Subgroups — C₂×#20245590 #26371069

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
2	#20245590	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688948	C₂	1	2	yes	maximal, complemented
4	C₂×#15688926	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
8	C₂×#13094003	—	2	1	no	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15689364	—	2	2	no	—
11	#15689364	C₂²	1	4	yes	—
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×#13093949	—	2	1	no	—
14	#15688948	C₂²	1	8	yes	complemented
15	#15688926	C₂²	1	8	yes	complemented
16	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
19	C₂×#107×C₄₇	—	2	2	no	—
20	C₂×#107×C₄₇	—	2	1	no	—
21	C₂×#107×C₄₇	C₂²	1	4	yes	—
22	C₂×#107×C₄₇	C₂²	1	2	yes	—
23	C₂²×Dic₉₄	—	2	1	no	—
24	C₂×Dic₁₈₈	—	2	4	no	—
25	C₂×Dic₁₈₈	—	2	1	no	—
26	C₂×Dic₁₈₈	C₂³	1	2	yes	—
27	#13094003	—	2	4	no	—
28	#13094003	—	4	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×Dic₄₇⋊C₄	—	4	1	no	—
31	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
32	C₂×C₄₇⋊C₁₆	—	2	4	no	—
33	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
34	#13093958	—	2	1	no	—
35	#13093958	C₂³	1	2	yes	—
36	#13093949	—	2	4	no	—
37	C₂³×C₄×C₄₇	—	4	1	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	2	4	no	—
40	C₂²×C₈×C₄₇	D₄	1	2	yes	—
41	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
47	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
48	#107×C₄₇	—	2	4	no	—
49	#107×C₄₇	—	2	2	no	—
50	#107×C₄₇	C₂³	1	4	yes	—
51	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	8	4	no	—
57	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	D₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	4	1	no	—
62	C₂²×Dic₄₇	—	4	1	no	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×Dic₉₄	—	4	2	no	—
66	C₂×Dic₉₄	—	4	1	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	4	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₂²⋊Dic₄₇	—	4	4	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	C₄⋊Dic₄₇	—	4	1	no	—
80	Dic₄₇⋊C₄	—	4	4	no	—
81	Dic₁₈₈	—	2	4	no	—
82	Dic₁₈₈	—	4	2	no	—
83	C₄₇⋊C₁₆	—	2	4	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	Q₃₂	1	4	yes	—
87	C₂³×C₄₇	C₂×D₈	1	2	yes	—
88	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	8	8	no	—
95	C₂×C₄×C₄₇	—	8	4	no	—
96	C₂×C₄×C₄₇	#123	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#137	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×Dic₄₇	—	4	2	no	—
102	C₂×Dic₄₇	—	4	2	no	—
103	C₂×Dic₄₇	—	8	2	no	—
104	C₂×Dic₄₇	—	8	2	no	—
105	Dic₉₄	—	4	2	no	—
106	Dic₉₄	—	4	2	no	—
107	Dic₉₄	—	8	2	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	#123	1	2	yes	—
110	C₈×C₄₇	C₂×#41	1	2	yes	—
111	C₂×#2236	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	8	no	—
115	C₂²×C₄₇	#480	1	2	yes	—
116	C₂²×C₄₇	C₂×Q₃₂	1	4	yes	—
117	C₂²×C₄₇	C₂×Q₃₂	1	2	yes	—
118	C₂²×C₄₇	#509	1	2	yes	—
119	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	8	4	no	—
122	C₄×C₄₇	#480	1	2	yes	—
123	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
124	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
125	Dic₄₇	—	8	2	no	—
126	Dic₄₇	—	8	2	no	—
127	C₂×#467	—	47	1	no	—
128	C₂²×#107	D₄₇	1	1	yes	characteristic
129	#2236	—	47	8	no	—
130	C₂×#366	—	47	2	no	—
131	C₂×#358	—	47	2	no	—
132	C₂×C₂²⋊C₁₆	—	47	1	no	—
133	C₂×C₄₇	—	2	4	no	—
134	C₂×C₄₇	#2236	1	4	yes	complemented
135	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
136	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
138	C₂²×Q₁₆	—	47	1	no	—
139	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₂²⋊Q₈	—	94	1	no	—
141	C₂²×C₄⋊C₄	—	2	1	no	—
142	C₂²×C₁₆	—	47	2	no	—
