Subgroups — C₂²×#15689378 #27083402

Order 12032.

2878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10625324	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13094000	C₂	1	2	yes	maximal
3	C₂×#15689378	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13093948	C₂	1	2	yes	maximal, complemented
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂²×#10625324	—	2	8	no	—
9	C₂²×#10625324	C₂²	1	12	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#13094000	—	2	12	no	—
12	C₂×#13094000	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
15	C₂²×(C₂×D₄₇)⋊C₄	—	2	2	no	—
16	#15689378	C₂²	1	64	yes	direct product factor
17	C₂×#13093956	C₂²	1	12	yes	complemented
18	C₂×#13093948	C₂²	1	24	yes	complemented
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₄₇	—	2	8	no	—
21	C₂²×Q₁₆×C₄₇	C₂²	1	12	yes	complemented
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×D₄₇	D₄	1	1	yes	characteristic
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	4	no	—
27	C₂²×C₄×D₄₇	—	2	4	no	—
28	C₂²×C₄×D₄₇	D₄	1	6	yes	—
29	C₂²×C₄×D₄₇	C₂³	1	6	yes	—
30	C₂×#10625324	—	2	48	no	—
31	C₂×#10625324	C₂³	1	16	yes	—
32	C₂²×C₄₇⋊C₈	—	2	4	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
34	#13094000	—	2	48	no	—
35	#13094000	C₂³	1	32	yes	—
36	C₂×C₄⋊Dic₄₇	—	2	6	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
38	C₂×Dic₄₇⋊C₄	—	2	24	no	—
39	C₂×(C₂×D₄₇)⋊C₄	—	2	24	no	—
40	#13093956	C₂³	1	16	yes	complemented
41	#13093948	C₂³	1	32	yes	complemented
42	C₂²×Q₈×C₄₇	—	2	6	no	—
43	C₂²×Q₈×C₄₇	—	2	8	no	—
44	C₂²×Q₈×C₄₇	—	2	8	no	—
45	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
46	C₂³×C₄×C₄₇	—	2	2	no	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₄₇	—	2	48	no	—
49	C₂×Q₁₆×C₄₇	C₂³	1	16	yes	complemented
50	C₂²×C₈×C₄₇	D₄	1	8	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂³×D₄₇	—	2	4	no	—
54	C₂³×D₄₇	—	2	8	no	—
55	C₂³×D₄₇	C₂×D₄	1	6	yes	—
56	C₂×Q₈×C₄₇	—	2	12	no	—
57	C₂×Q₈×C₄₇	—	2	24	no	—
58	C₂×Q₈×C₄₇	—	2	24	no	—
59	C₂×Q₈×C₄₇	—	4	24	no	—
60	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₄₇	—	2	12	no	—
62	C₂²×C₄×C₄₇	—	4	8	no	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	12	no	—
69	C₂²×Dic₄₇	—	4	8	no	—
70	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
71	C₂×C₄×D₄₇	—	2	24	no	—
72	C₂×C₄×D₄₇	—	2	12	no	—
73	C₂×C₄×D₄₇	—	2	12	no	—
74	C₂×C₄×D₄₇	C₂×D₄	1	12	yes	—
75	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
76	Q₁₆×C₄₇	—	2	32	no	—
77	C₂×C₈×C₄₇	C₂×D₄	1	24	yes	—
78	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
79	(C₂×D₄₇)⋊C₄	—	2	32	no	—
80	C₄⋊Dic₄₇	—	2	12	no	—
81	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
82	Dic₄₇⋊C₄	—	2	32	no	—
83	C₂×C₄₇⋊C₈	—	2	12	no	—
84	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
85	#10625324	—	2	32	no	—
86	C₂³×C₄₇	#121	1	4	yes	—
87	C₂³×C₄₇	#137	1	8	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
89	C₂²×D₄₇	—	2	12	no	—
90	C₂²×D₄₇	—	2	24	no	—
91	C₂²×D₄₇	—	4	16	no	—
