Subgroups — #26216923

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20243560	C₂	1	2	yes	maximal, complemented
3	#20243526	C₂	1	2	yes	maximal, complemented
4	#20243430	C₂	1	4	yes	maximal
5	#20243294	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689180	—	2	4	no	—
8	#15689176	—	2	4	no	—
9	C₂×#13093960	C₂²	1	8	yes	—
10	#15689147	C₂²	1	4	yes	—
11	#15689087	C₂²	1	4	yes	—
12	C₄×#10625329	C₂²	1	4	yes	—
13	#15688888	C₂²	1	8	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₄₇	C₄	1	4	yes	—
17	C₄×M₁₆×C₄₇	C₄	1	4	yes	—
18	C₂×#97×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	C₂³	1	8	yes	—
22	#13093960	—	2	16	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
25	#13093935	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₄₇	—	2	8	no	—
30	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	4	yes	—
36	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	#97×C₄₇	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	M₁₆×C₄₇	—	2	16	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₄₇	#41	1	8	yes	—
60	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	#10625329	—	2	16	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	—	4	2	no	—
71	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	8	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	4	4	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
87	C₈×C₄₇	—	2	4	no	—
88	C₈×C₄₇	#117	1	8	yes	—
89	C₄₇⋊C₈	—	4	8	no	—
90	#20831	—	47	1	no	maximal, Sylow
91	C₂²×C₄₇	—	2	2	no	—
92	C₂²×C₄₇	—	2	4	no	—
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	4	4	no	—
95	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₄₇	#533	1	2	yes	—
97	C₄×C₄₇	—	2	4	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	#430	1	4	yes	—
101	C₄×C₄₇	C₂×#117	1	2	yes	—
102	C₄×C₄₇	#532	1	2	yes	—
103	#2920	—	47	2	no	—
104	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
105	#1848	—	47	2	no	—
106	#1837	—	47	2	no	—
107	#1811	—	47	4	no	—
108	#1778	—	47	4	no	—
109	C₂×C₄₇	—	2	2	no	—
110	C₂×C₄₇	—	4	2	no	—
111	C₂×C₄₇	#2285	1	1	yes	characteristic
112	C₂×C₄₇	C₂×#430	1	2	yes	—
113	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
114	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
115	#432	—	94	4	no	—
116	#431	—	94	4	no	—
117	C₂×#108	—	47	8	no	—
118	#423	—	47	4	no	—
119	#407	—	47	4	no	—
120	C₄×M₁₆	—	47	4	no	—
121	C₄×M₁₆	Dic₄₇	1	4	yes	—
122	C₄×M₁₆	Dic₄₇	1	4	yes	—
123	C₂×#97	C₂×D₄₇	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
126	#341	—	47	8	no	—
127	C₄₇	#20831	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₄	Dic₉₄	1	2	yes	—
129	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
130	C₂×M₁₆	—	2	8	no	—
131	C₂×M₁₆	—	47	8	no	—
132	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
133	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
134	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
135	C₂×C₄²	Dic₉₄	1	2	yes	—
136	C₂×C₄²	D₁₈₈	1	4	yes	—
137	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
138	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
139	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	#108	—	94	16	no	—
142	C₄⋊C₈	—	94	8	no	—
143	C₂²⋊C₈	—	94	8	no	—
144	#97	—	94	4	no	—
145	#97	C₂×Dic₄₇	1	8	yes	—
146	C₄×C₈	—	94	4	no	—
147	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
148	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
149	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
150	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
151	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
156	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
158	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
159	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
160	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
162	M₁₆	—	2	16	no	—
163	M₁₆	—	94	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	94	8	no	—
166	C₂×C₈	#10625350	1	8	yes	—
167	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
168	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
169	C₄²	C₄⋊Dic₄₇	1	4	yes	—
170	C₄²	C₄⋊Dic₄₇	1	4	yes	—
171	C₄²	C₂×D₁₈₈	1	2	yes	—
172	C₄²	C₂×D₁₈₈	1	2	yes	—
173	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
174	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
175	C₄²	C₂²×Dic₄₇	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	4	2	no	—
178	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
179	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
189	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
191	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#10625350	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	188	8	no	—
196	C₈	#13093983	1	8	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
202	C₂²	#15689680	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	#15689172	1	4	yes	—
207	C₄	C₂×#13093983	1	2	yes	—
208	C₄	#15689673	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	4	2	no	—
211	C₂	#20245795	1	1	yes	characteristic
212	C₂	C₂×#15689172	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20831
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20243560	C₂	semidirect
3. #20243526	C₂	semidirect
5. #20243294	C₂	semidirect
127. C₄₇	#20831	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20243560	2
#20243526	2
#20243430	2
#20243294	2
C₂×C₄×M₁₆×C₄₇	2
#20831	47