Subgroups — #26369466

Order 12032.

690 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	C₂×#15689541	C₂	1	2	yes	maximal, complemented
2	#20252101	C₂	1	1	yes	maximal, characteristic
3	#20245219	C₂	1	4	yes	maximal, complemented
4	#20245207	C₂	1	4	yes	maximal, complemented
5	#20245003	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	2	yes	—
9	#15689673	C₄	1	1	yes	characteristic
10	#15689541	—	2	8	no	—
11	C₂×#13093963	C₂²	1	4	yes	—
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₄₇	C₂²	1	4	yes	—
15	#15689202	C₄	1	4	yes	—
16	C₈×D₁₈₈	C₂²	1	4	yes	—
17	#15689147	C₄	1	1	yes	characteristic
18	#15689089	C₄	1	2	yes	—
19	(C₂×D₄₇)⋊C₁₆	C₂²	1	8	yes	—
20	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
21	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
22	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₂×#10625350	C₂×C₄	1	2	yes	—
26	#13094030	—	2	4	no	—
27	#13094030	—	2	4	no	—
28	C₂×#10625329	C₂×C₄	1	2	yes	—
29	C₂×#10625324	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₄₇	C₂³	1	4	yes	—
31	#13094024	C₈	1	1	yes	characteristic
32	#13094021	C₈	1	1	yes	characteristic
33	#13094012	C₈	1	2	yes	—
34	#13093998	C₈	1	4	yes	—
35	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
36	#13093963	—	2	8	no	—
37	C₂×C₄₇⋊C₁₆	—	2	2	no	—
38	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
39	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
40	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
42	C₁₆×D₄₇	—	2	8	no	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₄₇	—	2	4	no	—
49	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₇	C₂×C₈	1	4	yes	—
55	#10625350	—	2	4	no	—
56	#10625350	—	2	4	no	—
57	C₂×D₁₈₈	C₂×C₈	1	4	yes	—
58	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
60	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₇	#41	1	2	yes	—
66	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	(C₂×D₄₇)⋊C₄	C₂×C₈	1	8	yes	—
71	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
72	#10625329	—	2	4	no	—
73	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
74	#10625324	—	2	8	no	—
75	C₈×D₄₇	—	2	4	no	—
76	C₈×D₄₇	—	2	4	no	—
77	C₁₆×C₄₇	—	2	4	no	—
78	C₄₇⋊C₁₆	—	2	2	no	—
79	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
80	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
81	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₄₇	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₄₇	#131	1	2	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₄₇	—	2	8	no	—
91	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
92	D₁₈₈	—	2	4	no	—
93	D₁₈₈	—	2	8	no	—
94	C₄×D₄₇	—	2	4	no	—
95	C₄×D₄₇	—	2	4	no	—
96	Dic₉₄	—	2	4	no	—
97	C₈×C₄₇	—	2	2	no	—
98	C₈×C₄₇	C₄×D₄	1	2	yes	—
99	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
101	C₄₇⋊C₈	—	2	4	no	—
102	#33434	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	C₈×D₄	1	2	yes	—
105	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₄₇	—	2	8	no	—
107	C₂×D₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	C₈×D₄	1	2	yes	—
110	C₄×C₄₇	#504	1	4	yes	complemented
111	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
113	Dic₄₇	—	2	4	no	—
114	C₂×#504	—	47	2	no	—
115	#2931	—	47	1	no	—
116	#2169	—	47	4	no	—
117	D₄×C₁₆	—	47	4	no	—
118	#2150	D₄₇	1	1	yes	characteristic, complemented
119	#2113	—	47	2	no	—
120	C₂×C₄×C₁₆	—	47	1	no	—
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₄₇	C₂×#504	1	2	yes	—
124	D₄₇	—	4	4	no	—
125	C₂×#131	—	47	2	no	—
126	#532	—	47	1	no	—
127	#504	—	94	8	no	—
128	C₂×M₃₂	—	47	4	no	—