92	C₂²×D₄₇	C₂²×D₄	1	4	yes	—
93	Q₈×C₄₇	—	2	8	no	—
94	Q₈×C₄₇	—	2	8	no	—
95	Q₈×C₄₇	—	4	24	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	4	24	no	—
98	C₂×C₄×C₄₇	#121	1	12	yes	—
99	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
102	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
103	C₂×Dic₄₇	—	2	12	no	—
104	C₂×Dic₄₇	—	2	8	no	—
105	C₂×Dic₄₇	—	4	24	no	—
106	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
107	C₄×D₄₇	—	2	24	no	—
108	C₄×D₄₇	—	2	4	no	—
109	C₄×D₄₇	—	2	4	no	—
110	C₈×C₄₇	C₂²×D₄	1	8	yes	—
111	C₄₇⋊C₈	—	2	4	no	—
112	C₂²×#470	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	#470	1	16	yes	complemented
114	C₂²×C₄₇	C₂×#121	1	6	yes	—
115	C₂²×C₄₇	C₂×#137	1	12	yes	—
116	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₂×D₄₇	—	2	4	no	—
118	C₂×D₄₇	—	2	8	no	—
119	C₂×D₄₇	—	4	24	no	—
120	C₄×C₄₇	—	4	8	no	—
121	C₄×C₄₇	C₂×#121	1	6	yes	—
122	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
123	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
124	Dic₄₇	—	2	4	no	—
125	Dic₄₇	—	4	8	no	—
126	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
127	C₂³×M₁₆	—	47	1	no	—
128	C₂²×C₂²⋊Q₈	—	47	2	no	—
129	C₂×#470	—	47	24	no	—
130	C₂²×#106	—	47	1	no	—
131	C₂²×#103	—	47	2	no	—
132	C₂×C₄₇	C₂×#470	1	12	yes	complemented
133	C₂×C₄₇	C₂²×#121	1	1	yes	characteristic
134	C₂×C₄₇	C₂²×#137	1	2	yes	—
135	D₄₇	—	4	4	no	—
136	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
137	C₂⁴×C₄	—	47	1	no	—
138	C₂²×Q₁₆	—	2	8	no	—
139	C₂²×Q₁₆	C₂×D₄₇	1	12	yes	complemented
140	C₂²×M₁₆	—	47	12	no	—
141	C₂²×M₁₆	—	94	8	no	—
142	C₂³×C₈	—	47	1	no	—
143	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	47	24	no	—
145	C₂×C₂²⋊Q₈	—	94	12	no	—
146	C₂²×C₄⋊C₄	—	47	2	no	—
147	C₂²×C₄⋊C₄	—	94	2	no	—
148	C₂²×C₂²⋊C₄	—	94	2	no	—
149	#470	—	47	64	no	—
150	C₂×#106	—	47	12	no	—
151	C₂×#103	—	47	24	no	—
152	C₄₇	C₂²×#470	1	1	yes	characteristic, complemented, Sylow
153	C₂⁵	—	47	1	no	—
154	C₂²×Q₈	—	2	6	no	—
155	C₂²×Q₈	—	2	8	no	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂³×C₄	—	47	1	no	—
160	C₂³×C₄	—	47	6	no	—
161	C₂³×C₄	—	47	6	no	—
162	C₂³×C₄	—	94	4	no	—
163	C₂³×C₄	—	94	4	no	—
164	C₂³×C₄	—	94	2	no	—
165	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×Q₁₆	—	2	48	no	—
167	C₂×Q₁₆	C₂²×D₄₇	1	16	yes	complemented
168	C₂×M₁₆	—	47	16	no	—
169	C₂×M₁₆	—	94	48	no	—
170	C₂²×C₈	—	47	6	no	—
171	C₂²×C₈	—	94	4	no	—
172	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
174	C₂²⋊Q₈	—	47	32	no	—
175	C₂²⋊Q₈	—	94	48	no	—
176	C₂×C₄⋊C₄	—	47	12	no	—
177	C₂×C₄⋊C₄	—	94	24	no	—
178	C₂×C₄⋊C₄	—	94	6	no	—
179	C₂×C₂²⋊C₄	—	94	24	no	—
180	#106	—	47	16	no	—
181	#103	—	47	32	no	—
182	C₂⁴	—	47	6	no	—
183	C₂⁴	—	94	4	no	—
184	C₂⁴	—	94	8	no	—
185	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
186	C₂×Q₈	—	2	12	no	—
187	C₂×Q₈	—	2	24	no	—
188	C₂×Q₈	—	2	24	no	—
189	C₂×Q₈	—	4	24	no	—
190	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
