Subgroups — C₂×#20249333 #26383429

Order 12032.

3326 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
2	#20249333	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689037	C₂	1	8	yes	maximal, complemented
4	C₂×#15689026	C₂	1	4	yes	maximal, complemented
5	C₂×#15688969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094016	—	2	8	no	—
8	C₂×#13094016	C₂²	1	8	yes	complemented
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₄₇	C₂²	1	16	yes	—
12	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
13	#15689037	C₂²	1	64	yes	complemented
14	#15689026	C₂²	1	32	yes	complemented
15	#15688969	C₂²	1	8	yes	complemented
16	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
18	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
19	C₂×#127×C₄₇	—	2	8	no	—
20	C₂×#127×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₇	C₂³	1	4	yes	—
23	C₂³×Dic₄₇	C₂³	1	4	yes	—
24	#13094016	—	2	32	no	—
25	C₂×C₄⋊Dic₄₇	—	2	8	no	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
27	C₂×Dic₄₇⋊C₄	—	2	16	no	—
28	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
29	C₂×C₂²⋊Dic₄₇	—	4	32	no	—
30	C₂×C₂²⋊Dic₄₇	C₂³	1	64	yes	—
31	C₂×C₄×Dic₄₇	—	2	8	no	—
32	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
33	C₄₇⋊#95	C₂³	1	16	yes	—
34	C₂⁵×C₄₇	C₂³	1	4	yes	—
35	C₂²×D₄×C₄₇	—	2	16	no	—
36	C₂²×D₄×C₄₇	—	2	8	no	—
37	C₂²×D₄×C₄₇	—	2	16	no	—
38	C₂²×D₄×C₄₇	—	4	16	no	—
39	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
40	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	#127×C₄₇	—	2	32	no	—
44	C₂×C₄²×C₄₇	D₄	1	8	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
46	C₂⁴×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	—	2	16	no	—
48	C₂⁴×C₄₇	—	2	32	no	—
49	C₂⁴×C₄₇	C₂⁴	1	8	yes	—
50	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₄×C₄₇	—	2	32	no	—
52	C₂×D₄×C₄₇	—	2	16	no	—
53	C₂×D₄×C₄₇	—	2	32	no	—
54	C₂×D₄×C₄₇	—	4	32	no	—
55	C₂×D₄×C₄₇	—	4	64	no	—
56	C₂×D₄×C₄₇	—	4	32	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₄₇	#41	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	2	8	no	—
65	C₂²×Dic₄₇	—	2	16	no	—
66	C₂²×Dic₄₇	—	2	16	no	—
67	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
68	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
69	C₄²×C₄₇	C₂×D₄	1	16	yes	—
70	C₂²⋊Dic₄₇	—	4	64	no	—
71	C₄⋊Dic₄₇	—	2	16	no	—
72	Dic₄₇⋊C₄	—	2	32	no	—
73	C₄×Dic₄₇	—	2	16	no	—
74	C₂³×C₄₇	—	2	8	no	—
75	C₂³×C₄₇	—	2	16	no	—
76	C₂³×C₄₇	—	2	32	no	—
77	C₂³×C₄₇	—	4	16	no	—
78	C₂³×C₄₇	—	4	32	no	—
79	C₂³×C₄₇	—	4	64	no	—
80	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
81	C₂³×C₄₇	C₂×#41	1	4	yes	—
82	C₂³×C₄₇	H(5,𝔽₂)	1	8	yes	—
83	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
84	D₄×C₄₇	—	4	64	no	—
85	D₄×C₄₇	—	4	32	no	—
86	C₂×C₄×C₄₇	—	2	16	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	#124	1	16	yes	—
89	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
90	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	—	2	16	no	—
95	C₂×Dic₄₇	—	2	16	no	—
96	C₂×Dic₄₇	—	4	16	no	—
97	C₂×Dic₄₇	—	4	16	no	—
98	C₂×#2641	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	4	8	no	—
100	C₂²×C₄₇	—	4	16	no	—
101	C₂²×C₄₇	—	4	32	no	—
102	C₂²×C₄₇	—	8	32	no	—
103	C₂²×C₄₇	C₂×#124	1	4	yes	—
104	C₂²×C₄₇	#535	1	8	yes	—
105	C₂²×C₄₇	#540	1	16	yes	—
106	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
107	C₂²×C₄₇	C₂²×#41	1	2	yes	—
108	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	4	yes	—
109	C₄×C₄₇	—	2	8	no	—
110	C₄×C₄₇	C₂×#124	1	8	yes	—
111	Dic₄₇	—	4	8	no	—
112	Dic₄₇	—	4	8	no	—
113	C₂²×#127	D₄₇	1	1	yes	characteristic
114	C₂²×#125	—	47	1	no	—
115	#2641	—	47	16	no	—
116	C₂×#394	—	47	8	no	—
117	C₂×#390	—	47	4	no	—
118	C₂×#376	—	47	1	no	—
119	C₂×C₄₇	—	8	8	no	—
120	C₂×C₄₇	#2641	1	8	yes	complemented
121	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#535	1	2	yes	—
123	C₂×C₄₇	C₂×#540	1	4	yes	—