129	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
130	C₂²×C₁₆	—	47	1	no	—
131	C₂²×C₁₆	—	47	1	no	—
132	C₂²×C₁₆	—	47	4	no	—
133	#436	—	47	4	no	—
134	C₈×D₄	—	47	4	no	—
135	#423	—	47	1	no	—
136	#408	—	47	2	no	—
137	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	—
139	C₂²⋊C₁₆	—	47	8	no	—
140	C₄×C₁₆	—	47	2	no	—
141	C₄×C₁₆	—	47	2	no	—
142	C₄₇	#33434	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	47	2	no	—
144	#131	—	94	4	no	—
145	#131	—	94	4	no	—
146	C₂×M₁₆	—	47	2	no	—
147	C₂×M₁₆	—	47	4	no	—
148	C₂²×C₈	—	47	4	no	—
149	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊Q₈	—	47	1	no	—
152	#127	—	47	1	no	—
153	#124	—	47	2	no	—
154	#121	—	47	4	no	—
155	C₄×D₄	—	47	4	no	—
156	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	94	8	no	—
159	C₂×C₁₆	—	47	4	no	—
160	C₂×C₁₆	—	47	1	no	—
161	C₂×C₁₆	—	47	1	no	—
162	C₂×C₁₆	—	47	2	no	—
163	C₂×C₁₆	—	94	8	no	—
164	C₂×C₁₆	—	94	2	no	—
165	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
166	C₄⋊C₈	—	47	4	no	—
167	C₂²⋊C₈	—	47	8	no	—
168	C₄×C₈	C₄×D₄₇	1	2	yes	—
169	C₄×C₈	C₂²×D₄₇	1	2	yes	—
170	#41	—	94	4	no	—
171	#41	—	94	4	no	—
172	C₂×Q₈	—	47	2	no	—
173	C₂×D₄	—	47	4	no	—
174	C₂×D₄	—	47	4	no	—
175	C₂×D₄	—	47	2	no	—
176	C₂²×C₄	—	47	4	no	—
177	C₂²×C₄	C₈×D₄₇	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
179	M₁₆	—	94	4	no	—
180	M₁₆	—	94	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	47	4	no	—
183	C₂×C₈	—	94	4	no	—
184	C₂×C₈	—	94	4	no	—
185	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
187	C₂×C₈	D₄×D₄₇	1	2	yes	—
188	C₂×C₈	#10625354	1	2	yes	—
189	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
190	C₄⋊C₄	—	47	4	no	—
191	C₂²⋊C₄	—	47	8	no	—
192	C₄²	C₈×D₄₇	1	1	yes	characteristic
193	C₄²	C₈×D₄₇	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₄₇	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	47	2	no	—
197	C₁₆	—	47	2	no	—
198	C₁₆	—	94	2	no	—
199	C₂³	—	47	4	no	—
200	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
201	Q₈	—	94	4	no	—
202	D₄	—	94	8	no	—
203	D₄	—	94	8	no	—
204	D₄	—	94	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	47	4	no	—
207	C₂×C₄	—	94	4	no	—
208	C₂×C₄	—	94	4	no	—
209	C₂×C₄	#13093986	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
213	C₂×C₄	#13094032	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	94	4	no	—
217	C₈	#13093986	1	2	yes	—
218	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
219	C₈	C₂×#10625354	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	94	4	no	—
222	C₂²	—	94	8	no	—
223	C₂²	#15689188	1	2	yes	—
224	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	94	4	no	—
227	C₄	#15689188	1	2	yes	—
228	C₄	#15689541	1	4	yes	complemented
229	C₄	C₂×#13093986	1	1	yes	characteristic
230	C₄	C₂×#13094032	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	188	4	no	—
233	C₂	C₂×#15689188	1	1	yes	characteristic
234	C₂	C₂×#15689541	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2150	C₂
prime radical	#15689673	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#33434
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#15689541	C₂	semidirect
3. #20245219	C₂	semidirect
4. #20245207	C₂	semidirect
5. #20245003	C₂	semidirect
6. C₂×C₄×C₄₇⋊C₁₆	C₂	semidirect
7. C₄₇×#2150	C₂	semidirect
38. C₂×C₄₇⋊C₁₆	D₄	semidirect
110. C₄×C₄₇	#504	semidirect
118. #2150	D₄₇	semidirect
142. C₄₇	#33434	semidirect
228. C₄	#15689541	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689541	2
#20252101	2
#20245219	2
#20245207	2
#20245003	2
C₂×C₄×C₄₇⋊C₁₆	2
C₄₇×#2150	2
#33434	47