Subgroups — #26216761

Order 12032.

890 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20243542	C₂	1	2	yes	maximal, complemented
3	#20243508	C₂	1	2	yes	maximal, complemented
4	#20243372	C₂	1	4	yes	maximal
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689176	—	2	8	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	#15688878	C₂²	1	8	yes	—
13	C₂×#117×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
16	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
17	C₄₇×#376	C₂²	1	1	yes	characteristic
18	C₄₇×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	C₄×C₄₇⋊C₈	—	2	8	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	#117×C₄₇	—	2	8	no	—
28	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	4	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
38	#95×C₄₇	C₂×C₄	1	4	yes	—
39	#95×C₄₇	C₂×C₄	1	4	yes	—
40	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	8	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	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	16	no	—
55	C₄²×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₄²×C₄₇	C₂×D₄	1	2	yes	—
59	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
60	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
61	C₄²×C₄₇	#41	1	8	yes	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	2	16	no	—
64	C₂³×C₄₇	—	2	2	no	—
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	16	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	8	no	—
76	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₄₇	#117	1	8	yes	—
80	C₂×C₄×C₄₇	#131	1	8	yes	—
81	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
84	C₄₇⋊C₈	—	4	8	no	—
85	#20809	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	4	no	—
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	—	4	4	no	—
89	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×#117	1	2	yes	—
91	C₂²×C₄₇	C₂×#131	1	4	yes	—
92	C₄×C₄₇	—	2	4	no	—
93	C₄×C₄₇	—	2	8	no	—
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	4	4	no	—
96	C₄×C₄₇	#424	1	8	yes	—
97	C₄×C₄₇	#532	1	2	yes	—
98	C₄×C₄₇	#533	1	2	yes	—
99	C₂×#431	—	47	2	no	—
100	C₄×#117	D₄₇	1	1	yes	characteristic
101	#1843	—	47	2	no	—
102	#1832	—	47	2	no	—
103	#1798	—	47	4	no	—
104	C₄×C₄⋊C₈	—	47	4	no	—
105	C₂×C₄₇	—	4	2	no	—
106	C₂×C₄₇	#2285	1	1	yes	characteristic
107	C₂×C₄₇	C₂×#424	1	2	yes	—
108	C₂×C₄₇	#2922	1	4	yes	—
109	C₂×#117	C₂×D₄₇	1	2	yes	—
110	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	#431	—	94	8	no	—
112	C₂×C₄⋊C₈	—	47	8	no	—
113	C₂×C₄⋊C₈	—	47	4	no	—
114	C₂×C₂²⋊C₈	—	47	4	no	—
115	C₂×C₄×C₈	—	47	4	no	—
116	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
117	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
118	#376	C₂×D₄₇	1	1	yes	characteristic
119	#376	C₂×D₄₇	1	1	yes	characteristic
120	C₄³	C₂×D₄₇	1	2	yes	—
121	#336	—	47	8	no	—
122	C₄₇	#20809	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
124	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂²×C₈	—	47	8	no	—
126	#117	—	2	8	no	—
127	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄²	Dic₉₄	1	4	yes	—
130	C₂×C₄²	D₁₈₈	1	4	yes	—
131	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
132	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
133	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊C₈	—	94	8	no	—
138	C₄⋊C₈	—	94	16	no	—
139	C₂²⋊C₈	—	94	8	no	—
140	C₄×C₈	—	94	8	no	—
141	#95	C₂×Dic₄₇	1	4	yes	—
142	#95	C₂×Dic₄₇	1	4	yes	—
143	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
147	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
148	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
151	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
152	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
153	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
154	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
155	C₂×C₈	—	94	8	no	—
156	C₂×C₈	—	94	16	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	8	no	—
159	C₄²	—	2	16	no	—
160	C₄²	—	2	4	no	—
161	C₄²	C₄⋊Dic₄₇	1	8	yes	—
162	C₄²	C₂×Dic₉₄	1	2	yes	—
163	C₄²	C₂×Dic₉₄	1	2	yes	—
164	C₄²	C₂×D₁₈₈	1	2	yes	—
165	C₄²	C₂×D₁₈₈	1	2	yes	—
166	C₄²	#10625350	1	8	yes	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
170	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
182	C₂×C₄	#13093983	1	8	yes	—
183	C₂×C₄	#13094031	1	8	yes	—
184	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
186	C₂×C₄	C₂×#10625350	1	4	yes	—
187	C₈	—	188	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
192	C₂²	C₂×#13093983	1	2	yes	—
193	C₂²	C₂×#13094031	1	4	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	4	4	no	—
198	C₄	#15689150	1	8	yes	—
199	C₄	#15689673	1	2	yes	—
200	C₄	#15689680	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	#20245795	1	1	yes	characteristic
203	C₂	C₂×#15689150	1	2	yes	—
204	C₂	#20252032	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20809
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243542	C₂	semidirect
3. #20243508	C₂	semidirect
5. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
122. C₄₇	#20809	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20243542	2
#20243508	2
#20243372	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄×#117×C₄₇	2
#20809	47