Subgroups — C₂×#20245249 #26369564

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	C₂²×#13093968	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20245249	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688898	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×#13093989	—	2	1	no	—
11	C₂×D₄×Dic₄₇	—	2	1	no	—
12	C₂×#13093968	C₄	1	8	yes	complemented
13	C₂×#13093968	C₂²	1	4	yes	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#15689217	—	2	2	no	—
16	#15689217	C₂²	1	4	yes	complemented
17	D₈×Dic₄₇	—	2	2	no	—
18	D₈×Dic₄₇	C₂²	1	4	yes	—
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093947	—	2	1	no	—
21	C₂×#13093945	—	2	1	no	—
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	#15688948	C₂²	1	4	yes	complemented
24	#15688928	C₂²	1	4	yes	complemented
25	#15688898	C₂²	1	4	yes	complemented
26	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#357	C₂²	1	4	yes	—
30	C₂²×Dic₉₄	—	2	1	no	—
31	C₂³×Dic₄₇	—	4	1	no	—
32	C₂×Dic₁₈₈	—	2	1	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	complemented
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
37	#13093989	—	2	4	no	—
38	#13093989	—	4	2	no	—
39	D₄×Dic₄₇	—	2	4	no	—
40	D₄×Dic₄₇	—	4	2	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×Dic₄₇⋊C₄	—	4	1	no	—
43	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
44	C₂×C₄×Dic₄₇	—	4	1	no	—
45	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
46	#13093968	C₂×C₄	1	16	yes	complemented
47	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
49	#13093958	—	2	1	no	—
50	#13093958	C₂³	1	2	yes	—
51	#13093947	—	2	4	no	—
52	#13093945	—	2	4	no	—
53	C₈×Dic₄₇	D₄	1	2	yes	—
54	C₈×Dic₄₇	C₂³	1	2	yes	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	C₂×D₈×C₄₇	—	2	1	no	—
57	C₂×D₈×C₄₇	—	2	2	no	—
58	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×D₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
62	C₂×C₁₆×C₄₇	—	2	2	no	—
63	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
64	#107×C₄₇	—	2	1	no	—
65	#107×C₄₇	C₂³	1	2	yes	—
66	C₂⁴×C₄₇	—	4	1	no	—
67	C₂×D₄×C₄₇	—	2	4	no	—
68	C₂×D₄×C₄₇	—	2	2	no	—
69	C₂×D₄×C₄₇	—	4	1	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	1	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	4	1	no	—
74	C₂²×Dic₄₇	—	4	1	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	8	4	no	—
77	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
78	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
79	C₂×Dic₉₄	—	2	4	no	—
80	C₂×Dic₉₄	—	2	2	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	D₈×C₄₇	—	2	4	no	—
84	D₈×C₄₇	—	2	2	no	—
85	D₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	4	1	no	—
92	C₂²⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	4	1	no	—
95	Dic₄₇⋊C₄	—	4	4	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₄×Dic₄₇	D₈	1	2	yes	—
98	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
101	Dic₁₈₈	—	2	4	no	—
102	Dic₁₈₈	—	2	2	no	—
103	Dic₁₈₈	C₂²×C₄	1	4	yes	complemented
104	C₁₆×C₄₇	—	2	2	no	—
105	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	complemented
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	D₄×C₄₇	—	2	4	no	—
112	D₄×C₄₇	—	4	2	no	—
113	D₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#135	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	C₂×Dic₄₇	—	8	8	no	—
126	C₂×Dic₄₇	—	8	2	no	—
127	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
128	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
129	Dic₉₄	—	2	4	no	—
130	Dic₉₄	—	4	2	no	—
131	Dic₉₄	—	4	4	no	—
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	—	8	2	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	—	4	4	no	—
152	Dic₄₇	—	8	2	no	—
153	C₂²×SD₃₂	—	47	1	no	—
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	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
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	—	47	8	no	—
171	C₂×SD₃₂	—	47	4	no	—
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	C₂×D₄₇	1	1	yes	characteristic
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	C₂×D₄₇	1	4	yes	—
186	#346	—	47	4	no	—
187	C₄₇	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	94	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂³×C₄	—	188	1	no	—
191	C₂×Q₁₆	—	47	4	no	—
192	C₂×Q₁₆	—	47	2	no	—
193	C₂×Q₁₆	—	94	1	no	—
194	C₂×Q₁₆	—	94	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₄×D₄₇	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	—	2	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	—	47	16	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	47	4	no	—
214	C₂×C₁₆	—	47	2	no	—
215	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
216	#107	—	2	1	no	—
217	#107	—	47	2	no	—
218	#107	—	94	1	no	—
219	#107	C₂²×D₄₇	1	2	yes	—
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	—	94	4	no	—
226	C₂×Q₈	—	94	2	no	—
227	C₂×Q₈	—	188	1	no	—
228	C₂×Q₈	—	188	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	—	47	4	no	—
242	Q₁₆	—	94	4	no	—
243	Q₁₆	—	94	2	no	—
244	D₈	—	2	4	no	—
245	D₈	—	2	2	no	—
246	D₈	C₂×C₄×D₄₇	1	4	yes	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₈	#10625352	1	2	yes	—
252	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
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	C₄²	—	47	2	no	—
260	C₄²	—	47	2	no	—
261	C₄²	—	188	4	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	47	4	no	—
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	—	94	4	no	—
270	Q₈	—	188	2	no	—
271	Q₈	—	188	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	—	8	2	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	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
287	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
288	C₂×C₄	#13094045	1	2	yes	—
289	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
290	C₈	—	94	2	no	—
291	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
292	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
293	C₈	C₂×#10625352	1	1	yes	characteristic
294	C₂²	—	4	4	no	—
295	C₂²	—	8	8	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	#15689203	1	2	yes	—
299	C₂²	#15689557	1	2	yes	—
300	C₂²	#15689562	1	2	yes	—
301	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
302	C₄	—	8	2	no	—
303	C₄	—	94	2	no	—
304	C₄	—	94	2	no	—
305	C₄	—	376	2	no	—
306	C₄	—	188	4	no	—
307	C₄	#15689203	1	2	yes	—
308	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
309	C₄	C₂×#13094045	1	1	yes	characteristic
310	C₂	—	8	4	no	—
311	C₂	#20245249	1	4	yes	direct product factor
312	C₂	C₂×#15689203	1	1	yes	characteristic
313	C₂	C₂×#15689557	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	C₂×C₄₇×#357	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	C₂×#2175
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. C₂×#15689217	C₂	semidirect
4. #20245249	C₂	direct (N × H)
5. C₂×#15688948	C₂	semidirect
6. C₂×#15688928	C₂	semidirect
7. C₂×#15688898	C₂	semidirect
12. C₂×#13093968	C₄	semidirect
16. #15689217	C₂²	semidirect
23. #15688948	C₂²	semidirect
24. #15688928	C₂²	semidirect
25. #15688898	C₂²	semidirect
34. C₂×Dic₁₈₈	C₂×C₄	semidirect
46. #13093968	C₂×C₄	semidirect
47. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
103. Dic₁₈₈	C₂²×C₄	semidirect
105. C₄₇⋊C₁₆	C₂²×C₄	semidirect
162. C₂×C₄₇	#2175	semidirect
187. C₄₇	C₂×#2175	semidirect
311. C₂	#20245249	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×#13093968	2
C₂×#15689217	2
C₂×D₈×Dic₄₇	2
#20245249	2
C₂×#15688948	2
C₂×#15688928	2
C₂×#15688898	2
C₂×C₄₇×#357	2
C₂×#2175	47

Subgroups — C2×#20245249 #26369564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245249 #26369564