Subgroups — C₂²×#15689185 #27079996

Order 12032.

2310 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093981	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689185	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093981	C₄	1	12	yes	—
10	C₂×#13093981	C₂²	1	12	yes	—
11	C₂²×Dic₄₇⋊C₄	C₄	1	2	yes	—
12	C₂²×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689185	C₂²	1	64	yes	direct product factor
15	C₂×Dic₄₇⋊C₈	C₂²	1	24	yes	complemented
16	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	24	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	12	yes	complemented
18	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#97×C₄₇	C₂²	1	12	yes	complemented
21	C₂⁴×D₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂³×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×D₄₇	—	2	8	no	—
25	C₂²×C₄×D₄₇	C₂×C₄	1	6	yes	—
26	C₂²×C₄×D₄₇	C₂³	1	6	yes	—
27	C₂²×C₄₇⋊C₈	—	2	8	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
29	#13093981	C₂×C₄	1	48	yes	—
30	#13093981	C₂³	1	16	yes	—
31	C₂×Dic₄₇⋊C₄	C₂×C₄	1	24	yes	—
32	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	24	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
35	Dic₄₇⋊C₈	C₂³	1	32	yes	complemented
36	(C₂×D₄₇)⋊C₈	C₂³	1	32	yes	—
37	C₄×C₄₇⋊C₈	C₂³	1	16	yes	complemented
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	8	no	—
41	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
44	#97×C₄₇	C₂³	1	16	yes	complemented
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂³×D₄₇	—	2	4	no	—
47	C₂³×D₄₇	—	2	8	no	—
48	C₂³×D₄₇	C₂²×C₄	1	6	yes	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₄₇	#41	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	C₂²×C₄	1	12	yes	—
57	C₂²×Dic₄₇	C₂²×C₄	1	6	yes	—
58	C₂×C₄×D₄₇	—	2	24	no	—
59	C₂×C₄×D₄₇	—	2	24	no	—
60	C₂×C₄×D₄₇	C₂²×C₄	1	12	yes	—
61	C₂×C₄×D₄₇	C₂⁴	1	4	yes	—
62	C₂×C₈×C₄₇	—	2	24	no	—
63	C₂×C₈×C₄₇	C₂⁴	1	8	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
65	C₄²×C₄₇	C₂⁴	1	4	yes	—
66	(C₂×D₄₇)⋊C₄	C₂²×C₄	1	32	yes	—
67	Dic₄₇⋊C₄	C₂²×C₄	1	32	yes	—
68	C₄×Dic₄₇	C₂²×C₄	1	12	yes	—
69	C₄×Dic₄₇	C₂⁴	1	4	yes	—
70	C₂×C₄₇⋊C₈	—	2	24	no	—
71	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
72	C₂³×C₄₇	#117	1	4	yes	—
73	C₂³×C₄₇	#131	1	8	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	3	yes	—
75	C₂²×D₄₇	—	2	12	no	—
76	C₂²×D₄₇	—	2	24	no	—
77	C₂²×D₄₇	—	4	16	no	—
78	C₂²×D₄₇	C₂³×C₄	1	4	yes	—
79	C₂×C₄×C₄₇	—	2	24	no	—
80	C₂×C₄×C₄₇	#117	1	12	yes	—
81	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
82	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
84	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
85	C₂×Dic₄₇	—	2	12	no	—
86	C₂×Dic₄₇	—	2	24	no	—
87	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
88	C₂×Dic₄₇	C₂³×C₄	1	4	yes	—
89	C₄×D₄₇	—	2	24	no	—
90	C₄×D₄₇	—	2	8	no	—
91	C₈×C₄₇	—	2	8	no	—
92	C₄₇⋊C₈	—	2	8	no	—
93	C₂²×#433	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	#433	1	16	yes	complemented
95	C₂²×C₄₇	C₂×#117	1	6	yes	—
96	C₂²×C₄₇	C₂×#131	1	12	yes	—
97	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
98	C₂×D₄₇	—	2	4	no	—
99	C₂×D₄₇	—	2	8	no	—
100	C₂×D₄₇	—	4	24	no	—
101	C₄×C₄₇	—	2	8	no	—
102	C₄×C₄₇	C₂×#117	1	6	yes	—
103	C₄×C₄₇	C₂²×#41	1	2	yes	—
104	Dic₄₇	—	2	4	no	—
105	Dic₄₇	—	2	8	no	—
106	C₂²×#117	—	47	1	no	—
107	C₂×#433	—	47	24	no	—
108	C₂²×C₄⋊C₈	—	47	2	no	—
109	C₂²×C₂²⋊C₈	—	47	2	no	—
110	C₂²×#97	D₄₇	1	1	yes	characteristic, complemented
111	C₂²×C₄×C₈	—	47	1	no	—
112	C₂×C₄₇	C₂×#433	1	12	yes	complemented
113	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
114	C₂×C₄₇	C₂²×#131	1	2	yes	—
115	D₄₇	—	4	4	no	—
116	C₂⁴×C₄	—	47	1	no	—
117	C₂³×C₈	—	47	2	no	—
118	C₂³×C₈	C₂×D₄₇	1	2	yes	—
119	C₂×#117	—	47	12	no	—
120	C₂×#117	—	47	12	no	—
121	C₂²×C₄⋊C₄	—	47	2	no	—
