Subgroups — C₂×#20251764 #26422289

Order 12032.

2814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094023	C₂	1	2	yes	maximal
2	#20251764	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689027	C₂	1	8	yes	maximal
4	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688988	C₂	1	2	yes	maximal
6	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₉₄	C₂²	1	4	yes	—
9	C₂×#13094023	—	2	16	no	—
10	C₂×#13094023	C₂²	1	16	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	#15689027	C₂²	1	64	yes	—
17	#15688989	C₂²	1	8	yes	—
18	#15688988	C₂²	1	16	yes	—
19	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
20	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
21	C₄²×Dic₄₇	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#382	C₂²	1	8	yes	—
25	C₂×#95×C₄₇	C₂²	1	4	yes	—
26	C₂²×Dic₉₄	—	2	16	no	—
27	C₂²×Dic₉₄	—	2	32	no	—
28	C₂²×Dic₉₄	C₂³	1	16	yes	—
29	C₂³×Dic₄₇	C₂³	1	4	yes	—
30	C₂³×Dic₄₇	C₂³	1	2	yes	—
31	C₂³×Dic₄₇	C₂³	1	2	yes	—
32	#13094023	—	2	64	no	—
33	C₂×C₄⋊Dic₄₇	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	16	no	—
36	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
37	C₂×C₄×Dic₄₇	—	2	16	no	—
38	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
39	C₂×C₄×Dic₄₇	Q₈	1	8	yes	complemented
40	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
43	C₄₇⋊#95	C₂³	1	32	yes	—
44	C₄₇⋊#95	C₂³	1	16	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
50	C₂×C₄²×C₄₇	Q₈	1	8	yes	complemented
51	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
52	#95×C₄₇	C₂³	1	16	yes	—
53	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	16	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
64	C₂²×Dic₄₇	—	2	8	no	—
65	C₂²×Dic₄₇	—	2	8	no	—
66	C₂²×Dic₄₇	—	2	16	no	—
67	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
68	C₂²×Dic₄₇	C₂×Q₈	1	8	yes	—
69	C₂²×Dic₄₇	#41	1	16	yes	—
70	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
71	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
73	C₂×Dic₉₄	—	2	32	no	—
74	C₂×Dic₉₄	—	2	64	no	—
75	C₂×Dic₉₄	—	4	64	no	—
76	C₄⋊C₄×C₄₇	—	2	16	no	—
77	C₄²×C₄₇	C₂×Q₈	1	16	yes	complemented
78	C₄⋊Dic₄₇	—	2	16	no	—
79	Dic₄₇⋊C₄	—	2	32	no	—
80	C₄×Dic₄₇	—	2	32	no	—
81	C₄×Dic₄₇	C₂×D₄	1	16	yes	—
82	C₄×Dic₄₇	C₂×Q₈	1	16	yes	complemented
83	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
84	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
85	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
86	C₂³×C₄₇	C₂×#41	1	8	yes	—
87	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	16	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	16	no	—
92	C₂×C₄×C₄₇	—	4	16	no	—
93	C₂×C₄×C₄₇	C₄⋊Q₈	1	16	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
95	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
97	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
98	C₂×Dic₄₇	—	2	16	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	2	8	no	—
101	C₂×Dic₄₇	—	2	16	no	—
102	C₂×Dic₄₇	—	4	16	no	—
103	C₂×Dic₄₇	#124	1	32	yes	—
104	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
105	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
106	C₂×Dic₄₇	C₂×#41	1	16	yes	—
107	Dic₉₄	—	4	64	no	—
108	C₂×#2862	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×#124	1	8	yes	—
110	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
111	C₂²×C₄₇	#533	1	8	yes	—
112	C₂²×C₄₇	#533	1	8	yes	—
113	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
115	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
116	C₂²×C₄₇	C₂²×#41	1	4	yes	—
117	C₄×C₄₇	—	2	8	no	—
118	C₄×C₄₇	—	4	8	no	—
119	C₄×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
120	Dic₄₇	—	2	8	no	—
121	Dic₄₇	—	4	8	no	—
122	Dic₄₇	C₂×#124	1	16	yes	—
123	C₂²×C₄⋊Q₈	—	47	2	no	—
124	#2862	—	47	16	no	—
125	C₂×#391	—	47	8	no	—
126	C₂×#382	—	47	1	no	—
127	C₂×#382	—	47	2	no	—
128	C₂×#382	D₄₇	1	1	yes	characteristic
129	C₂×C₄³	—	47	1	no	—
130	C₂×C₄₇	#2862	1	8	yes	complemented
131	C₂×C₄₇	C₂²×#124	1	2	yes	—
132	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#533	1	2	yes	—
134	C₂×C₄₇	C₂×#533	1	2	yes	—
135	C₂³×Q₈	—	47	4	no	—
136	C₂×C₄⋊Q₈	—	47	16	no	—
137	C₂×C₄⋊Q₈	—	94	16	no	—
138	C₂²×C₄⋊C₄	—	47	4	no	—
139	C₂²×C₄⋊C₄	—	47	2	no	—
