Subgroups — C₂×#20245750 #26374498

Order 12032.

4694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₈₈	C₂	1	3	yes	maximal
2	#20245750	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₄₇	C₂	1	3	yes	maximal, complemented
4	C₂×#15688970	C₂	1	6	yes	maximal, complemented
5	C₂×#95×D₄₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₄₇	C₂²	1	6	yes	—
9	C₂×C₄×D₁₈₈	C₄	1	48	yes	complemented
10	C₂×C₄×D₁₈₈	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	—
12	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	24	yes	complemented
15	#15688970	C₂²	1	48	yes	complemented
16	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
17	#95×D₄₇	C₂²	1	16	yes	complemented
18	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
19	C₄₇×#376	C₂²	1	8	yes	complemented
20	C₂×#95×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#95×C₄₇	C₂²	1	3	yes	—
22	C₂⁴×D₄₇	C₂³	1	2	yes	—
23	C₂²×D₁₈₈	C₂×C₄	1	24	yes	—
24	C₂²×D₁₈₈	C₂³	1	4	yes	—
25	C₂³×Dic₄₇	C₂³	1	6	yes	—
26	C₂²×C₄×D₄₇	—	4	24	no	—
27	C₂²×C₄×D₄₇	C₂×C₄	1	48	yes	complemented
28	C₂²×C₄×D₄₇	C₂³	1	24	yes	—
29	C₄×D₁₈₈	C₂×C₄	1	192	yes	complemented
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	24	yes	complemented
31	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
32	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	48	yes	complemented
33	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	24	yes	—
34	C₂×C₄×Dic₄₇	—	4	12	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
36	C₄₇⋊#95	C₂³	1	24	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	24	yes	complemented
41	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
42	#95×C₄₇	C₂³	1	4	yes	—
43	#95×C₄₇	C₂³	1	12	yes	—
44	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂³×D₄₇	—	2	12	no	—
46	C₂³×D₄₇	—	2	4	no	—
47	C₂³×D₄₇	C₂²×C₄	1	24	yes	—
48	C₂³×D₄₇	C₂⁴	1	4	yes	—
49	C₂²×C₄×C₄₇	—	2	6	no	—
50	C₂²×C₄×C₄₇	—	2	12	no	—
51	C₂²×C₄×C₄₇	—	2	6	no	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
58	C₂²×Dic₄₇	—	2	24	no	—
59	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
60	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
61	C₂×D₁₈₈	C₄²	1	64	yes	complemented
62	C₂×D₁₈₈	C₂²×C₄	1	48	yes	—
63	C₂×C₄×D₄₇	—	4	96	no	—
64	C₂×C₄×D₄₇	—	4	48	no	—
65	C₂×C₄×D₄₇	C₂²×C₄	1	96	yes	complemented
66	C₄²×C₄₇	C₂²×C₄	1	48	yes	complemented
67	(C₂×D₄₇)⋊C₄	C₂²×C₄	1	96	yes	complemented
68	C₄⋊Dic₄₇	C₂²×C₄	1	48	yes	complemented
69	C₄×Dic₄₇	—	4	24	no	—
70	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
71	C₂³×C₄₇	H(5,𝔽₂)	1	6	yes	—
72	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
73	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
74	C₂²×D₄₇	—	2	48	no	—
75	C₂²×D₄₇	—	2	48	no	—
76	C₂²×D₄₇	—	2	12	no	—
77	C₂²×D₄₇	—	2	4	no	—
78	C₂²×D₄₇	C₂×C₄²	1	32	yes	—
79	C₂²×D₄₇	C₂³×C₄	1	24	yes	—
80	C₂×C₄×C₄₇	—	2	24	no	—
81	C₂×C₄×C₄₇	—	2	24	no	—