191	C₂²×C₄	—	2	12	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	—	47	6	no	—
194	C₂²×C₄	—	47	12	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	94	4	no	—
197	C₂²×C₄	—	94	24	no	—
198	C₂²×C₄	—	94	12	no	—
199	C₂²×C₄	—	94	12	no	—
200	C₂²×C₄	—	94	12	no	—
201	C₂²×C₄	—	188	8	no	—
202	C₂²×C₄	D₄×D₄₇	1	3	yes	—
203	C₂²×C₄	#10625352	1	4	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
206	Q₁₆	—	2	32	no	—
207	M₁₆	—	94	32	no	—
208	C₂×C₈	—	47	4	no	—
209	C₂×C₈	—	94	12	no	—
210	C₂×C₈	C₂×C₂²⋊D₄₇	1	24	yes	—
211	C₂×C₈	C₂³×D₄₇	1	4	yes	—
212	C₄⋊C₄	—	47	8	no	—
213	C₄⋊C₄	—	94	32	no	—
214	C₄⋊C₄	—	94	12	no	—
215	C₂²⋊C₄	—	94	32	no	—
216	C₂³	—	47	4	no	—
217	C₂³	—	94	12	no	—
218	C₂³	—	94	24	no	—
219	C₂³	—	188	16	no	—
220	C₂³	#13093995	1	4	yes	—
221	C₂³	#13094056	1	8	yes	—
222	C₂³	C₂×D₄×D₄₇	1	3	yes	—
223	Q₈	—	2	8	no	—
224	Q₈	—	2	8	no	—
225	Q₈	—	4	24	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	4	24	no	—
228	C₂×C₄	—	47	4	no	—
229	C₂×C₄	—	94	12	no	—
230	C₂×C₄	—	94	24	no	—
231	C₂×C₄	—	94	4	no	—
232	C₂×C₄	—	94	4	no	—
233	C₂×C₄	—	94	8	no	—
234	C₂×C₄	—	188	24	no	—
235	C₂×C₄	#13093995	1	12	yes	—
236	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
237	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
238	C₂×C₄	C₂×#10625352	1	6	yes	—
239	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
240	C₈	—	94	4	no	—
241	C₈	C₂²×C₂²⋊D₄₇	1	8	yes	—
242	C₂²	—	94	8	no	—
243	C₂²	—	94	4	no	—
244	C₂²	—	188	24	no	—
245	C₂²	#15689378	1	16	yes	direct product factor
246	C₂²	C₂×#13093995	1	6	yes	—
247	C₂²	C₂×#13094056	1	12	yes	—
248	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
249	C₄	—	4	8	no	—
250	C₄	—	94	4	no	—
251	C₄	—	188	8	no	—
252	C₄	C₂×#13093995	1	6	yes	—
253	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₄	C₂²×#10625352	1	1	yes	characteristic
255	C₂	—	188	4	no	—
256	C₂	C₂×#15689378	1	12	yes	direct product factor
257	C₂	C₂²×#13093995	1	1	yes	characteristic
258	C₂	C₂²×#13094056	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁴×D₄₇	D₄
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂²×#470
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#15689378	C₂	direct (N × H)
4. C₂²×#13093956	C₂	semidirect
5. C₂²×#13093948	C₂	semidirect
6. C₂³×Q₁₆×C₄₇	C₂	semidirect
16. #15689378	C₂²	direct (N × H)
17. C₂×#13093956	C₂²	semidirect
18. C₂×#13093948	C₂²	semidirect
21. C₂²×Q₁₆×C₄₇	C₂²	semidirect
40. #13093956	C₂³	semidirect
41. #13093948	C₂³	semidirect
49. C₂×Q₁₆×C₄₇	C₂³	semidirect
113. C₂²×C₄₇	#470	semidirect
126. C₂³×Q₁₆	D₄₇	semidirect
132. C₂×C₄₇	C₂×#470	semidirect
139. C₂²×Q₁₆	C₂×D₄₇	semidirect
152. C₄₇	C₂²×#470	semidirect
167. C₂×Q₁₆	C₂²×D₄₇	semidirect
245. C₂²	#15689378	direct (N × H)
256. C₂	C₂×#15689378	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×#10625324	2
C₂²×#13094000	2
C₂×#15689378	2
C₂²×#13093956	2
C₂²×#13093948	2
C₂³×Q₁₆×C₄₇	2
C₂²×#470	47

Subgroups — C22×#15689378 #27083402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689378 #27083402