Subgroups — C₂×#20242869 #26198875

Order 12032.

850 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×#15689320	C₂	1	2	yes	maximal, complemented
2	C₂×#15689253	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242869	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688867	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625335	—	2	2	no	—
8	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
9	#15689320	—	2	4	no	—
10	#15689320	C₂²	1	8	yes	complemented
11	#15689253	—	2	2	no	—
12	#15689253	C₂²	1	4	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₈	—	2	1	no	—
16	C₂×#13093945	—	2	2	no	—
17	C₂×#13093934	—	2	1	no	—
18	#15688874	C₂²	1	4	yes	complemented
19	#15688867	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₄₇	—	2	1	no	—
21	C₂²×D₈×C₄₇	—	2	1	no	—
22	C₂×#127×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#493	—	2	2	no	—
24	C₄₇×#493	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
27	C₂×#10625335	—	2	8	no	—
28	C₂×#10625335	—	4	8	no	—
29	C₂²×C₄₇⋊C₈	—	2	1	no	—
30	C₂²×C₄₇⋊C₈	—	2	2	no	—
31	C₄×Dic₉₄	—	2	1	no	—
32	C₄×Dic₉₄	—	2	1	no	—
33	C₄×Dic₉₄	D₄	1	2	yes	—
34	C₄×Dic₉₄	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	C₂×Dic₄₇⋊C₄	—	2	1	no	—
37	C₂×C₄×Dic₄₇	—	2	1	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
42	Dic₄₇⋊C₈	—	2	4	no	—
43	#13093945	—	2	8	no	—
44	#13093934	—	2	4	no	—
45	C₂²×D₄×C₄₇	—	2	2	no	—
46	C₂²×D₄×C₄₇	—	2	2	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂×D₈×C₄₇	—	2	4	no	—
49	C₂×D₈×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	—	4	4	no	—
51	C₂×D₈×C₄₇	—	4	4	no	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	#127×C₄₇	—	2	2	no	—
55	#127×C₄₇	—	2	2	no	—
56	#127×C₄₇	—	2	2	no	—
57	#127×C₄₇	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₇	D₄	1	2	yes	—
60	C₄×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	—	4	2	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂×D₄×C₄₇	—	4	4	no	—
69	C₂×D₄×C₄₇	—	4	4	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	—	4	4	no	—
72	C₂×D₄×C₄₇	—	8	4	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	1	no	—
77	C₂²×Dic₄₇	—	2	1	no	—
78	C₂×Dic₉₄	—	2	4	no	—
79	C₂×Dic₉₄	—	2	1	no	—
80	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
81	D₈×C₄₇	—	4	8	no	—
82	D₈×C₄₇	—	4	8	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	#41	1	1	yes	characteristic
90	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
91	#10625335	—	4	16	no	—
92	C₄⋊Dic₄₇	—	2	1	no	—
93	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
94	Dic₄₇⋊C₄	—	2	2	no	—
95	Dic₄₇⋊C₄	—	2	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂×C₄₇⋊C₈	—	4	4	no	—
102	C₂³×C₄₇	—	4	4	no	—
103	C₂³×C₄₇	—	8	4	no	—
104	C₂³×C₄₇	—	8	4	no	—
105	C₂³×C₄₇	—	8	4	no	—
106	C₂³×C₄₇	#121	1	1	yes	characteristic
107	D₄×C₄₇	—	4	4	no	—
108	D₄×C₄₇	—	4	4	no	—
109	D₄×C₄₇	—	4	4	no	—
110	D₄×C₄₇	—	4	4	no	—
111	D₄×C₄₇	—	8	4	no	—
112	D₄×C₄₇	—	8	4	no	—
113	D₄×C₄₇	—	8	4	no	—
114	D₄×C₄₇	—	8	4	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	2	no	—
126	C₂×Dic₄₇	—	2	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	Dic₉₄	—	2	4	no	—
130	Dic₉₄	—	4	2	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	C₂×SD₁₆	1	4	yes	complemented
133	C₄₇⋊C₈	—	4	2	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	C₂×#1688	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	8	4	no	—
137	C₂²×C₄₇	—	8	4	no	—
138	C₂²×C₄₇	—	8	4	no	—
139	C₂²×C₄₇	—	16	8	no	—
140	C₂²×C₄₇	#460	1	2	yes	—
141	C₂²×C₄₇	#469	1	2	yes	—
142	C₂²×C₄₇	#472	1	2	yes	—
143	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	#460	1	2	yes	—
147	C₄×C₄₇	#469	1	2	yes	—
148	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
149	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	Dic₄₇	—	4	2	no	—
152	Dic₄₇	—	4	2	no	—
153	C₂×#493	D₄₇	1	1	yes	characteristic
154	C₂×#460	—	47	2	no	—
155	C₂×#446	—	47	1	no	—
156	#1688	—	47	8	no	—
157	C₂×#334	—	47	1	no	—
158	C₂×#331	—	47	2	no	—
159	C₂×C₄₇	—	16	4	no	—
160	C₂×C₄₇	#1688	1	4	yes	complemented
161	C₂×C₄₇	C₂×#460	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#469	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
164	C₂²×SD₁₆	—	94	2	no	—
165	C₂²×D₈	—	2	1	no	—
166	C₂²×D₈	—	2	1	no	—
