Subgroups — C₂²×#15689859 #27123330

Order 12032.

8710 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689859	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13094011	C₂	1	4	yes	maximal, complemented
3	C₂²×#13094005	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×Dic₉₄	C₂	1	2	yes	maximal, complemented
5	C₂²×D₄×Dic₄₇	C₂	1	4	yes	maximal, complemented
6	C₂²×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	2	yes	—
8	C₂⁴×Dic₄₇	C₂²	1	4	yes	—
9	#15689859	C₂²	1	256	yes	direct product factor
10	C₂×#13094011	C₂²	1	96	yes	complemented
11	C₂×#13094005	C₂²	1	96	yes	complemented
12	C₂×C₄×Dic₉₄	C₂²	1	48	yes	complemented
13	C₂×D₄×Dic₄₇	C₂²	1	96	yes	complemented
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
17	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
19	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
20	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
21	C₂×#127×C₄₇	C₂²	1	24	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₉₄	—	2	16	no	—
24	C₂²×Dic₉₄	C₂³	1	24	yes	—
25	C₂³×Dic₄₇	—	4	16	no	—
26	C₂³×Dic₄₇	C₂³	1	4	yes	—
27	C₂³×Dic₄₇	C₂³	1	4	yes	—
28	C₂³×Dic₄₇	C₂³	1	48	yes	—
29	#13094011	C₂³	1	256	yes	complemented
30	#13094005	C₂³	1	256	yes	complemented
31	C₄×Dic₉₄	C₂³	1	128	yes	complemented
32	D₄×Dic₄₇	C₂³	1	256	yes	complemented
33	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
34	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂³	1	96	yes	—
36	C₂×C₂²⋊Dic₄₇	C₂³	1	48	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
38	C₂⁵×C₄₇	C₂³	1	4	yes	—
39	C₂²×D₄×C₄₇	—	2	32	no	—
40	C₂²×D₄×C₄₇	—	4	8	no	—
41	C₂²×D₄×C₄₇	C₂³	1	48	yes	—
42	C₂²×D₄×C₄₇	C₂³	1	24	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
45	#127×C₄₇	C₂³	1	64	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
47	C₂⁴×C₄₇	—	2	32	no	—
48	C₂⁴×C₄₇	—	2	16	no	—
49	C₂⁴×C₄₇	C₂⁴	1	24	yes	—
50	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×D₄×C₄₇	—	2	192	no	—
52	C₂×D₄×C₄₇	—	4	48	no	—
53	C₂×D₄×C₄₇	C₂⁴	1	64	yes	—
54	C₂×D₄×C₄₇	C₂⁴	1	32	yes	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	#41	1	16	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
59	C₂²×Dic₄₇	—	2	16	no	—
60	C₂²×Dic₄₇	—	2	16	no	—
61	C₂²×Dic₄₇	—	4	96	no	—
62	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
63	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
64	C₂²×Dic₄₇	C₂⁴	1	64	yes	—
65	C₂×Dic₉₄	—	2	96	no	—
66	C₂×Dic₉₄	C₂⁴	1	32	yes	—
67	C₄²×C₄₇	C₂⁴	1	16	yes	—
68	C₂²⋊Dic₄₇	C₂⁴	1	128	yes	—
69	C₂²⋊Dic₄₇	C₂⁴	1	64	yes	—
70	C₄⋊Dic₄₇	C₂⁴	1	32	yes	—
71	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