124	C₂³×D₄	C₂×D₄₇	1	2	yes	—
125	C₂³×D₄	C₂×D₄₇	1	4	yes	—
126	C₂×#127	—	2	8	no	—
127	C₂×#127	C₂×D₄₇	1	8	yes	—
128	C₂×#125	—	47	8	no	—
129	C₂×#125	—	94	8	no	—
130	C₂²×C₄⋊C₄	—	47	4	no	—
131	C₂²×C₄⋊C₄	—	47	2	no	—
132	C₂²×C₂²⋊C₄	—	47	16	no	—
133	C₂²×C₄²	—	47	2	no	—
134	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
135	#394	—	47	64	no	—
136	#390	—	47	32	no	—
137	#376	—	47	8	no	—
138	C₂×#95	—	47	4	no	—
139	C₄₇	C₂×#2641	1	1	yes	characteristic, complemented, Sylow
140	C₂⁵	C₂²×D₄₇	1	4	yes	—
141	C₂²×D₄	—	2	16	no	—
142	C₂²×D₄	—	2	16	no	—
143	C₂²×D₄	—	2	8	no	—
144	C₂²×D₄	—	4	16	no	—
145	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
146	C₂²×D₄	C₂²×D₄₇	1	16	yes	—
147	C₂³×C₄	—	47	4	no	—
148	C₂³×C₄	—	47	4	no	—
149	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
150	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	#127	—	2	32	no	—
152	#125	—	94	32	no	—
153	C₂×C₄⋊C₄	—	47	16	no	—
154	C₂×C₄⋊C₄	—	47	8	no	—
155	C₂×C₄⋊C₄	—	94	16	no	—
156	C₂×C₄⋊C₄	—	94	8	no	—
157	C₂×C₂²⋊C₄	—	47	64	no	—
158	C₂×C₂²⋊C₄	—	188	32	no	—
159	C₂×C₄²	—	47	8	no	—
160	C₂×C₄²	—	94	8	no	—
161	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
163	#95	—	47	16	no	—
164	C₂⁴	—	2	8	no	—
165	C₂⁴	—	2	16	no	—
166	C₂⁴	—	2	32	no	—
167	C₂⁴	C₂³×D₄₇	1	8	yes	—
168	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
169	C₂×D₄	—	2	32	no	—
170	C₂×D₄	—	2	16	no	—
171	C₂×D₄	—	2	32	no	—
172	C₂×D₄	—	4	32	no	—
173	C₂×D₄	—	4	64	no	—
174	C₂×D₄	—	4	32	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	47	8	no	—
177	C₂²×C₄	—	47	8	no	—
178	C₂²×C₄	—	94	8	no	—
179	C₂²×C₄	—	94	8	no	—
180	C₂²×C₄	—	94	16	no	—
181	C₂²×C₄	—	94	16	no	—
182	C₂²×C₄	#10625352	1	8	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
184	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
187	C₄⋊C₄	—	94	32	no	—
188	C₄⋊C₄	—	94	16	no	—
189	C₂²⋊C₄	—	188	64	no	—
190	C₄²	—	94	16	no	—
191	C₄²	C₂×C₂²⋊D₄₇	1	16	yes	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	16	no	—
194	C₂³	—	2	32	no	—
195	C₂³	—	4	16	no	—
196	C₂³	—	4	32	no	—
197	C₂³	—	4	64	no	—
198	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
199	C₂³	C₂×#10625352	1	4	yes	—
200	C₂³	#13094069	1	8	yes	—
201	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
202	D₄	—	4	64	no	—
203	D₄	—	4	32	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	94	8	no	—
207	C₂×C₄	—	94	8	no	—
208	C₂×C₄	—	94	16	no	—
209	C₂×C₄	—	94	16	no	—
210	C₂×C₄	—	188	16	no	—
211	C₂×C₄	—	188	16	no	—
212	C₂×C₄	#13094011	1	16	yes	—
213	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
215	C₂×C₄	C₂×#10625352	1	8	yes	—
216	C₂²	—	4	8	no	—
217	C₂²	—	4	16	no	—
218	C₂²	—	4	32	no	—
219	C₂²	—	8	32	no	—
220	C₂²	C₂×#13094011	1	4	yes	—
221	C₂²	#15689690	1	8	yes	—
222	C₂²	#15689727	1	16	yes	—
223	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²	C₂²×#10625352	1	2	yes	—
225	C₂²	C₂×#13094069	1	4	yes	—
226	C₄	—	2	8	no	—
227	C₄	—	188	8	no	—
228	C₄	—	188	8	no	—
229	C₄	C₂×#13094011	1	8	yes	—
230	C₂	—	8	8	no	—
231	C₂	#20249333	1	8	yes	direct product factor
232	C₂	C₂²×#13094011	1	1	yes	characteristic
233	C₂	C₂×#15689690	1	2	yes	—
234	C₂	C₂×#15689727	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂×#2641
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×#13094016	C₂	semidirect
2. #20249333	C₂	direct (N × H)
3. C₂×#15689037	C₂	semidirect
4. C₂×#15689026	C₂	semidirect
5. C₂×#15688969	C₂	semidirect
8. C₂×#13094016	C₂²	semidirect
13. #15689037	C₂²	semidirect
14. #15689026	C₂²	semidirect
15. #15688969	C₂²	semidirect
120. C₂×C₄₇	#2641	semidirect
139. C₄₇	C₂×#2641	semidirect
231. C₂	#20249333	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×#13094016	2
#20249333	2
C₂×#15689037	2
C₂×#15689026	2
C₂×#15688969	2
C₂²×#127×C₄₇	2
C₂×#2641	47

Subgroups — C2×#20249333 #26383429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20249333 #26383429