Subgroups — #26214448

Order 12032.

930 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	C₂×#15689180	C₂	1	2	yes	maximal
2	#20243245	C₂	1	4	yes	maximal, complemented
3	#20243154	C₂	1	4	yes	maximal
4	#20243137	C₂	1	2	yes	maximal, complemented
5	C₄×#13093935	C₂	1	2	yes	maximal, complemented
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689180	C₄	1	16	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13093935	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	4	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
13	#15688878	C₂²	1	8	yes	—
14	C₂×#117×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
17	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
18	C₄₇×#376	C₂²	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
22	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
23	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
24	#13093935	—	2	8	no	—
25	#13093935	C₂×C₄	1	16	yes	—
26	C₄×C₄₇⋊C₈	—	2	8	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	#117×C₄₇	—	2	8	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
40	#95×C₄₇	C₂×C₄	1	8	yes	—
41	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	4	no	—
44	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₄₇	—	2	8	no	—
53	C₂²⋊C₄×C₄₇	—	2	8	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	—	4	4	no	—
56	C₄²×C₄₇	—	4	4	no	—
57	C₄²×C₄₇	C₄²	1	8	yes	—
58	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
59	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
60	C₄²×C₄₇	#41	1	8	yes	—
61	C₂×C₄₇⋊C₈	—	2	8	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
64	C₂³×C₄₇	—	2	2	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
67	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	8	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	8	no	—
75	C₂×C₄×C₄₇	—	4	8	no	—
76	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
78	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₄₇	#117	1	8	yes	—
80	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
82	C₄₇⋊C₈	—	2	16	no	—
83	#20467	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	4	no	—
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	—	4	4	no	—
87	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#117	1	2	yes	—
89	C₂²×C₄₇	#568	1	4	yes	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	2	8	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	—	4	4	no	—
95	C₄×C₄₇	C₄×#41	1	4	yes	—
96	C₂×#432	—	47	2	no	—
97	C₄×#117	D₄₇	1	1	yes	characteristic
98	#1768	—	47	4	no	—
99	#1750	—	47	4	no	—
100	#1745	—	47	2	no	—
101	C₄×#97	—	47	2	no	—
102	C₂×C₄₇	—	4	2	no	—
103	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
104	C₂×C₄₇	#2873	1	2	yes	—
105	C₂×C₄₇	#2923	1	4	yes	—
106	C₂×#117	C₂×D₄₇	1	2	yes	—
107	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
108	#432	—	47	16	no	—
109	C₂×C₄⋊C₈	—	47	4	no	—
110	C₂×C₂²⋊C₈	—	47	4	no	—
111	C₂×#97	—	47	4	no	—
112	C₂×#97	—	47	4	no	—
113	C₂×C₄×C₈	—	47	4	no	—
114	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
115	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
116	#376	C₂×D₄₇	1	2	yes	—
117	C₄³	C₂×D₄₇	1	1	yes	characteristic
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	47	8	no	—
120	C₄₇	#20467	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
122	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
123	C₂²×C₈	—	47	8	no	—
124	#117	—	2	8	no	—
125	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
127	C₂×C₄²	C₄×D₄₇	1	8	yes	—
128	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
130	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
131	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
135	C₄⋊C₈	—	47	16	no	—
136	C₂²⋊C₈	—	47	16	no	—
137	#97	—	47	16	no	—
138	#97	—	94	8	no	—
139	C₄×C₈	—	94	8	no	—
140	#95	C₂×Dic₄₇	1	8	yes	—
141	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
145	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
146	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
151	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
152	C₂×C₈	—	47	16	no	—
153	C₂×C₈	—	94	8	no	—
154	C₂×C₈	—	94	8	no	—
155	C₄⋊C₄	—	2	8	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	4	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₄×Dic₄₇	1	8	yes	—
161	C₄²	C₂×C₄×D₄₇	1	4	yes	—
162	C₄²	C₂×C₄×D₄₇	1	4	yes	—
163	C₄²	#10625350	1	8	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	4	no	—
166	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
167	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
177	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
178	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
179	C₂×C₄	#13093983	1	8	yes	—
180	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
181	C₂×C₄	C₂×#10625350	1	4	yes	—
182	C₈	—	94	16	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
187	C₂²	C₂×#13093983	1	2	yes	—
188	C₂²	#15689890	1	4	yes	—
189	C₄	—	2	8	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	C₄×#10625350	1	4	yes	—
195	C₂	—	4	2	no	—
196	C₂	C₄×#13093983	1	1	yes	characteristic
197	C₂	#20251807	1	2	yes	—
198	C₂	#20252039	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20467
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243245	C₂	semidirect
4. #20243137	C₂	semidirect
5. C₄×#13093935	C₂	semidirect
7. #15689180	C₄	semidirect
120. C₄₇	#20467	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
#20243245	2
#20243154	2
#20243137	2
C₄×#13093935	2
C₄×#117×C₄₇	2
#20467	47