Subgroups — C₂²×C₄₇×#541 #26102379

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

	subgroup	G/H
center, socle	C₂⁴×C₄₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₄₇
prime radical	C₂⁵×C₄₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#541	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈×C₄₇	C₂	semidirect
4. C₂²×#125×C₄₇	C₂	semidirect
5. C₂²×#125×C₄₇	C₂	semidirect
7. C₂²×C₄×Q₈×C₄₇	C₂	semidirect
11. C₄₇×#541	C₂²	direct (N × H)
12. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
14. C₂×#125×C₄₇	C₂²	semidirect
15. C₂×#125×C₄₇	C₂²	semidirect
17. C₂×C₄×Q₈×C₄₇	C₂²	semidirect
41. C₄⋊Q₈×C₄₇	C₂³	semidirect
43. #125×C₄₇	C₂³	semidirect
44. #125×C₄₇	C₂³	semidirect
46. C₄×Q₈×C₄₇	C₂³	semidirect
111. C₂²×#541	C₄₇	direct (N × H)
115. C₂²×C₄₇	#541	direct (N × H)
125. C₂×#541	C₂×C₄₇	direct (N × H)
126. C₂²×C₄⋊Q₈	C₂×C₄₇	semidirect
128. C₂²×#125	C₂×C₄₇	semidirect
129. C₂²×#125	C₂×C₄₇	semidirect
131. C₂²×C₄×Q₈	C₂×C₄₇	semidirect
134. C₂×C₄₇	C₂×#541	direct (N × H)
139. #541	C₂²×C₄₇	direct (N × H)
140. C₂×C₄⋊Q₈	C₂²×C₄₇	semidirect
142. C₂×#125	C₂²×C₄₇	semidirect
143. C₂×#125	C₂²×C₄₇	semidirect
145. C₂×C₄×Q₈	C₂²×C₄₇	semidirect
158. C₄₇	C₂²×#541	direct (N × H)
170. C₄⋊Q₈	C₂³×C₄₇	semidirect
172. #125	C₂³×C₄₇	semidirect
173. #125	C₂³×C₄₇	semidirect
175. C₄×Q₈	C₂³×C₄₇	semidirect
243. C₂²	C₄₇×#541	direct (N × H)
254. C₂	C₂×C₄₇×#541	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×C₄₇×#541	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#126×C₄₇	2
C₂²×#125×C₄₇	2
C₂²×C₂²⋊Q₈×C₄₇	2
C₂²×C₄×Q₈×C₄₇	2
C₂²×#117×C₄₇	2
C₂²×#541	47

Subgroups — C22×C47×#541 #26102379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₇×#541 #26102379