Subgroups — C₂×#20253897 #27085878

Order 12032.

1942 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689678	C₂	1	2	yes	maximal
2	#20253897	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689526	C₂	1	2	yes	maximal, complemented
4	C₂×#15689523	C₂	1	4	yes	maximal, complemented
5	C₂×#15689521	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15689158	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
10	#15689678	—	2	8	no	—
11	#15689678	C₂²	1	16	yes	—
12	C₂×#13094024	C₂²	1	2	yes	—
13	C₂×#13094015	C₂²	1	4	yes	—
14	C₂×#13094004	—	2	2	no	—
15	C₂×#13094000	—	2	2	no	—
16	C₂×Q₈×Dic₄₇	—	2	2	no	—
17	C₂×#13093989	—	2	2	no	—
18	C₂×#13093982	C₂²	1	4	yes	—
19	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
20	#15689526	C₂²	1	16	yes	complemented
21	#15689523	C₂²	1	32	yes	complemented
22	#15689521	C₂²	1	8	yes	complemented
23	#15689158	C₂²	1	32	yes	—
24	C₂×#13093957	C₂²	1	8	yes	—
25	C₂×#13093956	C₂²	1	4	yes	—
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
28	C₄×C₈×D₄₇	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#498	C₂²	1	8	yes	complemented
31	C₂×#106×C₄₇	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₉₄	—	2	2	no	—
34	C₂²×C₄×D₄₇	D₄	1	2	yes	—
35	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂×C₈×D₄₇	Q₈	1	16	yes	complemented
38	C₂×C₈×D₄₇	C₂³	1	8	yes	—
39	#13094024	—	2	4	no	—
40	#13094024	C₂³	1	8	yes	—
41	#13094015	—	2	8	no	—
42	#13094015	C₂³	1	16	yes	—
43	#13094004	—	2	16	no	—
44	#13094000	—	2	16	no	—
45	Q₈×Dic₄₇	—	2	16	no	—
46	#13093989	—	2	16	no	—
47	#13093982	—	2	8	no	—
48	#13093982	C₂³	1	16	yes	—
49	C₂×C₄⋊Dic₄₇	—	2	2	no	—
50	C₂×C₄⋊Dic₄₇	—	2	4	no	—
51	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
52	C₂×Dic₄₇⋊C₄	—	2	4	no	—
53	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
54	C₂×C₄×Dic₄₇	—	2	4	no	—
55	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
56	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
57	C₄²×D₄₇	D₄	1	4	yes	—
58	C₄²×D₄₇	C₂³	1	4	yes	—
59	#13093957	C₂³	1	32	yes	—
60	#13093956	C₂³	1	16	yes	—
61	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
62	C₈×Dic₄₇	C₂³	1	8	yes	—
63	C₂²×Q₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₇	—	2	4	no	—
66	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	#106×C₄₇	C₂³	1	16	yes	—
71	C₄×C₈×C₄₇	C₂³	1	4	yes	—
72	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₄₇	—	2	8	no	—
74	C₂×Q₈×C₄₇	—	4	8	no	—
75	C₂²×C₄×C₄₇	—	2	4	no	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	4	no	—
79	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
80	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₄₇	—	2	8	no	—
82	C₂×C₄×D₄₇	C₂×D₄	1	8	yes	—
83	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₂×Q₈	1	8	yes	—
85	C₂×C₄×D₄₇	C₂⁴	1	2	yes	—
86	C₂×Dic₉₄	—	2	8	no	—
87	C₂×Dic₉₄	—	4	8	no	—
88	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
90	C₄⋊C₄×C₄₇	—	2	8	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
93	C₄²×C₄₇	C₂×D₄	1	2	yes	—
94	C₄²×C₄₇	C₂⁴	1	2	yes	—
95	(C₂×D₄₇)⋊C₄	—	2	16	no	—
96	C₄⋊Dic₄₇	—	2	8	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	16	no	—
99	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
100	Dic₄₇⋊C₄	—	2	16	no	—
101	C₄×Dic₄₇	—	2	16	no	—
102	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
