Subgroups — C₂×#20245322 #26370032

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093970	C₂	1	2	yes	maximal, complemented
2	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
3	#20245322	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688927	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
8	C₂×#13094003	—	2	1	no	—
9	C₂×#13093970	—	2	8	no	—
10	C₂×#13093970	C₂²	1	8	yes	complemented
11	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	#15689364	—	2	2	no	—
13	#15689364	C₂²	1	4	yes	—
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093949	—	2	1	no	—
16	#15688927	C₂²	1	8	yes	complemented
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
18	C₂³×Q₈×C₄₇	—	2	1	no	—
19	C₂²×Q₁₆×C₄₇	—	2	8	no	—
20	C₂²×Q₁₆×C₄₇	—	2	2	no	—
21	C₂²×Q₁₆×C₄₇	—	2	1	no	—
22	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	—	2	1	no	—
26	C₂×Dic₁₈₈	—	2	4	no	—
27	C₂×Dic₁₈₈	—	2	1	no	—
28	C₂×Dic₁₈₈	C₂³	1	2	yes	—
29	#13094003	—	2	4	no	—
30	#13094003	—	4	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×Dic₄₇⋊C₄	—	4	1	no	—
33	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
34	#13093970	—	2	16	no	—
35	C₂×C₄₇⋊C₁₆	—	2	4	no	—
36	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
37	#13093958	—	2	1	no	—
38	#13093958	C₂³	1	2	yes	—
39	#13093949	—	2	4	no	—
40	C₂²×Q₈×C₄₇	—	2	8	no	—
41	C₂²×Q₈×C₄₇	—	2	4	no	—
42	C₂²×Q₈×C₄₇	—	2	2	no	—
43	C₂²×Q₈×C₄₇	—	4	4	no	—
44	C₂²×Q₈×C₄₇	—	4	1	no	—
45	C₂²×Q₈×C₄₇	—	4	2	no	—
46	C₂³×C₄×C₄₇	—	4	1	no	—
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₄₇	—	2	8	no	—
49	C₂×Q₁₆×C₄₇	—	2	16	no	—
50	C₂×Q₁₆×C₄₇	—	2	16	no	—
51	C₂×Q₁₆×C₄₇	—	2	4	no	—
52	C₂×Q₁₆×C₄₇	—	2	2	no	—
53	C₂×Q₁₆×C₄₇	—	2	4	no	—
54	C₂×Q₁₆×C₄₇	D₄	1	8	yes	—
55	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₇	—	2	4	no	—
57	C₂²×C₈×C₄₇	D₄	1	2	yes	—
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	16	no	—
63	C₂×Q₈×C₄₇	—	2	8	no	—
64	C₂×Q₈×C₄₇	—	2	4	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂×Q₈×C₄₇	—	4	8	no	—
67	C₂×Q₈×C₄₇	—	4	16	no	—
68	C₂×Q₈×C₄₇	—	4	4	no	—
69	C₂×Q₈×C₄₇	—	4	2	no	—
70	C₂×Q₈×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	4	8	no	—
74	C₂²×C₄×C₄₇	—	4	4	no	—
75	C₂²×C₄×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	D₈	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	4	1	no	—
81	C₂²×Dic₄₇	—	4	1	no	—
82	C₂×Dic₉₄	—	2	2	no	—
83	C₂×Dic₉₄	—	4	2	no	—
84	C₂×Dic₉₄	—	4	2	no	—
85	C₂×Dic₉₄	—	4	1	no	—
86	Q₁₆×C₄₇	—	2	16	no	—
87	Q₁₆×C₄₇	—	2	8	no	—
88	Q₁₆×C₄₇	—	2	4	no	—
89	Q₁₆×C₄₇	C₂×D₄	1	8	yes	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	4	no	—
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	—	4	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	4	1	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	Dic₁₈₈	—	2	4	no	—
102	Dic₁₈₈	—	4	2	no	—
103	C₄₇⋊C₁₆	—	2	4	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	Q₃₂	1	4	yes	complemented
107	C₂³×C₄₇	C₂×D₈	1	2	yes	—
108	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₄₇	—	2	8	no	—
110	Q₈×C₄₇	—	4	16	no	—
111	Q₈×C₄₇	—	4	4	no	—
112	Q₈×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	8	no	—
120	C₂×C₄×C₄₇	#120	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	8	2	no	—
128	C₂×Dic₄₇	—	8	2	no	—
129	Dic₉₄	—	4	2	no	—
130	Dic₉₄	—	4	2	no	—
131	Dic₉₄	—	8	2	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	#120	1	2	yes	—
134	C₈×C₄₇	C₂²×D₄	1	2	yes	—
135	C₂×#2188	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	8	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	#447	1	2	yes	—
140	C₂²×C₄₇	C₂×Q₃₂	1	2	yes	—
141	C₂²×C₄₇	C₂×Q₃₂	1	4	yes	complemented
142	C₂²×C₄₇	#510	1	2	yes	—
143	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	4	8	no	—
146	C₄×C₄₇	#447	1	2	yes	—
147	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
148	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	Dic₄₇	—	8	2	no	—
150	Dic₄₇	—	8	2	no	—
151	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
152	C₂²×Q₃₂	—	47	2	no	—
153	C₂×#467	—	47	1	no	—
154	#2188	—	47	8	no	—
155	C₂×#358	—	47	2	no	—
156	C₂×C₂²⋊C₁₆	—	47	1	no	—
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	#2188	1	4	yes	complemented
159	C₂×C₄₇	C₂×#447	1	1	yes	characteristic
160	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
162	C₂³×Q₈	—	2	1	no	—
163	C₂²×Q₁₆	—	2	8	no	—
164	C₂²×Q₁₆	—	2	1	no	—
165	C₂²×Q₁₆	—	2	2	no	—
166	C₂²×Q₁₆	—	47	1	no	—
