Subgroups — C₂²×#15689851 #27123321

Order 12032.

7414 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689851	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13094022	C₂	1	2	yes	maximal
4	C₂²×#13094016	C₂	1	2	yes	maximal
5	C₂²×#13094014	C₂	1	4	yes	maximal
6	C₂²×C₄×Dic₉₄	C₂	1	2	yes	maximal
7	C₂²×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13093989	C₂	1	2	yes	maximal
9	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₉₄	C₂²	1	2	yes	—
11	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic
12	#15689851	C₂²	1	256	yes	direct product factor
13	C₂×#13094024	C₂²	1	24	yes	—
14	C₂×#13094022	C₂²	1	48	yes	—
15	C₂×#13094016	C₂²	1	48	yes	—
16	C₂×#13094014	C₂²	1	96	yes	—
17	C₂×C₄×Dic₉₄	C₂²	1	48	yes	—
18	C₂×C₄×Dic₉₄	C₂²	1	24	yes	—
19	C₂×#13093989	C₂²	1	48	yes	—
20	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
24	C₂²×Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
26	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
28	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
29	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	24	yes	—
31	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
34	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	16	no	—
36	C₂²×Dic₉₄	Q₈	1	16	yes	complemented
37	C₂²×Dic₉₄	C₂³	1	24	yes	—
38	C₂²×Dic₉₄	C₂³	1	12	yes	—
39	C₂³×Dic₄₇	C₂³	1	2	yes	—
40	C₂³×Dic₄₇	C₂³	1	4	yes	—
41	C₂³×Dic₄₇	C₂³	1	2	yes	—
42	#13094024	C₂³	1	64	yes	—
43	#13094022	C₂³	1	128	yes	—
44	#13094016	C₂³	1	128	yes	—
45	#13094014	C₂³	1	256	yes	—
46	C₄×Dic₉₄	C₂³	1	128	yes	—
47	C₄×Dic₉₄	C₂³	1	64	yes	—
48	#13093989	C₂³	1	128	yes	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
50	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
51	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
52	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
53	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
54	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
55	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
56	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
57	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
58	C₂²×Q₈×C₄₇	—	2	8	no	—
59	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₄₇	C₂³	1	64	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
67	C₂×C₄⋊C₄×C₄₇	C₂³	1	24	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	24	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
70	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂×Q₈×C₄₇	—	2	48	no	—
72	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
73	C₂²×C₄×C₄₇	—	2	8	no	—
74	C₂²×C₄×C₄₇	—	2	8	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
77	C₂²×C₄×C₄₇	#41	1	8	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
83	C₂²×Dic₄₇	—	2	16	no	—
84	C₂²×Dic₄₇	—	2	8	no	—
85	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
86	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
87	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
88	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
89	C₂×Dic₉₄	—	2	96	no	—
90	C₂×Dic₉₄	C₂×Q₈	1	96	yes	complemented
91	C₂×Dic₉₄	C₂⁴	1	32	yes	—
92	C₂×Dic₉₄	C₂⁴	1	16	yes	—
93	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
94	C₄⋊C₄×C₄₇	C₂⁴	1	32	yes	—
95	C₄²×C₄₇	C₂⁴	1	32	yes	—
96	C₄²×C₄₇	C₂⁴	1	16	yes	—
97	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
98	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
99	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