103	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
104	C₄×Dic₄₇	C₂⁴	1	2	yes	—
105	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
106	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
107	C₈×D₄₇	C₂×Q₈	1	32	yes	complemented
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	4	no	—
110	C₂²×D₄₇	C₄⋊Q₈	1	4	yes	—
111	C₂²×D₄₇	C₂²×D₄	1	2	yes	—
112	Q₈×C₄₇	—	4	16	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	4	8	no	—
115	C₂×C₄×C₄₇	#135	1	8	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
119	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	8	no	—
123	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
124	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
125	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
126	C₄×D₄₇	—	2	16	no	—
127	C₄×D₄₇	C₄⋊Q₈	1	8	yes	—
128	C₄×D₄₇	C₂²×Q₈	1	8	yes	—
129	Dic₉₄	—	4	16	no	—
130	C₈×C₄₇	C₂²×Q₈	1	8	yes	—
131	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
132	C₂×#3160	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
134	C₂²×C₄₇	C₂×#135	1	4	yes	—
135	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
136	C₂×D₄₇	—	2	8	no	—
137	C₂×D₄₇	—	2	4	no	—
138	C₂×D₄₇	C₂×C₄⋊Q₈	1	4	yes	—
139	C₄×C₄₇	—	4	8	no	—
140	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
141	C₄×C₄₇	C₂×#135	1	8	yes	—
142	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	4	8	no	—
145	Dic₄₇	C₂×C₄⋊Q₈	1	4	yes	—
146	C₂×#533	—	47	2	no	—
147	#3160	—	47	16	no	—
148	C₂×#500	—	47	2	no	—
149	C₂×#499	—	47	4	no	—
150	C₂×#498	—	47	1	no	—
151	C₂×#498	D₄₇	1	1	yes	characteristic, complemented
152	C₂×#427	—	47	4	no	—
153	C₂²×C₄×C₈	—	47	1	no	—
154	C₂×C₄₇	#3160	1	4	yes	complemented
155	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
156	C₂×C₄₇	C₂²×#135	1	2	yes	—
157	D₄₇	—	2	4	no	—
158	C₂³×C₈	—	47	2	no	—
159	#533	—	47	16	no	—
160	#533	—	94	8	no	—
161	C₂×C₄⋊Q₈	—	47	2	no	—
162	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
163	C₂×#125	—	47	4	no	—
164	C₂×C₂²⋊Q₈	—	94	2	no	—
165	C₂×C₂²⋊Q₈	—	94	2	no	—
166	C₂×C₄×Q₈	—	94	2	no	—
167	C₂×C₄×Q₈	—	94	2	no	—
168	C₂×#117	—	47	4	no	—
169	C₂²×C₄²	—	47	1	no	—
170	#500	—	47	16	no	—
171	#499	—	47	32	no	—
172	#498	—	47	8	no	—
173	#498	C₂×D₄₇	1	8	yes	complemented
174	#427	—	47	32	no	—
175	C₂×#107	—	47	8	no	—
176	C₂×#106	—	47	4	no	—
177	C₂×#106	C₂×D₄₇	1	4	yes	—
178	C₂×C₄×C₈	—	47	2	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	C₂×C₄×C₈	—	47	8	no	—
181	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₄₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	94	2	no	—
185	C₂³×C₄	—	47	1	no	—
186	C₂³×C₄	—	47	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	16	no	—
189	C₂²×C₈	—	47	8	no	—
190	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	47	8	no	—
193	C₄⋊Q₈	—	94	4	no	—
194	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
195	#125	—	47	16	no	—
196	#125	—	94	8	no	—
197	C₂²⋊Q₈	—	94	16	no	—
198	C₂²⋊Q₈	—	94	16	no	—
199	C₄×Q₈	—	94	16	no	—
200	C₄×Q₈	—	94	16	no	—
201	#117	—	47	16	no	—
202	#117	—	94	8	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	47	4	no	—
205	C₂×C₄⋊C₄	—	94	4	no	—
206	C₂×C₄⋊C₄	—	94	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
209	C₂×C₂²⋊C₄	—	94	4	no	—
210	C₂×C₄²	—	47	2	no	—
211	C₂×C₄²	—	47	1	no	—
212	C₂×C₄²	—	47	4	no	—
213	C₂×C₄²	—	47	4	no	—