82	C₂×C₄×C₄₇	—	2	6	no	—
83	C₂×C₄×C₄₇	—	2	12	no	—
84	C₂×C₄×C₄₇	—	2	6	no	—
85	C₂×C₄×C₄₇	C₂×C₄²	1	16	yes	—
86	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
89	C₂×Dic₄₇	—	2	48	no	—
90	C₂×Dic₄₇	—	2	24	no	—
91	C₂×Dic₄₇	C₂³×C₄	1	24	yes	—
92	D₁₈₈	C₂×C₄²	1	64	yes	complemented
93	C₄×D₄₇	—	4	96	no	—
94	C₂×#2275	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	C₂²×C₄²	1	4	yes	—
96	C₂²×C₄₇	#518	1	12	yes	—
97	C₂²×C₄₇	#520	1	12	yes	—
98	C₂²×C₄₇	C₂⁴×C₄	1	3	yes	—
99	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	3	yes	—
100	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×D₄₇	—	2	24	no	—
102	C₂×D₄₇	—	2	24	no	—
103	C₂×D₄₇	—	2	64	no	—
104	C₂×D₄₇	C₂²×C₄²	1	16	yes	—
105	C₄×C₄₇	—	2	12	no	—
106	C₄×C₄₇	—	2	12	no	—
107	C₄×C₄₇	C₂²×C₄²	1	8	yes	—
108	Dic₄₇	—	2	24	no	—
109	C₂²×C₄×D₄	—	47	3	no	—
110	#2275	—	47	16	no	—
111	C₂×C₄×C₄⋊C₄	—	47	3	no	—
112	C₂×C₄×C₂²⋊C₄	—	47	6	no	—
113	C₂×#376	D₄₇	1	1	yes	characteristic, complemented
114	C₂²×#95	—	47	2	no	—
115	C₂×C₄₇	#2275	1	8	yes	complemented
116	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
117	C₂×C₄₇	C₂×#518	1	3	yes	—
118	C₂×C₄₇	C₂×#520	1	3	yes	—
119	D₄₇	—	2	16	no	—
120	C₂³×D₄	—	47	1	no	—
121	C₂⁴×C₄	—	47	6	no	—
122	C₂×C₄×D₄	—	47	48	no	—
123	C₂×C₄×D₄	—	47	24	no	—
124	C₂²×C₄⋊C₄	—	47	3	no	—
125	C₂²×C₂²⋊C₄	—	47	6	no	—
126	C₂²×C₄²	—	47	6	no	—
127	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
128	C₄×C₄⋊C₄	—	47	24	no	—
129	C₄×C₂²⋊C₄	—	47	48	no	—
130	#376	C₂×D₄₇	1	8	yes	complemented
131	C₂×#95	—	47	6	no	—
132	C₂×#95	—	47	16	no	—
133	C₂×#95	C₂×D₄₇	1	1	yes	characteristic
134	C₂×#95	C₂×D₄₇	1	3	yes	—
135	C₄₇	C₂×#2275	1	1	yes	characteristic, complemented, Sylow
136	C₂⁵	—	47	2	no	—
137	C₂²×D₄	—	47	24	no	—
138	C₂²×D₄	—	47	4	no	—
139	C₂³×C₄	—	47	6	no	—
140	C₂³×C₄	—	47	48	no	—
141	C₂³×C₄	—	47	24	no	—
142	C₂³×C₄	—	188	24	no	—
143	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
144	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
145	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₄×D₄	—	47	192	no	—
147	C₂×C₄⋊C₄	—	47	24	no	—
148	C₂×C₄⋊C₄	—	47	12	no	—
149	C₂×C₂²⋊C₄	—	47	48	no	—
150	C₂×C₂²⋊C₄	—	47	24	no	—
151	C₂×C₄²	—	47	24	no	—
152	C₂×C₄²	—	188	12	no	—
153	C₂×C₄²	C₄×D₄₇	1	24	yes	complemented
154	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
155	#95	—	47	24	no	—
156	#95	C₂²×D₄₇	1	4	yes	—
157	#95	C₂²×D₄₇	1	12	yes	—
158	C₂⁴	—	47	24	no	—
159	C₂⁴	—	47	4	no	—
160	C₂⁴	—	94	12	no	—
161	C₂⁴	—	94	4	no	—
162	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
163	C₂×D₄	—	47	64	no	—
164	C₂×D₄	—	47	48	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	47	24	no	—
169	C₂²×C₄	—	47	12	no	—
170	C₂²×C₄	—	47	96	no	—
171	C₂²×C₄	—	94	24	no	—
172	C₂²×C₄	—	188	96	no	—
173	C₂²×C₄	—	188	48	no	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
176	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