72	C₄×Dic₄₇	C₂⁴	1	64	yes	—
73	C₂³×C₄₇	—	2	96	no	—
74	C₂³×C₄₇	—	2	48	no	—
75	C₂³×C₄₇	—	4	64	no	—
76	C₂³×C₄₇	C₂×#41	1	8	yes	—
77	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
78	C₂³×C₄₇	C₂⁵	1	16	yes	—
79	C₂³×C₄₇	C₂⁵	1	3	yes	—
80	D₄×C₄₇	—	2	128	no	—
81	D₄×C₄₇	—	4	32	no	—
82	C₂×C₄×C₄₇	—	2	12	no	—
83	C₂×C₄×C₄₇	C₂×#41	1	48	yes	—
84	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
85	C₂×C₄×C₄₇	C₂⁵	1	8	yes	—
86	C₂×Dic₄₇	—	2	48	no	—
87	C₂×Dic₄₇	—	2	48	no	—
88	C₂×Dic₄₇	—	4	64	no	—
89	C₂×Dic₄₇	C₂⁵	1	16	yes	—
90	C₂×Dic₄₇	C₂⁵	1	16	yes	—
91	Dic₉₄	—	2	64	no	—
92	C₂²×#559	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	32	no	—
94	C₂²×C₄₇	—	2	16	no	—
95	C₂²×C₄₇	—	4	96	no	—
96	C₂²×C₄₇	#559	1	16	yes	complemented
97	C₂²×C₄₇	C₂²×#41	1	12	yes	—
98	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	6	yes	—
99	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	C₂²×#41	1	16	yes	—
102	Dic₄₇	—	2	16	no	—
103	Dic₄₇	—	2	16	no	—
104	C₂×#559	—	47	48	no	—
105	C₂²×#127	D₄₇	1	1	yes	characteristic
106	C₂²×#124	—	47	4	no	—
107	C₂²×#123	—	47	4	no	—
108	C₂²×C₄×Q₈	—	47	2	no	—
109	C₂²×C₄×D₄	—	47	4	no	—
110	C₂×C₄₇	—	4	16	no	—
111	C₂×C₄₇	C₂×#559	1	12	yes	complemented
112	C₂×C₄₇	C₂³×#41	1	2	yes	—
113	C₂×C₄₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
114	C₂³×Q₈	—	47	2	no	—
115	C₂³×D₄	C₂×D₄₇	1	4	yes	—
116	C₂³×D₄	C₂×D₄₇	1	2	yes	—
117	C₂⁴×C₄	—	47	4	no	—
118	#559	—	47	256	no	—
119	C₂×#127	C₂×D₄₇	1	24	yes	—
120	C₂×#124	—	47	96	no	—
121	C₂×#123	—	47	96	no	—
122	C₂×C₄×Q₈	—	47	48	no	—
123	C₂×C₄×D₄	—	47	96	no	—
124	C₂²×C₄⋊C₄	—	47	2	no	—
125	C₂²×C₄⋊C₄	—	47	4	no	—
126	C₂²×C₂²⋊C₄	—	47	8	no	—
127	C₂²×C₂²⋊C₄	—	47	4	no	—
128	C₂²×C₄²	—	47	4	no	—
129	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
130	C₄₇	C₂²×#559	1	1	yes	characteristic, complemented, Sylow
131	C₂⁵	C₂²×D₄₇	1	4	yes	—
132	C₂²×Q₈	—	47	24	no	—
133	C₂²×Q₈	—	94	16	no	—
134	C₂²×D₄	—	2	32	no	—
135	C₂²×D₄	—	4	8	no	—
136	C₂²×D₄	C₂²×D₄₇	1	48	yes	—
137	C₂²×D₄	C₂²×D₄₇	1	24	yes	—
138	C₂³×C₄	—	47	4	no	—
139	C₂³×C₄	—	47	4	no	—
140	C₂³×C₄	—	47	48	no	—
141	C₂³×C₄	—	188	16	no	—
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
144	#127	C₂²×D₄₇	1	64	yes	—
145	#124	—	47	256	no	—
146	#123	—	47	256	no	—
147	C₄×Q₈	—	47	128	no	—
148	C₄×D₄	—	47	256	no	—
149	C₂×C₄⋊C₄	—	47	24	no	—
150	C₂×C₄⋊C₄	—	47	48	no	—
151	C₂×C₂²⋊C₄	—	47	96	no	—
152	C₂×C₂²⋊C₄	—	47	48	no	—
153	C₂×C₄²	—	47	48	no	—
154	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
155	C₂⁴	—	2	32	no	—
156	C₂⁴	—	2	16	no	—
157	C₂⁴	C₂³×D₄₇	1	24	yes	—
158	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