122	C₂²×C₂²⋊C₄	—	47	2	no	—
123	C₂²×C₄²	—	47	1	no	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	#433	—	47	64	no	—
126	C₂×C₄⋊C₈	—	47	24	no	—
127	C₂×C₂²⋊C₈	—	47	24	no	—
128	C₂×#97	C₂×D₄₇	1	12	yes	complemented
129	C₂×C₄×C₈	—	47	12	no	—
130	C₄₇	C₂²×#433	1	1	yes	characteristic, complemented, Sylow
131	C₂⁵	—	47	1	no	—
132	C₂³×C₄	—	47	2	no	—
133	C₂³×C₄	—	47	1	no	—
134	C₂³×C₄	—	47	6	no	—
135	C₂³×C₄	—	47	6	no	—
136	C₂³×C₄	—	94	8	no	—
137	C₂³×C₄	C₄×D₄₇	1	2	yes	—
138	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	—	47	12	no	—
141	C₂²×C₈	—	94	8	no	—
142	C₂²×C₈	C₂²×D₄₇	1	12	yes	—
143	#117	—	47	48	no	—
144	#117	—	47	16	no	—
145	C₂×C₄⋊C₄	—	47	24	no	—
146	C₂×C₂²⋊C₄	—	47	24	no	—
147	C₂×C₄²	—	47	6	no	—
148	C₂×C₄²	—	47	6	no	—
149	C₂×C₄²	C₄×D₄₇	1	6	yes	—
150	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
151	C₄⋊C₈	—	47	32	no	—
152	C₂²⋊C₈	—	47	32	no	—
153	#97	C₂²×D₄₇	1	16	yes	complemented
154	C₄×C₈	—	47	16	no	—
155	C₂⁴	—	47	6	no	—
156	C₂⁴	—	94	4	no	—
157	C₂⁴	—	94	8	no	—
158	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	47	12	no	—
161	C₂²×C₄	—	47	6	no	—
162	C₂²×C₄	—	47	12	no	—
163	C₂²×C₄	—	47	4	no	—
164	C₂²×C₄	—	94	4	no	—
165	C₂²×C₄	—	94	8	no	—
166	C₂²×C₄	—	94	24	no	—
167	C₂²×C₄	—	94	24	no	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
170	C₂²×C₄	#10625350	1	8	yes	—
171	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
172	C₂×C₈	—	2	24	no	—
173	C₂×C₈	—	47	8	no	—
174	C₂×C₈	—	94	24	no	—
175	C₂×C₈	C₂³×D₄₇	1	8	yes	—
176	C₄⋊C₄	—	47	32	no	—
177	C₂²⋊C₄	—	47	32	no	—
178	C₄²	—	47	12	no	—
179	C₄²	—	47	4	no	—
180	C₄²	C₂×C₄×D₄₇	1	12	yes	—
181	C₄²	C₂³×D₄₇	1	4	yes	—
182	C₂³	—	47	4	no	—
183	C₂³	—	94	12	no	—
184	C₂³	—	94	24	no	—
185	C₂³	—	188	16	no	—
186	C₂³	#13093981	1	4	yes	—
187	C₂³	#13094032	1	8	yes	—
188	C₂³	C₂²×C₄×D₄₇	1	3	yes	—
189	C₂×C₄	—	2	24	no	—
190	C₂×C₄	—	47	8	no	—
191	C₂×C₄	—	47	4	no	—
192	C₂×C₄	—	94	12	no	—
193	C₂×C₄	—	94	24	no	—
194	C₂×C₄	—	94	24	no	—
195	C₂×C₄	—	94	8	no	—
196	C₂×C₄	#13093981	1	12	yes	—
197	C₂×C₄	C₂²×C₄×D₄₇	1	8	yes	—
198	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
199	C₂×C₄	C₂×#10625350	1	12	yes	—
200	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
201	C₈	—	2	8	no	—
202	C₈	—	94	8	no	—
203	C₂²	—	94	4	no	—
204	C₂²	—	94	8	no	—
205	C₂²	—	188	24	no	—
206	C₂²	#15689185	1	16	yes	direct product factor
207	C₂²	C₂×#13093981	1	6	yes	—
208	C₂²	C₂×#13094032	1	12	yes	—
209	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
210	C₄	—	2	8	no	—
211	C₄	—	94	4	no	—
212	C₄	—	94	8	no	—
213	C₄	C₂×#13093981	1	6	yes	—
214	C₄	C₂²×#10625350	1	2	yes	—
215	C₂	—	188	4	no	—
216	C₂	C₂×#15689185	1	12	yes	direct product factor
217	C₂	C₂²×#13093981	1	1	yes	characteristic
218	C₂	C₂²×#13094032	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁴×D₄₇	C₂×C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂²×#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#15689185	C₂	direct (N × H)
3. C₂²×Dic₄₇⋊C₈	C₂	semidirect
5. C₂²×C₄×C₄₇⋊C₈	C₂	semidirect
6. C₂²×#97×C₄₇	C₂	semidirect
14. #15689185	C₂²	direct (N × H)
15. C₂×Dic₄₇⋊C₈	C₂²	semidirect
17. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
20. C₂×#97×C₄₇	C₂²	semidirect
35. Dic₄₇⋊C₈	C₂³	semidirect
37. C₄×C₄₇⋊C₈	C₂³	semidirect
44. #97×C₄₇	C₂³	semidirect
94. C₂²×C₄₇	#433	semidirect
110. C₂²×#97	D₄₇	semidirect
112. C₂×C₄₇	C₂×#433	semidirect
128. C₂×#97	C₂×D₄₇	semidirect
130. C₄₇	C₂²×#433	semidirect
153. #97	C₂²×D₄₇	semidirect
206. C₂²	#15689185	direct (N × H)
216. C₂	C₂×#15689185	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×#13093981	2
C₂×#15689185	2
C₂²×Dic₄₇⋊C₈	2
C₂²×(C₂×D₄₇)⋊C₈	2
C₂²×C₄×C₄₇⋊C₈	2
C₂²×#97×C₄₇	2
C₂²×#433	47

Subgroups — C22×#15689185 #27079996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689185 #27079996