140	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
141	C₂²×C₄²	—	47	1	no	—
142	C₂²×C₄²	—	47	1	no	—
143	C₂²×C₄²	—	47	4	no	—
144	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
145	#391	—	47	64	no	—
146	#382	—	47	16	no	—
147	#382	—	47	8	no	—
148	#382	C₂×D₄₇	1	8	yes	—
149	C₂×#95	—	47	8	no	—
150	C₂×#95	—	47	4	no	—
151	C₂×#95	C₂×D₄₇	1	4	yes	—
152	C₄³	—	47	8	no	—
153	C₄₇	C₂×#2862	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	47	16	no	—
155	C₂²×Q₈	—	94	16	no	—
156	C₂²×Q₈	—	94	32	no	—
157	C₂³×C₄	—	47	2	no	—
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	—	47	4	no	—
160	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
161	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	94	64	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	—	47	16	no	—
166	C₂×C₄⋊C₄	—	47	8	no	—
167	C₂×C₄⋊C₄	—	94	16	no	—
168	C₂×C₄⋊C₄	—	94	8	no	—
169	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
170	C₂×C₄²	—	47	8	no	—
171	C₂×C₄²	—	47	8	no	—
172	C₂×C₄²	—	47	4	no	—
173	C₂×C₄²	—	47	4	no	—
174	C₂×C₄²	—	47	16	no	—
175	C₂×C₄²	—	94	16	no	—
176	C₂×C₄²	Dic₉₄	1	8	yes	complemented
177	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
178	#95	—	47	32	no	—
179	#95	—	47	16	no	—
180	#95	C₂²×D₄₇	1	16	yes	—
181	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
182	C₂×Q₈	—	94	32	no	—
183	C₂×Q₈	—	94	64	no	—
184	C₂×Q₈	—	188	64	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	47	8	no	—
189	C₂²×C₄	—	47	16	no	—
190	C₂²×C₄	—	47	4	no	—
191	C₂²×C₄	—	47	8	no	—
192	C₂²×C₄	—	47	4	no	—
193	C₂²×C₄	—	47	8	no	—
194	C₂²×C₄	—	94	8	no	—
195	C₂²×C₄	—	94	8	no	—
196	C₂²×C₄	—	94	16	no	—
197	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
198	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
199	C₂²×C₄	D₄×D₄₇	1	4	yes	—
200	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
203	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
204	C₄⋊C₄	—	2	16	no	—
205	C₄⋊C₄	—	94	32	no	—
206	C₄⋊C₄	—	94	16	no	—
207	C₄²	—	47	16	no	—
208	C₄²	—	47	16	no	—
209	C₄²	—	94	32	no	—
210	C₄²	C₂×Dic₉₄	1	16	yes	complemented
211	C₂³	C₂×D₄×D₄₇	1	2	yes	—
212	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
213	C₂³	C₂²×Dic₉₄	1	2	yes	—
214	C₂³	#41×D₄₇	1	8	yes	—
215	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
216	Q₈	—	188	64	no	—
217	C₂×C₄	—	2	16	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	4	16	no	—
222	C₂×C₄	—	47	32	no	—
223	C₂×C₄	—	47	8	no	—
224	C₂×C₄	—	47	16	no	—
225	C₂×C₄	—	47	8	no	—
226	C₂×C₄	—	94	16	no	—
227	C₂×C₄	—	94	8	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	—	94	16	no	—
230	C₂×C₄	—	188	16	no	—
231	C₂×C₄	#13094023	1	16	yes	—
232	C₂×C₄	C₂×D₄×D₄₇	1	4	yes	—
233	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
234	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
235	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
236	C₂²	#124×D₄₇	1	8	yes	—
237	C₂²	C₂×#13094023	1	4	yes	—
238	C₂²	#15689676	1	8	yes	—
239	C₂²	#15689679	1	8	yes	—
240	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
241	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
242	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
243	C₂²	C₂×#41×D₄₇	1	4	yes	—
244	C₄	—	2	8	no	—
245	C₄	—	4	8	no	—
246	C₄	—	47	16	no	—
247	C₄	—	94	8	no	—
248	C₄	—	188	8	no	—
249	C₄	C₂×#13094023	1	8	yes	—
250	C₂	#20251764	1	8	yes	direct product factor
251	C₂	C₂×#124×D₄₇	1	2	yes	—
252	C₂	C₂²×#13094023	1	1	yes	characteristic
253	C₂	C₂×#15689676	1	2	yes	—
254	C₂	C₂×#15689679	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	C₂×#2862
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
2. #20251764	C₂	direct (N × H)
39. C₂×C₄×Dic₄₇	Q₈	semidirect
50. C₂×C₄²×C₄₇	Q₈	semidirect
77. C₄²×C₄₇	C₂×Q₈	semidirect
82. C₄×Dic₄₇	C₂×Q₈	semidirect
130. C₂×C₄₇	#2862	semidirect
153. C₄₇	C₂×#2862	semidirect
176. C₂×C₄²	Dic₉₄	semidirect
210. C₄²	C₂×Dic₉₄	semidirect
250. C₂	#20251764	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×#13094023	2
#20251764	2
C₂×#15689027	2
C₂×#15688989	2
C₂×#15688988	2
C₂×C₄²×Dic₄₇	2
C₂×C₄₇×#382	2
C₂×#2862	47

Subgroups — C2×#20251764 #26422289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251764 #26422289