159	C₂×Q₈	—	47	32	no	—
160	C₂×Q₈	—	94	96	no	—
161	C₂×D₄	—	2	192	no	—
162	C₂×D₄	—	4	48	no	—
163	C₂×D₄	C₂³×D₄₇	1	64	yes	—
164	C₂×D₄	C₂³×D₄₇	1	32	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	47	24	no	—
167	C₂²×C₄	—	47	24	no	—
168	C₂²×C₄	—	47	64	no	—
169	C₂²×C₄	—	94	16	no	—
170	C₂²×C₄	—	94	16	no	—
171	C₂²×C₄	—	188	96	no	—
172	C₂²×C₄	#10625352	1	16	yes	—
173	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
175	C₄⋊C₄	—	47	64	no	—
176	C₄⋊C₄	—	47	32	no	—
177	C₂²⋊C₄	—	47	128	no	—
178	C₂²⋊C₄	—	47	64	no	—
179	C₄²	—	47	64	no	—
180	C₄²	C₂³×D₄₇	1	16	yes	—
181	C₂³	—	2	96	no	—
182	C₂³	—	2	48	no	—
183	C₂³	—	4	64	no	—
184	C₂³	C₂×#10625352	1	8	yes	—
185	C₂³	#13094069	1	4	yes	—
186	C₂³	C₂⁴×D₄₇	1	16	yes	—
187	C₂³	C₂⁴×D₄₇	1	3	yes	—
188	Q₈	—	94	64	no	—
189	D₄	—	2	128	no	—
190	D₄	—	4	32	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	47	16	no	—
193	C₂×C₄	—	47	16	no	—
194	C₂×C₄	—	94	48	no	—
195	C₂×C₄	—	94	48	no	—
196	C₂×C₄	—	188	64	no	—
197	C₂×C₄	C₂×#10625352	1	48	yes	—
198	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
199	C₂×C₄	C₂⁴×D₄₇	1	8	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	16	no	—
202	C₂²	—	4	96	no	—
203	C₂²	#15689859	1	16	yes	direct product factor
204	C₂²	C₂²×#10625352	1	12	yes	—
205	C₂²	C₂×#13094069	1	6	yes	—
206	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	94	16	no	—
209	C₄	—	94	16	no	—
210	C₄	C₂²×#10625352	1	16	yes	—
211	C₂	—	4	16	no	—
212	C₂	C₂×#15689859	1	12	yes	direct product factor
213	C₂	C₂³×#10625352	1	2	yes	—
214	C₂	C₂²×#13094069	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting, prime radical	C₂²×#127×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×#127	D₄₇
hypocenter	C₄₇	C₂²×#559
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#15689859	C₂	direct (N × H)
2. C₂²×#13094011	C₂	semidirect
3. C₂²×#13094005	C₂	semidirect
4. C₂²×C₄×Dic₉₄	C₂	semidirect
5. C₂²×D₄×Dic₄₇	C₂	semidirect
9. #15689859	C₂²	direct (N × H)
10. C₂×#13094011	C₂²	semidirect
11. C₂×#13094005	C₂²	semidirect
12. C₂×C₄×Dic₉₄	C₂²	semidirect
13. C₂×D₄×Dic₄₇	C₂²	semidirect
29. #13094011	C₂³	semidirect
30. #13094005	C₂³	semidirect
31. C₄×Dic₉₄	C₂³	semidirect
32. D₄×Dic₄₇	C₂³	semidirect
96. C₂²×C₄₇	#559	semidirect
111. C₂×C₄₇	C₂×#559	semidirect
130. C₄₇	C₂²×#559	semidirect
203. C₂²	#15689859	direct (N × H)
212. C₂	C₂×#15689859	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689859	2
C₂²×#13094011	2
C₂²×#13094005	2
C₂²×C₄×Dic₉₄	2
C₂²×D₄×Dic₄₇	2
C₂²×#127×C₄₇	2
C₂²×#559	47

Subgroups — C22×#15689859 #27123330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689859 #27123330