Subgroups — C₂×#20246574 #26378927

Order 12032.

2198 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246574	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689022	C₂	1	4	yes	maximal
3	C₂×#15689009	C₂	1	4	yes	maximal
4	C₂×#15689009	C₂	1	4	yes	maximal
5	C₂×#15688989	C₂	1	2	yes	maximal
6	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
11	#15689022	C₂²	1	32	yes	—
12	#15689009	C₂²	1	32	yes	—
13	#15689009	C₂²	1	32	yes	—
14	#15688989	C₂²	1	16	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
19	C₂×#125×C₄₇	C₄	1	16	yes	complemented
20	C₂×#125×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	C₂³	1	4	yes	—
25	C₂³×Dic₄₇	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	4	8	no	—
28	C₂×C₄⋊Dic₄₇	—	4	4	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
32	C₂×C₄×Dic₄₇	—	4	8	no	—
33	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
34	C₄₇⋊#95	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	16	yes	—
37	C₄₇⋊#95	C₂³	1	16	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
41	#125×C₄₇	C₂×C₄	1	64	yes	complemented
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
48	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	2	8	no	—
65	C₂²×Dic₄₇	—	2	8	no	—
66	C₂²×Dic₄₇	—	2	8	no	—
67	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
68	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
69	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
70	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
72	C₄⋊Dic₄₇	—	4	8	no	—
73	C₄⋊Dic₄₇	—	4	16	no	—
74	C₄⋊Dic₄₇	—	4	8	no	—
75	C₄×Dic₄₇	—	4	16	no	—
76	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
77	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
78	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
79	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
80	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
81	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	32	no	—
84	C₂×C₄×C₄₇	—	2	16	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
94	C₂×Dic₄₇	—	2	8	no	—
95	C₂×Dic₄₇	—	2	8	no	—
96	C₂×Dic₄₇	—	2	8	no	—
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	8	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	4	16	no	—
101	C₂×Dic₄₇	—	4	16	no	—
102	C₂×#2382	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	#519	1	4	yes	—
104	C₂²×C₄₇	#520	1	8	yes	—
105	C₂²×C₄₇	#562	1	4	yes	—
106	C₂²×C₄₇	#563	1	8	yes	—
107	C₂²×C₄₇	#564	1	4	yes	—
108	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
111	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	—	2	16	no	—
115	C₄×C₄₇	—	2	8	no	—
116	Dic₄₇	—	4	8	no	—
117	Dic₄₇	—	4	8	no	—
118	C₂²×#125	D₄₇	1	1	yes	characteristic
119	#2382	—	47	16	no	—
120	C₂×#389	—	47	4	no	—
121	C₂×#387	—	47	4	no	—
122	C₂×#387	—	47	4	no	—
123	C₂×#382	—	47	2	no	—
124	C₂×C₄₇	#2382	1	8	yes	complemented
125	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#520	1	2	yes	—
127	C₂×C₄₇	C₂×#562	1	1	yes	characteristic
128	C₂×C₄₇	C₂×#563	1	2	yes	—
129	C₂×C₄₇	C₂×#564	1	1	yes	characteristic
130	C₂×#125	Dic₄₇	1	16	yes	complemented
131	C₂×#125	C₂×D₄₇	1	8	yes	—
132	C₂²×C₄⋊C₄	—	47	2	no	—
133	C₂²×C₄⋊C₄	—	47	2	no	—
134	C₂²×C₄⋊C₄	—	47	4	no	—
135	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
136	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
137	C₂²×C₄²	—	47	4	no	—
138	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
139	#389	—	47	32	no	—
140	#387	—	47	32	no	—
141	#387	—	47	32	no	—
142	#382	—	47	16	no	—
143	C₂×#95	—	47	4	no	—
144	C₂×#95	—	47	4	no	—
145	C₂×#95	—	47	4	no	—
146	C₂×#95	—	47	4	no	—