167	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
168	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
169	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₂²⋊Q₈	—	94	1	no	—
171	C₂×Q₃₂	—	47	8	no	—
172	C₂×Q₃₂	—	94	8	no	—
173	C₂²×C₁₆	—	47	2	no	—
174	#467	—	47	4	no	—
175	#467	—	94	2	no	—
176	C₂×#107	—	47	1	no	—
177	C₂×#103	—	94	1	no	—
178	#358	—	47	8	no	—
179	C₂²⋊C₁₆	—	47	4	no	—
180	C₄₇	C₂×#2188	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	2	8	no	—
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	4	4	no	—
185	C₂²×Q₈	—	4	2	no	—
186	C₂²×Q₈	—	4	1	no	—
187	C₂²×Q₈	—	94	1	no	—
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×Q₁₆	—	2	8	no	—
191	C₂×Q₁₆	—	2	16	no	—
192	C₂×Q₁₆	—	2	16	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	2	2	no	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×Q₁₆	—	47	2	no	—
197	C₂×Q₁₆	—	94	4	no	—
198	C₂×Q₁₆	—	94	1	no	—
199	C₂×Q₁₆	C₂²⋊D₄₇	1	8	yes	—
200	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
201	C₂²×C₈	—	2	4	no	—
202	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
204	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
205	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²⋊Q₈	—	94	4	no	—
207	C₂²⋊Q₈	—	188	2	no	—
208	C₂×C₄⋊C₄	—	94	1	no	—
209	C₂×C₄⋊C₄	—	188	1	no	—
210	C₂×C₂²⋊C₄	—	188	1	no	—
211	Q₃₂	—	94	16	no	—
212	C₂×C₁₆	—	47	4	no	—
213	C₂×C₁₆	—	94	4	no	—
214	#107	—	47	2	no	—
215	#107	—	94	1	no	—
216	#103	—	94	4	no	—
217	C₂⁴	#10625334	1	1	yes	characteristic
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	4	8	no	—
222	C₂×Q₈	—	4	16	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	94	2	no	—
228	C₂×Q₈	—	188	2	no	—
229	C₂×Q₈	—	188	2	no	—
230	C₂×Q₈	—	188	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	188	1	no	—
237	C₂²×C₄	—	188	1	no	—
238	C₂²×C₄	#10625334	1	2	yes	—
239	C₂²×C₄	#10625334	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	Q₁₆	—	2	16	no	—
243	Q₁₆	—	2	8	no	—
244	Q₁₆	—	2	4	no	—
245	Q₁₆	—	94	4	no	—
246	Q₁₆	—	188	2	no	—
247	Q₁₆	C₂×C₂²⋊D₄₇	1	8	yes	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
254	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	188	4	no	—
257	C₄⋊C₄	—	188	1	no	—
258	C₂²⋊C₄	—	188	4	no	—
259	C₁₆	—	94	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	#13093970	1	4	yes	complemented
263	C₂³	C₂×#10625334	1	2	yes	—
264	C₂³	C₂×#10625334	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	Q₈	—	4	16	no	—
267	Q₈	—	4	8	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	376	2	no	—
270	Q₈	—	188	2	no	—
271	Q₈	—	188	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	16	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	376	2	no	—
280	C₂×C₄	—	376	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	#13093993	1	2	yes	—
284	C₂×C₄	C₂×#10625334	1	2	yes	—
285	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
286	C₂×C₄	#13094050	1	2	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	#13093993	1	2	yes	—
290	C₈	C₂²×C₂²⋊D₄₇	1	2	yes	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	#15689257	1	2	yes	—
295	C₂²	C₂×#13093970	1	2	yes	—
296	C₂²	C₂×#13093970	1	4	yes	complemented
297	C₂²	#15689564	1	2	yes	—
298	C₂²	C₂²×#10625334	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	8	no	—
301	C₄	—	376	2	no	—
302	C₄	—	376	2	no	—
303	C₄	#15689257	1	2	yes	—
304	C₄	C₂×#13093993	1	1	yes	characteristic
305	C₄	C₂×#13094050	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	#20245322	1	4	yes	direct product factor
308	C₂	C₂×#15689257	1	1	yes	characteristic
309	C₂	C₂²×#13093970	1	1	yes	characteristic
310	C₂	C₂×#15689564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2188
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093970	C₂	semidirect
3. #20245322	C₂	direct (N × H)
4. C₂×#15688927	C₂	semidirect
10. C₂×#13093970	C₂²	semidirect
16. #15688927	C₂²	semidirect
106. C₂³×C₄₇	Q₃₂	semidirect
141. C₂²×C₄₇	C₂×Q₃₂	semidirect
158. C₂×C₄₇	#2188	semidirect
180. C₄₇	C₂×#2188	semidirect
262. C₂³	#13093970	semidirect
296. C₂²	C₂×#13093970	semidirect
307. C₂	#20245322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093970	2
C₂×#15689364	2
#20245322	2
C₂×#15688927	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂³×Q₁₆×C₄₇	2
C₂×#2188	47

Subgroups — C2×#20245322 #26370032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245322 #26370032