100	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
101	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
102	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
103	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
104	C₄×Dic₄₇	C₂⁴	1	64	yes	—
105	C₄×Dic₄₇	C₂⁴	1	32	yes	—
106	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
107	C₂³×C₄₇	C₂×#41	1	4	yes	—
108	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
109	C₂³×C₄₇	C₂⁵	1	3	yes	—
110	Q₈×C₄₇	—	2	32	no	—
111	C₂×C₄×C₄₇	—	2	24	no	—
112	C₂×C₄×C₄₇	—	2	24	no	—
113	C₂×C₄×C₄₇	—	2	12	no	—
114	C₂×C₄×C₄₇	C₂²×Q₈	1	24	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	24	yes	—
116	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
117	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
118	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
119	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
120	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
121	C₂×Dic₄₇	—	2	48	no	—
122	C₂×Dic₄₇	—	2	24	no	—
123	C₂×Dic₄₇	C₂²×Q₈	1	48	yes	—
124	C₂×Dic₄₇	C₂⁵	1	8	yes	—
125	C₂×Dic₄₇	C₂⁵	1	16	yes	—
126	C₂×Dic₄₇	C₂⁵	1	8	yes	—
127	Dic₉₄	—	2	64	no	—
128	Dic₉₄	C₂²×Q₈	1	64	yes	complemented
129	C₂²×#557	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	#557	1	16	yes	complemented
131	C₂²×C₄₇	C₂³×Q₈	1	6	yes	—
132	C₂²×C₄₇	C₂²×#41	1	6	yes	—
133	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	6	yes	—
134	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
135	C₄×C₄₇	—	2	8	no	—
136	C₄×C₄₇	—	2	8	no	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	C₂³×Q₈	1	8	yes	—
139	C₄×C₄₇	C₂²×#41	1	8	yes	—
140	Dic₄₇	—	2	16	no	—
141	Dic₄₇	—	2	8	no	—
142	Dic₄₇	C₂³×Q₈	1	16	yes	—
143	C₂×#557	—	47	48	no	—
144	C₂²×C₄⋊Q₈	—	47	1	no	—
145	C₂²×C₄⋊Q₈	—	47	2	no	—
146	C₂²×#125	—	47	2	no	—
147	C₂²×#125	—	47	4	no	—
148	C₂²×C₄×Q₈	—	47	1	no	—
149	C₂²×C₄×Q₈	—	47	2	no	—
150	C₂²×C₄×Q₈	—	47	2	no	—
151	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#557	1	12	yes	complemented
153	C₂×C₄₇	C₂⁴×Q₈	1	1	yes	characteristic
154	C₂×C₄₇	C₂³×#41	1	1	yes	characteristic
155	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
156	C₂³×Q₈	—	47	1	no	—
157	C₂³×Q₈	—	47	2	no	—
158	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
159	#557	—	47	256	no	—
160	C₂×C₄⋊Q₈	—	47	48	no	—
161	C₂×C₄⋊Q₈	—	47	24	no	—
162	C₂×#125	—	47	96	no	—
163	C₂×#125	—	47	48	no	—
164	C₂×C₄×Q₈	—	47	48	no	—
165	C₂×C₄×Q₈	—	47	48	no	—
166	C₂×C₄×Q₈	—	47	24	no	—
167	C₂×C₄×Q₈	C₂×D₄₇	1	24	yes	—
168	C₂²×C₄⋊C₄	—	47	2	no	—
169	C₂²×C₄⋊C₄	—	47	4	no	—
170	C₂²×C₄⋊C₄	—	47	4	no	—
171	C₂²×C₄⋊C₄	—	47	1	no	—
172	C₂²×C₄⋊C₄	—	47	2	no	—
173	C₂²×C₄⋊C₄	—	47	4	no	—
174	C₂²×C₄⋊C₄	—	47	2	no	—
175	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
177	C₂²×C₄²	—	47	4	no	—
178	C₂²×C₄²	—	47	2	no	—
179	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	C₄₇	C₂²×#557	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	—	2	8	no	—
183	C₂²×Q₈	—	47	16	no	—
184	C₂²×Q₈	—	47	12	no	—
185	C₂²×Q₈	—	47	24	no	—
186	C₂²×Q₈	—	94	16	no	—
187	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
188	C₂³×C₄	—	47	2	no	—
189	C₂³×C₄	—	47	2	no	—
190	C₂³×C₄	—	47	4	no	—
191	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
193	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₄⋊Q₈	—	47	128	no	—
197	C₄⋊Q₈	—	47	64	no	—
198	#125	—	47	256	no	—
199	#125	—	47	128	no	—
200	C₄×Q₈	—	47	128	no	—
201	C₄×Q₈	—	47	128	no	—
202	C₄×Q₈	—	47	64	no	—
203	C₄×Q₈	C₂²×D₄₇	1	64	yes	—
