Subgroups — #26369266

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	2	yes	maximal
2	#20245147	C₂	1	8	yes	maximal, complemented
3	#20245144	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	—	2	8	no	—
6	C₂×#13093964	C₂²	1	4	yes	—
7	C₂×#13093964	C₂²	1	4	yes	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	16	yes	complemented
10	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#405	C₄	1	4	yes	—
14	C₄₇×#405	C₂²	1	4	yes	—
15	C₂×#97×C₄₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	#13093964	—	2	8	no	—
18	#13093964	—	2	8	no	—
19	C₂×C₄₇⋊C₁₆	—	2	8	no	—
20	C₂×C₄₇⋊C₁₆	C₂³	1	8	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₄₇	—	2	4	no	—
23	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₇	D₄	1	4	yes	—
28	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
29	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	#117×C₄₇	C₈	1	4	yes	—
32	#117×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₄₇	C₈	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	#97×C₄₇	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₄₇	C₂³	1	4	yes	—
39	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	M₁₆×C₄₇	—	2	8	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
61	C₄²×C₄₇	C₂×C₈	1	4	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
63	C₄₇⋊C₁₆	—	2	8	no	—
64	C₂³×C₄₇	—	4	2	no	—
65	C₂³×C₄₇	C₄⋊C₈	1	4	yes	—
66	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
67	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
74	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₄₇	—	2	4	no	—
80	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
84	#33397	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	—	2	4	no	—
87	C₂²×C₄₇	—	4	4	no	—
88	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
89	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
90	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₄₇	—	2	8	no	—
92	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
93	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
94	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
96	C₂²×M₃₂	—	47	2	no	—
97	C₂×#405	D₄₇	1	1	yes	characteristic
98	#2151	—	47	8	no	—
99	#2150	—	47	4	no	—
100	C₂×C₄₇	—	2	2	no	—
101	C₂×C₄₇	—	2	4	no	—
102	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₄₇	#3407	1	2	yes	—
104	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
106	C₂×#117	Dic₄₇	1	1	yes	characteristic
107	C₂×M₃₂	—	47	4	no	—
108	C₂×M₃₂	—	47	4	no	—
109	C₂×M₃₂	—	94	8	no	—
110	C₂²×C₁₆	—	47	4	no	—
111	#405	Dic₄₇	1	4	yes	—
112	#405	C₂×D₄₇	1	4	yes	—
113	C₂×#97	Dic₄₇	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
115	C₄⋊C₁₆	—	47	16	no	—
116	C₄₇	#33397	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
120	C₂²×C₈	Dic₉₄	1	4	yes	—
121	C₂²×C₈	D₁₈₈	1	4	yes	—
122	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
123	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
125	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
127	#117	C₄₇⋊C₈	1	4	yes	—
128	#117	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄⋊C₄	C₄₇⋊C₈	1	2	yes	—
130	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	2	yes	—
131	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
132	M₃₂	—	94	8	no	—
133	M₃₂	—	94	8	no	—
134	C₂×C₁₆	—	47	8	no	—
135	C₂×C₁₆	—	94	8	no	—
136	#97	C₂×Dic₄₇	1	8	yes	—
137	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
138	C₄×C₈	C₂²×D₄₇	1	4	yes	—
139	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
141	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
143	C₂²×C₄	#10625329	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
145	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
147	M₁₆	—	2	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
150	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
151	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
152	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
153	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
154	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
155	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
156	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
158	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
160	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
161	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
162	C₄²	C₂²×Dic₄₇	1	4	yes	—
163	C₁₆	—	94	8	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
166	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
167	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
170	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
173	C₂×C₄	C₂²×C₄₇⋊C₈	1	8	yes	—
174	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
176	C₂×C₄	C₂×#10625329	1	2	yes	—
177	C₂×C₄	C₂×#10625329	1	4	yes	—
178	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₄⋊Dic₄₇	1	4	yes	—
181	C₈	C₂×C₄⋊Dic₄₇	1	2	yes	—
182	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
183	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
188	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
189	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
192	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	4	yes	—
193	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
194	C₄	C₂²×#10625329	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
198	C₂	#20256491	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄₇×#405	C₂
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#33397
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20245147	C₂	semidirect
3. #20245144	C₂	semidirect
9. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
116. C₄₇	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
#20245147	2
#20245144	2
C₂×C₄₇×#405	2
#33397	47