Subgroups — C₂×#20245253 #26369568

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245253	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688924	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688898	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₄₇	—	2	1	no	—
10	C₂×D₄×Dic₄₇	—	2	1	no	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	Q₁₆×Dic₄₇	—	2	2	no	—
13	Q₁₆×Dic₄₇	C₂²	1	4	yes	complemented
14	D₈×Dic₄₇	—	2	2	no	—
15	D₈×Dic₄₇	C₂²	1	4	yes	—
16	C₂×#13093958	C₂²	1	1	yes	characteristic
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093948	—	2	1	no	—
19	C₂×#13093945	—	2	1	no	—
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688949	C₂²	1	4	yes	complemented
22	#15688927	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	—
24	#15688898	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×SD₃₂×C₄₇	C₄	1	8	yes	complemented
28	C₂×SD₃₂×C₄₇	C₂²	1	4	yes	—
29	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₇	—	4	1	no	—
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	Q₈×Dic₄₇	—	2	4	no	—
33	Q₈×Dic₄₇	—	4	2	no	—
34	D₄×Dic₄₇	—	2	4	no	—
35	D₄×Dic₄₇	—	4	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×C₄⋊Dic₄₇	—	4	1	no	—
39	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
40	C₂×C₄×Dic₄₇	—	4	1	no	—
41	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄₇⋊C₁₆	—	2	2	no	—
43	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
44	#13093958	—	2	1	no	—
45	#13093958	—	2	1	no	—
46	#13093958	C₂³	1	2	yes	—
47	#13093958	C₂³	1	2	yes	—
48	#13093948	—	2	4	no	—
49	#13093945	—	2	4	no	—
50	C₈×Dic₄₇	D₄	1	2	yes	—
51	C₈×Dic₄₇	C₂³	1	2	yes	—
52	C₂²×Q₈×C₄₇	—	2	1	no	—
53	C₂²×D₄×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	1	no	—
55	C₂×Q₁₆×C₄₇	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂×D₈×C₄₇	—	2	1	no	—
59	C₂×D₈×C₄₇	—	2	2	no	—
60	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×D₈×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	SD₃₂×C₄₇	C₂×C₄	1	16	yes	complemented
64	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
66	C₂⁴×C₄₇	—	4	1	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	—	2	2	no	—
69	C₂×Q₈×C₄₇	—	4	1	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂×D₄×C₄₇	—	2	4	no	—
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	4	1	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₇	—	4	1	no	—
78	C₂²×Dic₄₇	—	4	1	no	—
79	C₂²×Dic₄₇	—	4	4	no	—
80	C₂²×Dic₄₇	—	8	4	no	—
81	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
82	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
83	Q₁₆×C₄₇	—	2	4	no	—
84	Q₁₆×C₄₇	—	2	2	no	—
85	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	complemented
86	D₈×C₄₇	—	2	4	no	—
87	D₈×C₄₇	—	2	2	no	—
88	D₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊Dic₄₇	—	4	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	4	1	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄⋊Dic₄₇	—	4	4	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₄×Dic₄₇	D₈	1	2	yes	—
101	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
104	C₁₆×C₄₇	C₂²×C₄	1	4	yes	complemented
105	C₄₇⋊C₁₆	—	2	2	no	—
106	C₂³×C₄₇	—	4	2	no	—
107	C₂³×C₄₇	—	4	4	no	—
108	C₂³×C₄₇	—	8	2	no	—
109	C₂³×C₄₇	—	8	2	no	—
110	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₄₇	—	2	4	no	—
112	Q₈×C₄₇	—	4	2	no	—
113	Q₈×C₄₇	—	4	4	no	—
114	D₄×C₄₇	—	2	4	no	—
115	D₄×C₄₇	—	4	2	no	—
116	D₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#135	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	4	2	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	8	8	no	—
128	C₂×Dic₄₇	—	8	2	no	—
129	C₂×Dic₄₇	—	8	2	no	—
130	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
131	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
132	C₈×C₄₇	C₄×D₄	1	2	yes	—
133	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₄₇⋊C₈	—	2	2	no	—
136	C₂×#2175	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	8	2	no	—
139	C₂²×C₄₇	—	8	2	no	—
140	C₂²×C₄₇	—	8	8	no	—
141	C₂²×C₄₇	C₄×D₈	1	2	yes	—
142	C₂²×C₄₇	#509	1	2	yes	—
143	C₂²×C₄₇	#510	1	2	yes	—
144	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₄₇	—	4	4	no	—
146	C₄×C₄₇	C₄×D₈	1	2	yes	—
147	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
149	Dic₄₇	—	2	2	no	—
150	Dic₄₇	—	2	2	no	—
151	Dic₄₇	—	8	2	no	—
152	Dic₄₇	—	8	2	no	—
153	C₂²×SD₃₂	D₄₇	1	1	yes	characteristic
154	C₂×C₄×Q₁₆	—	47	1	no	—
155	C₂×C₄×D₈	—	47	1	no	—
156	#2175	—	47	8	no	—
157	C₂×#366	—	47	1	no	—
158	C₂×#358	—	47	1	no	—
159	C₂×#357	—	47	1	no	—
160	C₂×#346	—	47	1	no	—
161	C₂×C₄₇	—	8	4	no	—
162	C₂×C₄₇	#2175	1	4	yes	complemented
163	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
166	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
167	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄×Q₈	—	94	1	no	—
