Subgroups — #26215991

Order 12032.

1446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	#20243459	C₂	1	8	yes	maximal, complemented
3	#20243082	C₂	1	4	yes	maximal, complemented
4	C₂²×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	#15689087	—	2	8	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	#15688878	C₂²	1	16	yes	complemented
9	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₄₇	—	2	8	no	—
11	C₂×#117×C₄₇	C₄	1	4	yes	—
12	C₂×#117×C₄₇	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	—	2	16	no	—
18	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
20	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	—	2	8	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₇	D₄	1	8	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	#117×C₄₇	—	2	16	no	—
29	#117×C₄₇	—	2	16	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
33	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₄₇	—	2	8	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
37	C₂⁴×C₄₇	—	2	4	no	—
38	C₂⁴×C₄₇	—	4	4	no	—
39	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
40	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	8	no	—
44	C₂²×C₄×C₄₇	—	2	32	no	—
45	C₂²×C₄×C₄₇	—	2	8	no	—
46	C₂²×C₄×C₄₇	—	2	8	no	—
47	C₂²×C₄×C₄₇	—	4	8	no	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₇	#41	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₄₇	—	2	16	no	—
58	C₂²⋊C₄×C₄₇	—	2	16	no	—
59	C₄²×C₄₇	—	2	8	no	—
60	C₄²×C₄₇	—	2	8	no	—
61	#10625329	—	4	16	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	2	16	no	—
64	C₂³×C₄₇	—	2	4	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	—	2	8	no	—
67	C₂³×C₄₇	—	2	8	no	—
68	C₂³×C₄₇	—	4	8	no	—
69	C₂³×C₄₇	—	4	16	no	—
70	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
71	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂³×C₄₇	#117	1	4	yes	—
73	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	8	no	—
76	C₂×C₄×C₄₇	—	2	16	no	—
77	C₂×C₄×C₄₇	—	2	16	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	—	2	16	no	—
80	C₂×C₄×C₄₇	—	4	16	no	—
81	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	#117	1	4	yes	—
84	C₂×C₄×C₄₇	#123	1	16	yes	—
85	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
87	C₄₇⋊C₈	—	4	8	no	—
88	#20683	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	4	no	—
90	C₂²×C₄₇	—	2	8	no	—
91	C₂²×C₄₇	—	2	8	no	—
92	C₂²×C₄₇	—	2	4	no	—
93	C₂²×C₄₇	—	4	16	no	—
94	C₂²×C₄₇	—	4	8	no	—
95	C₂²×C₄₇	#381	1	4	yes	—
96	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#117	1	2	yes	—
98	C₂²×C₄₇	#568	1	4	yes	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	4	8	no	—
101	C₄×C₄₇	#381	1	4	yes	—
102	C₄×C₄₇	C₂×#123	1	4	yes	—
103	C₂²×#117	D₄₇	1	1	yes	characteristic
104	C₂×#407	—	47	2	no	—
105	#1818	—	47	8	no	—
106	#1728	—	47	4	no	—
107	C₂×C₄₇	—	2	4	no	—
108	C₂×C₄₇	—	4	4	no	—
109	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
110	C₂×C₄₇	#2879	1	2	yes	—
111	C₂×C₄₇	#2921	1	4	yes	—
112	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
113	C₂²×M₁₆	—	47	4	no	—
114	C₂×#117	—	2	8	no	—
115	C₂×#117	Dic₄₇	1	4	yes	—
116	C₂×#117	C₂×D₄₇	1	4	yes	—
117	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
118	C₂²×C₂²⋊C₄	Dic₄₇	1	2	yes	—
119	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
120	#407	—	94	8	no	—
121	C₂×C₂²⋊C₈	—	47	8	no	—
122	#336	—	47	16	no	—
123	C₄₇	#20683	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
127	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
128	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
129	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
130	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
131	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
132	C₂×M₁₆	—	94	16	no	—
133	C₂×M₁₆	—	94	8	no	—
134	C₂²×C₈	—	47	8	no	—
135	#117	—	2	16	no	—
136	#117	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
139	C₂×C₂²⋊C₄	—	2	8	no	—
140	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
141	C₂×C₄²	—	2	8	no	—
142	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
143	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
144	C₂²⋊C₈	—	94	16	no	—
145	C₂⁴	—	2	4	no	—
146	C₂⁴	—	4	4	no	—
147	C₂⁴	C₂²⋊Dic₄₇	1	4	yes	—
148	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
149	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	32	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
157	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
158	C₂²×C₄	#10625350	1	8	yes	—
159	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
160	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
164	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
165	M₁₆	—	188	16	no	—
166	C₂×C₈	—	94	8	no	—
167	C₂×C₈	—	94	16	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₂²⋊C₄	—	2	16	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	8	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	8	no	—
176	C₂³	—	4	8	no	—
177	C₂³	—	4	16	no	—
178	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
179	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
180	C₂³	#13093983	1	4	yes	—
181	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	4	16	no	—
189	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
190	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
191	C₂×C₄	#13093983	1	4	yes	—
192	C₂×C₄	#13094007	1	16	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
194	C₂×C₄	C₂×#10625350	1	4	yes	—
195	C₈	—	188	8	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	4	16	no	—
201	C₂²	—	4	8	no	—
202	C₂²	#15688986	1	4	yes	—
203	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
204	C₂²	C₂×#13093983	1	2	yes	—
205	C₂²	#15689890	1	4	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	#15688986	1	4	yes	—
209	C₄	C₂×#13094007	1	4	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	4	no	—
212	C₂	C₂×#15688986	1	1	yes	characteristic
213	C₂	#20251833	1	2	yes	—
214	C₂	#20252026	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂²×#117×C₄₇	C₂
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#117	D₄₇
hypocenter	C₄₇	#20683
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #20243459	C₂	semidirect
3. #20243082	C₂	semidirect
8. #15688878	C₂²	semidirect
123. C₄₇	#20683	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
#20243459	2
#20243082	2
C₂²×#117×C₄₇	2
#20683	47