177	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
179	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
180	C₄⋊C₄	—	47	48	no	—
181	C₂²⋊C₄	—	47	96	no	—
182	C₄²	—	188	24	no	—
183	C₄²	C₂×C₄×D₄₇	1	48	yes	complemented
184	C₂³	—	47	32	no	—
185	C₂³	—	47	24	no	—
186	C₂³	—	94	48	no	—
187	C₂³	—	94	48	no	—
188	C₂³	—	94	12	no	—
189	C₂³	—	94	4	no	—
190	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
191	C₂³	#13094070	1	6	yes	—
192	C₂³	#13094071	1	2	yes	—
193	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
194	D₄	—	47	64	no	—
195	C₂×C₄	—	2	24	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	47	24	no	—
201	C₂×C₄	—	94	48	no	—
202	C₂×C₄	—	94	24	no	—
203	C₂×C₄	—	188	96	no	—
204	C₂×C₄	C₄²×D₄₇	1	16	yes	—
205	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
206	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
207	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
208	C₂²	—	47	16	no	—
209	C₂²	—	94	24	no	—
210	C₂²	—	94	24	no	—
211	C₂²	—	94	64	no	—
212	C₂²	C₂×C₄²×D₄₇	1	4	yes	—
213	C₂²	#15689597	1	12	yes	—
214	C₂²	#15689603	1	12	yes	—
215	C₂²	C₂³×C₄×D₄₇	1	3	yes	—
216	C₂²	C₂×#13094070	1	3	yes	—
217	C₂²	C₂×#13094071	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	2	12	no	—
220	C₄	—	94	24	no	—
221	C₄	C₂×C₄²×D₄₇	1	8	yes	—
222	C₂	—	94	16	no	—
223	C₂	#20245750	1	8	yes	direct product factor
224	C₂	C₂²×C₄²×D₄₇	1	1	yes	characteristic
225	C₂	C₂×#15689597	1	3	yes	—
226	C₂	C₂×#15689603	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂³×D₁₈₈	C₂²
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	C₂×#2275
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #20245750	C₂	direct (N × H)
3. C₂×C₄×C₄⋊Dic₄₇	C₂	semidirect
4. C₂×#15688970	C₂	semidirect
5. C₂×#95×D₄₇	C₂	semidirect
6. C₂×C₄₇×#376	C₂	semidirect
9. C₂×C₄×D₁₈₈	C₄	semidirect
14. C₄×C₄⋊Dic₄₇	C₂²	semidirect
15. #15688970	C₂²	semidirect
17. #95×D₄₇	C₂²	semidirect
19. C₄₇×#376	C₂²	semidirect
27. C₂²×C₄×D₄₇	C₂×C₄	semidirect
29. C₄×D₁₈₈	C₂×C₄	semidirect
30. C₂×C₄⋊Dic₄₇	C₂×C₄	semidirect
32. C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	semidirect
40. C₂×C₄²×C₄₇	C₂×C₄	semidirect
61. C₂×D₁₈₈	C₄²	semidirect
65. C₂×C₄×D₄₇	C₂²×C₄	semidirect
66. C₄²×C₄₇	C₂²×C₄	semidirect
67. (C₂×D₄₇)⋊C₄	C₂²×C₄	semidirect
68. C₄⋊Dic₄₇	C₂²×C₄	semidirect
92. D₁₈₈	C₂×C₄²	semidirect
113. C₂×#376	D₄₇	semidirect
115. C₂×C₄₇	#2275	semidirect
130. #376	C₂×D₄₇	semidirect
135. C₄₇	C₂×#2275	semidirect
153. C₂×C₄²	C₄×D₄₇	semidirect
183. C₄²	C₂×C₄×D₄₇	semidirect
223. C₂	#20245750	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×C₄×D₁₈₈	2
#20245750	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂×#15688970	2
C₂×#95×D₄₇	2
C₂×C₄₇×#376	2
C₂×#2275	47

Subgroups — C2×#20245750 #26374498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245750 #26374498