214	C₂×C₄²	—	94	4	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	#107	—	47	32	no	—
217	#106	—	47	16	no	—
218	#106	C₂²×D₄₇	1	16	yes	—
219	C₄×C₈	—	47	8	no	—
220	C₄×C₈	—	47	4	no	—
221	C₄×C₈	C₂²×D₄₇	1	4	yes	—
222	C₂⁴	—	47	1	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	94	8	no	—
226	C₂×Q₈	—	188	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	47	1	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	8	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	—	47	2	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	—	94	8	no	—
235	C₂²×C₄	—	94	4	no	—
236	C₂²×C₄	D₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
238	C₂×C₈	—	47	8	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	32	no	—
241	C₂×C₈	Q₈×D₄₇	1	8	yes	—
242	C₂×C₈	C₂³×D₄₇	1	4	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	47	8	no	—
246	C₄⋊C₄	—	94	16	no	—
247	C₄⋊C₄	—	94	16	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	—	94	8	no	—
250	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
251	C₂²⋊C₄	—	94	16	no	—
252	C₄²	—	47	8	no	—
253	C₄²	—	47	2	no	—
254	C₄²	—	47	2	no	—
255	C₄²	—	94	16	no	—
256	C₄²	D₄×D₄₇	1	2	yes	—
257	C₄²	C₂³×D₄₇	1	2	yes	—
258	C₂³	—	47	4	no	—
259	C₂³	—	47	2	no	—
260	C₂³	—	94	4	no	—
261	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
262	Q₈	—	4	16	no	—
263	Q₈	—	188	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	47	4	no	—
267	C₂×C₄	—	47	2	no	—
268	C₂×C₄	—	47	8	no	—
269	C₂×C₄	—	47	8	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	16	no	—
273	C₂×C₄	—	94	8	no	—
274	C₂×C₄	—	188	8	no	—
275	C₂×C₄	#13094045	1	8	yes	—
276	C₂×C₄	C₂×D₄×D₄₇	1	4	yes	—
277	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
279	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
280	C₈	—	47	8	no	—
281	C₈	C₂×Q₈×D₄₇	1	8	yes	—
282	C₂²	—	47	4	no	—
283	C₂²	—	94	4	no	—
284	C₂²	—	94	8	no	—
285	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
286	C₂²	C₂×#13094045	1	4	yes	—
287	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
288	C₄	—	4	8	no	—
289	C₄	—	47	4	no	—
290	C₄	—	94	4	no	—
291	C₄	—	188	8	no	—
292	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
293	C₄	C₂×#13094045	1	8	yes	—
294	C₄	C₂²×Q₈×D₄₇	1	2	yes	—
295	C₂	—	94	4	no	—
296	C₂	#20253897	1	4	yes	direct product factor
297	C₂	C₂×C₄⋊Q₈×D₄₇	1	1	yes	characteristic
298	C₂	C₂²×#13094045	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#498	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#498	D₄₇
hypocenter	C₄₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. #20253897	C₂	direct (N × H)
3. C₂×#15689526	C₂	semidirect
4. C₂×#15689523	C₂	semidirect
5. C₂×#15689521	C₂	semidirect
8. C₂×C₄₇×#498	C₂	semidirect
20. #15689526	C₂²	semidirect
21. #15689523	C₂²	semidirect
22. #15689521	C₂²	semidirect
30. C₄₇×#498	C₂²	semidirect
37. C₂×C₈×D₄₇	Q₈	semidirect
107. C₈×D₄₇	C₂×Q₈	semidirect
151. C₂×#498	D₄₇	semidirect
154. C₂×C₄₇	#3160	semidirect
173. #498	C₂×D₄₇	semidirect
182. C₄₇	C₂×#3160	semidirect
296. C₂	#20253897	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×#15689678	2
#20253897	2
C₂×#15689526	2
C₂×#15689523	2
C₂×#15689521	2
C₂×#15689158	2
C₂×C₄×C₈×D₄₇	2
C₂×C₄₇×#498	2
C₂×#3160	47

Subgroups — C2×#20253897 #27085878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253897 #27085878