Subgroups — C₂×#20252490 #27081242

Order 12032.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252490	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689254	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈×C₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×#15689184	C₂	1	4	yes	maximal
5	C₂×#15689176	C₂	1	2	yes	maximal
6	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
7	#15689254	C₂²	1	32	yes	complemented
8	Q₈×C₄₇⋊C₈	C₂²	1	32	yes	complemented
9	#15689184	C₂²	1	32	yes	—
10	#15689176	C₂²	1	16	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
18	C₄₇×#533	C₄	1	8	yes	—
19	C₄₇×#533	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
21	C₂×#125×C₄₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
23	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
24	C₂×#117×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
29	C₄⋊(C₄₇⋊C₈)	C₂³	1	32	yes	—
30	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
31	C₂²⋊(C₄₇⋊C₈)	C₂³	1	16	yes	—
32	#13093935	C₂³	1	16	yes	—
33	C₄×C₄₇⋊C₈	C₂³	1	16	yes	—
34	C₄×C₄₇⋊C₈	C₂³	1	16	yes	—
35	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	—
39	#125×C₄₇	C₂×C₄	1	16	yes	—
40	C₂²⋊Q₈×C₄₇	C₂×C₄	1	32	yes	—
41	C₄×Q₈×C₄₇	C₂×C₄	1	16	yes	—
42	C₄×Q₈×C₄₇	C₂³	1	16	yes	—
43	#117×C₄₇	C₂×C₄	1	8	yes	—
44	#117×C₄₇	C₂³	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	—	2	16	no	—
57	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
69	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
70	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
71	C₂²⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
73	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
76	C₄²×C₄₇	C₂⁴	1	4	yes	—
77	C₄²×C₄₇	C₂⁴	1	8	yes	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	2	8	no	—
80	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
81	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
85	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₄₇	—	2	32	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	16	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	16	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	#131	1	8	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
99	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
100	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
101	C₄₇⋊C₈	—	2	8	no	—
102	C₄₇⋊C₈	—	2	8	no	—
103	C₂×#2986	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	4	4	no	—
107	C₂²×C₄₇	#519	1	2	yes	—
108	C₂²×C₄₇	C₂×#131	1	4	yes	—
109	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	2	16	no	—
113	C₄×C₄₇	#519	1	2	yes	—
114	C₄×C₄₇	C₂×#131	1	8	yes	—
115	C₄×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
116	C₂×#533	D₄₇	1	1	yes	characteristic
117	#2986	—	47	16	no	—
118	C₂×#446	—	47	4	no	—
119	C₂×C₈×Q₈	—	47	4	no	—
120	C₂×#433	—	47	4	no	—
121	C₂×#431	—	47	2	no	—
122	C₂×C₄₇	—	4	2	no	—
123	C₂×C₄₇	#2986	1	4	yes	complemented
124	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
125	C₂×C₄₇	C₂²×#131	1	2	yes	—
126	#533	Dic₄₇	1	8	yes	—
127	#533	C₂×D₄₇	1	8	yes	—
128	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
129	C₂×#125	Dic₄₇	1	2	yes	—
130	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
131	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
132	C₂×#117	C₂×D₄₇	1	2	yes	—
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	#446	—	47	32	no	—
135	C₈×Q₈	—	47	32	no	—
136	#433	—	47	32	no	—
137	#431	—	47	16	no	—
138	C₂×C₄⋊C₈	—	47	2	no	—
139	C₂×C₄⋊C₈	—	47	2	no	—
140	C₂×C₄⋊C₈	—	47	8	no	—
141	C₂×C₂²⋊C₈	—	47	4	no	—
142	C₂×#97	—	47	4	no	—
143	C₂×C₄×C₈	—	47	4	no	—
144	C₂×C₄×C₈	—	47	4	no	—
145	C₄₇	C₂×#2986	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
147	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
148	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
149	C₂²×C₈	—	47	4	no	—
150	C₂²×C₈	—	47	4	no	—
151	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	—
152	#125	C₂×Dic₄₇	1	16	yes	—
153	C₂²⋊Q₈	C₂×Dic₄₇	1	32	yes	—
154	C₄×Q₈	C₂×Dic₄₇	1	16	yes	—
155	C₄×Q₈	C₂²×D₄₇	1	16	yes	—
156	#117	C₂×Dic₄₇	1	8	yes	—
157	#117	C₂²×D₄₇	1	8	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
160	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
161	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
162	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
166	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
168	C₄⋊C₈	—	47	8	no	—
169	C₄⋊C₈	—	47	8	no	—
170	C₄⋊C₈	—	47	32	no	—
171	C₂²⋊C₈	—	47	16	no	—
172	#97	—	47	16	no	—
173	C₄×C₈	—	47	16	no	—
174	C₄×C₈	—	47	16	no	—
175	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
176	C₂×Q₈	—	2	16	no	—
177	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
187	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
188	C₂×C₈	—	47	8	no	—
189	C₂×C₈	—	47	8	no	—
190	C₂×C₈	—	94	8	no	—
191	C₂×C₈	—	94	8	no	—
192	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
193	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
194	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
195	C₂²⋊C₄	C₂²×Dic₄₇	1	16	yes	—
196	C₄²	C₂²×Dic₄₇	1	4	yes	—
197	C₄²	C₂²×Dic₄₇	1	4	yes	—
198	C₄²	C₂²×Dic₄₇	1	4	yes	—
199	C₄²	C₂²×Dic₄₇	1	8	yes	—
200	C₄²	C₂³×D₄₇	1	4	yes	—
201	C₄²	C₂³×D₄₇	1	8	yes	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂³×Dic₄₇	1	2	yes	—
205	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
206	Q₈	—	2	32	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#13094031	1	8	yes	—
214	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
215	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
216	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
217	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
219	C₂×C₄	#13094072	1	4	yes	—
220	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
221	C₈	—	94	8	no	—
222	C₈	—	94	8	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#15689601	1	2	yes	—
227	C₂²	C₂×#13094031	1	4	yes	—
228	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	2	no	—
232	C₄	#15689601	1	2	yes	—
233	C₄	C₂×#13094031	1	8	yes	—
234	C₄	C₂×#13094072	1	2	yes	—
235	C₂	—	4	2	no	—
236	C₂	#20252490	1	4	yes	direct product factor
237	C₂	C₂×#15689601	1	1	yes	characteristic
238	C₂	C₂²×#13094031	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#2986
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20252490	C₂	direct (N × H)
2. C₂×#15689254	C₂	semidirect
3. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
7. #15689254	C₂²	semidirect
8. Q₈×C₄₇⋊C₈	C₂²	semidirect
123. C₂×C₄₇	#2986	semidirect
145. C₄₇	C₂×#2986	semidirect
236. C₂	#20252490	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20252490	2
C₂×#15689254	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×#15689184	2
C₂×#15689176	2
C₂×C₄₇×#533	2
C₂×#2986	47

Subgroups — C2×#20252490 #27081242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20252490 #27081242