Subgroups — #26216883

Order 12032.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	2	yes	maximal
2	#20243552	C₂	1	2	yes	maximal, complemented
3	#20243519	C₂	1	2	yes	maximal, complemented
4	#20243421	C₂	1	4	yes	maximal
5	#20243285	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093958	C₂²	1	4	yes	—
8	C₂×#13093956	C₂²	1	4	yes	—
9	#15688991	—	2	2	no	—
10	#15688989	C₂²	1	4	yes	—
11	#15688986	—	2	2	no	—
12	#15688983	C₂²	1	4	yes	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
15	#15688969	—	2	2	no	—
16	#15688886	C₂²	1	8	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
20	C₂×#97×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄×Dic₄₇	—	2	4	no	—
26	#13093958	—	2	8	no	—
27	#13093956	—	2	8	no	—
28	C₄₇⋊#95	—	2	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₄₇⋊#95	—	2	4	no	—
31	C₄₇⋊#95	—	2	4	no	—
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	4	no	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#97×C₄₇	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	8	no	—
59	M₁₆×C₄₇	—	2	16	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	#41	1	4	yes	—
65	C₂×C₈×C₄₇	#41	1	4	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
70	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²⋊Dic₄₇	—	4	8	no	—
72	C₄⋊Dic₄₇	—	2	8	no	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄×Dic₄₇	—	4	8	no	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	4	8	no	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
93	C₂×Dic₄₇	—	4	8	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₂×Dic₄₇	—	4	8	no	—
96	C₈×C₄₇	—	2	4	no	—
97	C₈×C₄₇	#117	1	8	yes	—
98	#20825	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	4	4	no	—
102	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₇	#532	1	2	yes	—
104	C₂²×C₄₇	#575	1	4	yes	—
105	C₄×C₄₇	—	2	4	no	—
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	C₂×#117	1	2	yes	—
109	C₄×C₄₇	#533	1	1	yes	characteristic
110	C₄×C₄₇	#533	1	1	yes	characteristic
111	Dic₄₇	—	8	8	no	—
112	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
113	#2283	—	47	2	no	—
114	#1846	—	47	2	no	—
115	#1835	—	47	2	no	—
116	#1809	—	47	4	no	—
117	#1776	—	47	4	no	—
118	C₂×C₄₇	—	4	2	no	—
119	C₂×C₄₇	#2285	1	1	yes	characteristic
120	C₂×C₄₇	#2911	1	2	yes	—
121	C₂×C₄₇	#3183	1	4	yes	—
122	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	C₂×#107	—	47	4	no	—
125	C₂×#106	—	47	4	no	—
126	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
127	C₂×#97	C₂×D₄₇	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
130	#383	—	94	2	no	—
131	#382	—	47	4	no	—
132	#381	—	94	2	no	—
133	#380	—	47	4	no	—
134	C₄×C₄⋊C₄	—	47	4	no	—
135	C₄×C₂²⋊C₄	—	94	2	no	—
136	#376	—	94	2	no	—
137	#340	—	47	8	no	—
138	C₄₇	#20825	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	D₁₈₈	1	2	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
144	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
145	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
146	C₂×C₄⋊C₄	—	47	8	no	—
147	C₂×C₂²⋊C₄	—	94	4	no	—
148	C₂×C₄²	—	94	4	no	—
149	C₂×C₄²	Dic₉₄	1	4	yes	—
150	C₂×C₄²	D₁₈₈	1	2	yes	—
151	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
154	#107	—	94	8	no	—
155	#106	—	94	8	no	—
156	#97	C₂×Dic₄₇	1	8	yes	—
157	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
158	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
159	#95	—	94	4	no	—
160	#95	—	94	4	no	—
161	#95	—	94	4	no	—
162	#95	—	94	4	no	—
163	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	—	94	8	no	—
168	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
169	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
170	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
172	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	#10625350	1	4	yes	—
179	C₂×C₈	#10625350	1	4	yes	—
180	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
181	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
182	C₄⋊C₄	—	94	8	no	—
183	C₄⋊C₄	—	94	8	no	—
184	C₂²⋊C₄	—	188	8	no	—
185	C₄²	—	188	8	no	—
186	C₄²	C₄⋊Dic₄₇	1	8	yes	—
187	C₄²	C₂×Dic₉₄	1	2	yes	—
188	C₄²	C₂×Dic₉₄	1	2	yes	—
189	C₄²	C₂²×Dic₄₇	1	2	yes	—
190	C₄²	C₂²×Dic₄₇	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
194	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	188	8	no	—
203	C₂×C₄	—	188	8	no	—
204	C₂×C₄	—	188	8	no	—
205	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
208	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
210	C₂×C₄	C₂×#10625350	1	2	yes	—
211	C₂×C₄	C₂×#10625350	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	#13093983	1	8	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
218	C₂²	#15689673	1	2	yes	—
219	C₂²	#15689923	1	4	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	376	8	no	—
224	C₄	C₂×#13093983	1	2	yes	—
225	C₄	#15689680	1	1	yes	characteristic
226	C₄	#15689680	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#20245795	1	1	yes	characteristic
229	C₂	#20251978	1	2	yes	—
230	C₂	#20254117	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20825
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20243552	C₂	semidirect
3. #20243519	C₂	semidirect
5. #20243285	C₂	semidirect
19. C₄×M₁₆×C₄₇	C₄	semidirect
126. C₄×M₁₆	Dic₄₇	semidirect
138. C₄₇	#20825	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
#20243552	2
#20243519	2
#20243421	2
#20243285	2
C₂×C₄×M₁₆×C₄₇	2
#20825	47