147	C₄₇	C₂×#2382	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	47	4	no	—
149	C₂³×C₄	—	47	4	no	—
150	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
151	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
152	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
153	#125	C₂×Dic₄₇	1	64	yes	complemented
154	C₂×C₄⋊C₄	—	47	8	no	—
155	C₂×C₄⋊C₄	—	47	8	no	—
156	C₂×C₄⋊C₄	—	47	16	no	—
157	C₂×C₄⋊C₄	—	188	8	no	—
158	C₂×C₄⋊C₄	—	188	4	no	—
159	C₂×C₄⋊C₄	—	188	4	no	—
160	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
161	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
162	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
163	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
164	C₂×C₄²	—	47	16	no	—
165	C₂×C₄²	—	188	8	no	—
166	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
168	#95	—	47	16	no	—
169	#95	—	47	16	no	—
170	#95	—	47	16	no	—
171	#95	—	47	16	no	—
172	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	47	8	no	—
180	C₂²×C₄	—	47	8	no	—
181	C₂²×C₄	—	94	8	no	—
182	C₂²×C₄	—	94	8	no	—
183	C₂²×C₄	—	94	8	no	—
184	C₂²×C₄	—	94	8	no	—
185	C₂²×C₄	—	94	8	no	—
186	C₂²×C₄	—	94	8	no	—
187	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
188	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
189	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
190	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
191	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
193	C₄⋊C₄	—	188	16	no	—
194	C₄⋊C₄	—	188	8	no	—
195	C₄⋊C₄	—	188	8	no	—
196	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
197	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
198	C₄²	—	188	16	no	—
199	C₄²	C₂²×Dic₄₇	1	16	yes	—
200	C₂³	C₂³×Dic₄₇	1	2	yes	—
201	C₂³	#13094070	1	4	yes	—
202	C₂³	#13094071	1	4	yes	—
203	C₂³	#13094072	1	2	yes	—
204	C₂³	#13094072	1	2	yes	—
205	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	32	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	94	8	no	—
216	C₂×C₄	—	94	8	no	—
217	C₂×C₄	—	94	8	no	—
218	C₂×C₄	—	94	8	no	—
219	C₂×C₄	—	94	8	no	—
220	C₂×C₄	—	94	8	no	—
221	C₂×C₄	—	188	16	no	—
222	C₂×C₄	—	188	16	no	—
223	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
224	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
225	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
226	C₂²	#15689601	1	4	yes	—
227	C₂²	#15689606	1	8	yes	—
228	C₂²	#15689871	1	4	yes	—
229	C₂²	#15689877	1	8	yes	—
230	C₂²	C₄₇⋊#564	1	4	yes	—
231	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
232	C₂²	C₂×#13094070	1	2	yes	—
233	C₂²	C₂×#13094071	1	2	yes	—
234	C₂²	C₂×#13094072	1	1	yes	characteristic
235	C₂²	C₂×#13094072	1	1	yes	characteristic
236	C₄	—	2	4	no	—
237	C₄	—	2	16	no	—
238	C₄	—	2	8	no	—
239	C₄	—	188	8	no	—
240	C₄	—	188	8	no	—
241	C₂	#20246574	1	8	yes	direct product factor
242	C₂	C₂×#15689601	1	1	yes	characteristic
243	C₂	C₂×#15689606	1	2	yes	—
244	C₂	C₂×#15689871	1	1	yes	characteristic
245	C₂	C₂×#15689877	1	2	yes	—
246	C₂	C₂×C₄₇⋊#564	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#2382
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #20246574	C₂	direct (N × H)
19. C₂×#125×C₄₇	C₄	semidirect
41. #125×C₄₇	C₂×C₄	semidirect
124. C₂×C₄₇	#2382	semidirect
130. C₂×#125	Dic₄₇	semidirect
147. C₄₇	C₂×#2382	semidirect
153. #125	C₂×Dic₄₇	semidirect
241. C₂	#20246574	direct (N × H)

Maximal subgroups

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

subgroup	index
#20246574	2
C₂×#15689022	2
C₂×#15689009	2
C₂×#15688989	2
C₂²×#125×C₄₇	2
C₂×#2382	47

Subgroups — C2×#20246574 #26378927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246574 #26378927