169	C₂×C₄×D₄	—	94	1	no	—
170	C₂×SD₃₂	Dic₄₇	1	8	yes	complemented
171	C₂×SD₃₂	C₂×D₄₇	1	4	yes	—
172	C₂²×C₁₆	—	47	1	no	—
173	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
174	C₄×Q₁₆	—	47	4	no	—
175	C₄×Q₁₆	—	94	2	no	—
176	C₄×D₈	—	47	4	no	—
177	C₄×D₈	—	94	2	no	—
178	C₂×#107	—	47	1	no	—
179	C₂×#107	—	47	1	no	—
180	C₂×#103	—	94	1	no	—
181	C₂×#102	—	94	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	#366	—	47	4	no	—
184	#358	—	47	4	no	—
185	#357	—	47	4	no	—
186	#346	—	47	4	no	—
187	C₄₇	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂³×C₄	—	188	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	complemented
194	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
198	C₂×D₈	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₈	—	47	1	no	—
200	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₄×Q₈	—	94	4	no	—
202	C₄×Q₈	—	188	2	no	—
203	C₄×D₄	—	94	4	no	—
204	C₄×D₄	—	188	2	no	—
205	C₂×C₄⋊C₄	—	94	1	no	—
206	C₂×C₄⋊C₄	—	94	1	no	—
207	C₂×C₄⋊C₄	—	188	1	no	—
208	C₂×C₂²⋊C₄	—	188	1	no	—
209	C₂×C₄²	—	47	1	no	—
210	C₂×C₄²	—	188	1	no	—
211	SD₃₂	C₂×Dic₄₇	1	16	yes	complemented
212	C₂×C₁₆	—	47	2	no	—
213	C₂×C₁₆	—	94	2	no	—
214	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	complemented
215	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
216	#107	—	47	2	no	—
217	#107	—	47	2	no	—
218	#107	—	94	1	no	—
219	#107	—	94	1	no	—
220	#103	—	94	4	no	—
221	#102	—	94	4	no	—
222	C₄×C₈	—	47	2	no	—
223	C₄×C₈	—	47	2	no	—
224	C₂⁴	—	4	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	47	1	no	—
235	C₂²×C₄	—	47	1	no	—
236	C₂²×C₄	—	376	4	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	—	188	1	no	—
239	C₂²×C₄	—	188	1	no	—
240	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂²×Dic₄₇	1	4	yes	complemented
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂²×Dic₄₇	1	4	yes	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
250	C₂×C₈	#10625352	1	2	yes	—
251	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
252	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	188	4	no	—
256	C₄⋊C₄	—	188	1	no	—
257	C₄⋊C₄	—	188	1	no	—
258	C₂²⋊C₄	—	188	4	no	—
259	C₄²	—	47	2	no	—
260	C₄²	—	47	2	no	—
261	C₄²	—	188	4	no	—
262	C₁₆	—	94	2	no	—
263	C₁₆	C₂²×Dic₄₇	1	4	yes	complemented
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	8	2	no	—
267	C₂³	—	8	2	no	—
268	C₂³	D₈×D₄₇	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	47	2	no	—
278	C₂×C₄	—	47	2	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	376	8	no	—
282	C₂×C₄	—	376	2	no	—
283	C₂×C₄	—	376	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	2	no	—
286	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
288	C₂×C₄	#13094043	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
290	C₈	—	94	2	no	—
291	C₈	D₄×Dic₄₇	1	2	yes	—
292	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₈	C₂×#10625352	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	8	no	—
298	C₂²	D₈×Dic₄₇	1	2	yes	—
299	C₂²	#15689560	1	2	yes	—
300	C₂²	#15689562	1	2	yes	—
301	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	94	2	no	—
304	C₄	—	94	2	no	—
305	C₄	—	376	2	no	—
306	C₄	—	376	2	no	—
307	C₄	D₈×Dic₄₇	1	2	yes	—
308	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
309	C₄	C₂×#13094043	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	#20245253	1	4	yes	direct product factor
312	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic
313	C₂	C₂×#15689560	1	1	yes	characteristic
314	C₂	C₂×#15689562	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×SD₃₂×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×SD₃₂	D₄₇
hypocenter	C₄₇	C₂×#2175
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×Dic₄₇	C₂	semidirect
3. #20245253	C₂	direct (N × H)
4. C₂×#15688949	C₂	semidirect
5. C₂×#15688927	C₂	semidirect
7. C₂×#15688898	C₂	semidirect
13. Q₁₆×Dic₄₇	C₂²	semidirect
21. #15688949	C₂²	semidirect
22. #15688927	C₂²	semidirect
24. #15688898	C₂²	semidirect
27. C₂×SD₃₂×C₄₇	C₄	semidirect
56. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
63. SD₃₂×C₄₇	C₂×C₄	semidirect
64. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
85. Q₁₆×C₄₇	C₂²×C₄	semidirect
104. C₁₆×C₄₇	C₂²×C₄	semidirect
162. C₂×C₄₇	#2175	semidirect
170. C₂×SD₃₂	Dic₄₇	semidirect
187. C₄₇	C₂×#2175	semidirect
193. C₂×Q₁₆	C₂×Dic₄₇	semidirect
211. SD₃₂	C₂×Dic₄₇	semidirect
214. C₂×C₁₆	C₂×Dic₄₇	semidirect
243. Q₁₆	C₂²×Dic₄₇	semidirect
263. C₁₆	C₂²×Dic₄₇	semidirect
311. C₂	#20245253	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×Q₁₆×Dic₄₇	2
C₂×D₈×Dic₄₇	2
#20245253	2
C₂×#15688949	2
C₂×#15688927	2
C₂×#15688924	2
C₂×#15688898	2
C₂²×SD₃₂×C₄₇	2
C₂×#2175	47

Subgroups — C2×#20245253 #26369568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245253 #26369568