²×Q₈×Dic₄₇	2
C₂²×#13093981	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#541	47

Subgroups — C22×#15689734 #27123095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689734 #27123095