204	C₂×C₄⋊C₄	—	47	24	no	—
205	C₂×C₄⋊C₄	—	47	48	no	—
206	C₂×C₄⋊C₄	—	47	12	no	—
207	C₂×C₄⋊C₄	—	47	48	no	—
208	C₂×C₄⋊C₄	—	47	24	no	—
209	C₂×C₄⋊C₄	—	47	48	no	—
210	C₂×C₄⋊C₄	—	47	24	no	—
211	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
212	C₂×C₄⋊C₄	C₂²×D₄₇	1	24	yes	—
213	C₂×C₄²	—	47	48	no	—
214	C₂×C₄²	—	47	24	no	—
215	C₂×C₄²	C₂²×D₄₇	1	24	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
217	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
218	C₂×Q₈	—	2	48	no	—
219	C₂×Q₈	—	47	96	no	—
220	C₂×Q₈	—	47	16	no	—
221	C₂×Q₈	—	47	32	no	—
222	C₂×Q₈	—	94	96	no	—
223	C₂×Q₈	C₂³×D₄₇	1	16	yes	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	47	16	no	—
228	C₂²×C₄	—	47	12	no	—
229	C₂²×C₄	—	47	12	no	—
230	C₂²×C₄	—	47	24	no	—
231	C₂²×C₄	—	94	16	no	—
232	C₂²×C₄	—	94	8	no	—
233	C₂²×C₄	#10625350	1	8	yes	—
234	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
235	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
236	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
239	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
240	C₄⋊C₄	—	47	32	no	—
241	C₄⋊C₄	—	47	64	no	—
242	C₄⋊C₄	—	47	64	no	—
243	C₄⋊C₄	—	47	64	no	—
244	C₄⋊C₄	—	47	16	no	—
245	C₄⋊C₄	—	47	32	no	—
246	C₄⋊C₄	—	47	32	no	—
247	C₄⋊C₄	C₂³×D₄₇	1	16	yes	—
248	C₄⋊C₄	C₂³×D₄₇	1	32	yes	—
249	C₄²	—	47	64	no	—
250	C₄²	—	47	32	no	—
251	C₄²	C₂³×D₄₇	1	32	yes	—
252	C₄²	C₂³×D₄₇	1	16	yes	—
253	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
254	C₂³	C₂×#10625350	1	4	yes	—
255	C₂³	#13094071	1	4	yes	—
256	C₂³	C₂⁴×D₄₇	1	3	yes	—
257	Q₈	—	2	32	no	—
258	Q₈	—	47	64	no	—
259	Q₈	—	94	64	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	24	no	—
262	C₂×C₄	—	2	12	no	—
263	C₂×C₄	—	47	48	no	—
264	C₂×C₄	—	47	8	no	—
265	C₂×C₄	—	47	8	no	—
266	C₂×C₄	—	47	16	no	—
267	C₂×C₄	—	94	48	no	—
268	C₂×C₄	—	94	24	no	—
269	C₂×C₄	C₂×Q₈×D₄₇	1	24	yes	—
270	C₂×C₄	C₂×#10625350	1	24	yes	—
271	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
272	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
273	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
274	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
275	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
276	C₂²	#15689851	1	16	yes	direct product factor
277	C₂²	C₂²×Q₈×D₄₇	1	6	yes	—
278	C₂²	C₂²×#10625350	1	6	yes	—
279	C₂²	C₂×#13094071	1	6	yes	—
280	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
281	C₄	—	2	8	no	—
282	C₄	—	2	8	no	—
283	C₄	—	2	4	no	—
284	C₄	—	47	16	no	—
285	C₄	—	94	16	no	—
286	C₄	—	94	8	no	—
287	C₄	C₂²×Q₈×D₄₇	1	8	yes	—
288	C₄	C₂²×#10625350	1	8	yes	—
289	C₂	C₂×#15689851	1	12	yes	direct product factor
290	C₂	C₂³×Q₈×D₄₇	1	1	yes	characteristic
291	C₂	C₂³×#10625350	1	1	yes	characteristic
292	C₂	C₂²×#13094071	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂²×#557
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×#15689851	C₂	direct (N × H)
12. #15689851	C₂²	direct (N × H)
36. C₂²×Dic₉₄	Q₈	semidirect
90. C₂×Dic₉₄	C₂×Q₈	semidirect
128. Dic₉₄	C₂²×Q₈	semidirect
130. C₂²×C₄₇	#557	semidirect
152. C₂×C₄₇	C₂×#557	semidirect
181. C₄₇	C₂²×#557	semidirect
276. C₂²	#15689851	direct (N × H)
289. C₂	C₂×#15689851	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×#15689851	2
C₂²×#13094024	2
C₂²×#13094022	2
C₂²×#13094016	2
C₂²×#13094014	2
C₂²×C₄×Dic₉₄	2
C₂²×#13093989	2
C₂²×C₄×Q₈×C₄₇	2
C₂²×#557	47

Subgroups — C22×#15689851 #27123321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689851 #27123321