143	#467	—	47	4	no	—
144	#467	—	94	2	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×#107	—	2	2	no	—
147	C₂×#107	—	47	1	no	—
148	C₂×#107	C₂×D₄₇	1	2	yes	—
149	C₂×#107	C₂×D₄₇	1	4	yes	—
150	C₂×#103	—	94	1	no	—
151	#366	—	47	8	no	—
152	#358	—	47	8	no	—
153	C₂²⋊C₁₆	—	47	4	no	—
154	C₄₇	C₂×#2236	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	94	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	47	2	no	—
159	C₂×Q₁₆	—	94	4	no	—
160	C₂×Q₁₆	—	94	1	no	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
164	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
165	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
166	C₂²⋊Q₈	—	94	4	no	—
167	C₂²⋊Q₈	—	188	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	4	2	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	94	1	no	—
173	C₂×C₄⋊C₄	—	188	1	no	—
174	C₂×C₂²⋊C₄	—	188	1	no	—
175	C₂×C₁₆	—	47	4	no	—
176	C₂×C₁₆	—	94	4	no	—
177	#107	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	47	2	no	—
180	#107	—	94	1	no	—
181	#107	C₂²×D₄₇	1	4	yes	—
182	#103	—	94	4	no	—
183	C₂⁴	#10625334	1	1	yes	characteristic
184	C₂×Q₈	—	94	2	no	—
185	C₂×Q₈	—	188	2	no	—
186	C₂×Q₈	—	188	2	no	—
187	C₂×Q₈	—	188	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	8	4	no	—
193	C₂²×C₄	—	188	1	no	—
194	C₂²×C₄	—	188	1	no	—
195	C₂²×C₄	#10625334	1	2	yes	—
196	C₂²×C₄	#10625334	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	Q₁₆	—	94	4	no	—
200	Q₁₆	—	188	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	#10625350	1	4	yes	—
205	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	94	2	no	—
212	C₄⋊C₄	—	188	4	no	—
213	C₄⋊C₄	—	188	1	no	—
214	C₂²⋊C₄	—	188	4	no	—
215	C₁₆	—	94	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	#13093970	1	4	yes	—
219	C₂³	C₂×#10625334	1	2	yes	—
220	C₂³	C₂×#10625334	1	1	yes	characteristic
221	Q₈	—	376	2	no	—
222	Q₈	—	188	2	no	—
223	Q₈	—	188	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	8	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	376	2	no	—
232	C₂×C₄	—	376	2	no	—
233	C₂×C₄	—	188	2	no	—
234	C₂×C₄	—	188	2	no	—
235	C₂×C₄	#13094007	1	2	yes	—
236	C₂×C₄	C₂×#10625334	1	2	yes	—
237	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
238	C₂×C₄	#13094055	1	2	yes	—
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	#13094007	1	2	yes	—
242	C₈	C₂×#10625350	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	2	no	—
246	C₂²	#15689436	1	2	yes	—
247	C₂²	C₂×#13093970	1	4	yes	—
248	C₂²	C₂×#13093970	1	2	yes	—
249	C₂²	#15689559	1	2	yes	—
250	C₂²	C₂²×#10625334	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	4	no	—
253	C₄	—	376	2	no	—
254	C₄	—	376	2	no	—
255	C₄	#15689436	1	2	yes	—
256	C₄	C₂×#13094007	1	1	yes	characteristic
257	C₄	C₂×#13094055	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#20245590	1	4	yes	direct product factor
260	C₂	C₂×#15689436	1	1	yes	characteristic
261	C₂	C₂²×#13093970	1	1	yes	characteristic
262	C₂	C₂×#15689559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	C₂×#2236
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. #20245590	C₂	direct (N × H)
3. C₂×#15688948	C₂	semidirect
4. C₂×#15688926	C₂	semidirect
14. #15688948	C₂²	semidirect
15. #15688926	C₂²	semidirect
134. C₂×C₄₇	#2236	semidirect
154. C₄₇	C₂×#2236	semidirect
259. C₂	#20245590	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×#15689364	2
#20245590	2
C₂×#15688948	2
C₂×#15688926	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂²×#107×C₄₇	2
C₂×#2236	47

Subgroups — C2×#20245590 #26371069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245590 #26371069