167	C₂×#127	C₂×D₄₇	1	2	yes	—
168	C₂×C₄×Q₈	—	47	1	no	—
169	#493	—	2	2	no	—
170	#493	C₂×D₄₇	1	4	yes	—
171	#460	—	47	8	no	—
172	#460	—	94	4	no	—
173	#446	—	47	4	no	—
174	#446	—	94	2	no	—
175	C₂×C₄⋊C₈	—	47	1	no	—
176	C₂×C₄⋊C₈	—	47	2	no	—
177	C₂×C₄⋊C₈	—	94	1	no	—
178	C₂×#102	—	94	2	no	—
179	C₂×#97	—	94	1	no	—
180	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
181	#334	—	47	4	no	—
182	#331	—	47	8	no	—
183	C₄₇	C₂×#1688	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	47	1	no	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	—	2	2	no	—
187	C₂²×D₄	—	2	2	no	—
188	C₂×SD₁₆	—	94	8	no	—
189	C₂×SD₁₆	—	188	8	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂×D₈	—	4	4	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	—	94	1	no	—
196	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	2	2	no	—
201	#127	C₂²×D₄₇	1	4	yes	—
202	C₄×Q₈	—	47	2	no	—
203	C₄×Q₈	—	47	2	no	—
204	C₄×Q₈	—	94	1	no	—
205	C₄×Q₈	—	94	1	no	—
206	C₂×C₄⋊C₄	—	47	1	no	—
207	C₂×C₄⋊C₄	—	94	1	no	—
208	C₂×C₄²	—	94	1	no	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₄⋊C₈	—	47	2	no	—
211	C₄⋊C₈	—	47	4	no	—
212	C₄⋊C₈	—	94	4	no	—
213	C₄⋊C₈	—	94	2	no	—
214	C₄⋊C₈	—	94	1	no	—
215	#102	—	94	8	no	—
216	#97	—	94	4	no	—
217	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
218	C₄×C₈	C₂²×D₄₇	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	C₂×Q₈	—	47	2	no	—
221	C₂×Q₈	—	94	1	no	—
222	C₂×Q₈	—	94	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	94	1	no	—
235	C₂²×C₄	—	94	1	no	—
236	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	#10625352	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	SD₁₆	—	188	16	no	—
240	D₈	—	4	8	no	—
241	D₈	—	4	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	—	188	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	#10625335	1	4	yes	complemented
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₄⋊C₄	—	47	2	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	94	1	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄²	—	94	2	no	—
255	C₄²	—	94	2	no	—
256	C₄²	D₄×D₄₇	1	1	yes	characteristic
257	C₄²	#10625352	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₄²	C₂³×D₄₇	1	1	yes	characteristic
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	#13093995	1	1	yes	characteristic
265	Q₈	—	94	4	no	—
266	Q₈	—	188	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	#13093995	1	1	yes	characteristic
281	C₂×C₄	#13093995	1	1	yes	characteristic
282	C₂×C₄	#13093995	1	1	yes	characteristic
283	C₂×C₄	C₂×#10625335	1	2	yes	—
284	C₂×C₄	#13094048	1	4	yes	—
285	C₂×C₄	#13094050	1	2	yes	—
286	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	188	4	no	—
291	C₈	—	188	2	no	—
292	C₈	C₂×#10625335	1	4	yes	complemented
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	#15689320	1	2	yes	—
298	C₂²	#15689373	1	2	yes	—
299	C₂²	#15689390	1	2	yes	—
300	C₂²	C₂×#13093995	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	188	2	no	—
304	C₄	—	188	2	no	—
305	C₄	#15689320	1	2	yes	—
306	C₄	#15689373	1	2	yes	—
307	C₄	C₂²×#10625335	1	1	yes	characteristic
308	C₄	C₂×#13094048	1	2	yes	—
309	C₄	C₂×#13094050	1	1	yes	characteristic
310	C₂	—	16	4	no	—
311	C₂	#20242869	1	4	yes	direct product factor
312	C₂	C₂×#15689320	1	1	yes	characteristic
313	C₂	C₂×#15689373	1	1	yes	characteristic
314	C₂	C₂×#15689390	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	#121
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#1688
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689320	C₂	semidirect
2. C₂×#15689253	C₂	semidirect
3. #20242869	C₂	direct (N × H)
4. C₂×#15688874	C₂	semidirect
5. C₂×#15688867	C₂	semidirect
10. #15689320	C₂²	semidirect
12. #15689253	C₂²	semidirect
18. #15688874	C₂²	semidirect
19. #15688867	C₂²	semidirect
84. C₂×C₈×C₄₇	SD₁₆	semidirect
132. C₈×C₄₇	C₂×SD₁₆	semidirect
160. C₂×C₄₇	#1688	semidirect
183. C₄₇	C₂×#1688	semidirect
247. C₂×C₈	#10625335	semidirect
292. C₈	C₂×#10625335	semidirect
311. C₂	#20242869	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×#15689320	2
C₂×#15689253	2
#20242869	2
C₂×#15688874	2
C₂×#15688867	2
C₂×C₄₇×#493	2
C₂×#1688	47

Subgroups — C2×#20242869 #26198875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242869 #26198875