Subgroups — #26106364

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#20241271	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	4	no	—
5	#15689176	C₄	1	8	yes	complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂×#117×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#384	C₂²	1	2	yes	—
12	C₄₇×#382	C₄	1	2	yes	—
13	C₄₇×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
18	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
19	C₄×C₄₇⋊C₈	—	2	8	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	D₄	1	2	yes	—
22	#117×C₄₇	—	4	4	no	—
23	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄²×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#95×C₄₇	—	2	4	no	—
34	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	8	no	—
38	C₂²×C₄×C₄₇	—	2	4	no	—
39	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₄⋊C₄×C₄₇	—	4	4	no	—
48	C₄⋊C₄×C₄₇	—	4	8	no	—
49	C₂²⋊C₄×C₄₇	—	4	4	no	—
50	C₄²×C₄₇	—	4	4	no	—
51	C₄²×C₄₇	C₄²	1	2	yes	—
52	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄₇⋊C₈	—	2	8	no	—
59	C₂×C₄₇⋊C₈	—	4	8	no	—
60	C₂³×C₄₇	—	2	1	no	—
61	C₂³×C₄₇	—	2	1	no	—
62	C₂³×C₄₇	—	2	4	no	—
63	C₂³×C₄₇	#95	1	2	yes	—
64	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	8	no	—
72	C₂×C₄×C₄₇	—	4	4	no	—
73	C₂×C₄×C₄₇	—	4	8	no	—
74	C₂×C₄×C₄₇	#95	1	4	yes	—
75	C₂×C₄×C₄₇	#95	1	2	yes	—
76	C₂×C₄×C₄₇	#100	1	2	yes	—
77	C₂×C₄×C₄₇	#101	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#135	1	8	yes	—
83	C₄₇⋊C₈	—	4	8	no	—
84	#8038	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	4	no	—
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	4	2	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
92	C₂²×C₄₇	#416	1	4	yes	—
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	8	4	no	—
97	C₄×C₄₇	#343	1	4	yes	—
98	C₄×C₄₇	#416	1	4	yes	—
99	C₄×C₄₇	#427	1	4	yes	—
100	C₂×#431	—	47	2	no	—
101	#2285	D₄₇	1	1	yes	characteristic
102	#1296	—	47	4	no	—
103	C₂×C₄₇	—	4	2	no	—
104	C₂×C₄₇	#1543	1	2	yes	—
105	C₂×C₄₇	#1544	1	2	yes	—
106	C₂×C₄₇	#1735	1	2	yes	—
107	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
108	C₂×#117	Dic₄₇	1	2	yes	—
109	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
110	#431	—	47	8	no	—
111	#431	—	94	4	no	—
112	C₂×C₄⋊C₈	—	47	4	no	—
113	C₂×C₂²⋊C₈	—	94	2	no	—
114	C₂×C₄×C₈	—	94	2	no	—
115	#384	C₂×D₄₇	1	2	yes	—
116	#382	Dic₄₇	1	2	yes	—
117	#380	—	2	2	no	—
118	C₄×C₄⋊C₄	—	2	2	no	—
119	C₄₇	#8038	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
121	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
122	C₂²×C₈	—	94	4	no	—
123	#117	—	4	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₂²⋊C₄	—	2	2	no	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	Dic₉₄	1	2	yes	—
129	C₂×C₄²	C₄×D₄₇	1	4	yes	—
130	C₂×C₄²	D₁₈₈	1	2	yes	—
131	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₈	—	47	8	no	—
135	C₄⋊C₈	—	94	4	no	—
136	C₂²⋊C₈	—	94	8	no	—
137	C₄×C₈	—	94	8	no	—
138	#95	—	2	4	no	—
139	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
145	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
146	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
147	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
148	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
149	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
152	C₂×C₈	—	94	8	no	—
153	C₂×C₈	—	188	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₄×Dic₄₇	1	2	yes	—
159	C₄²	Dic₄₇⋊C₄	1	4	yes	—
160	C₄²	C₄⋊Dic₄₇	1	2	yes	—
161	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
162	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
163	C₄²	C₂×C₄×D₄₇	1	2	yes	—
164	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	C₄₇⋊#95	1	2	yes	—
169	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	C₄₇⋊#95	1	4	yes	—
180	C₂×C₄	C₄₇⋊#95	1	2	yes	—
181	C₂×C₄	#13093941	1	2	yes	—
182	C₂×C₄	#13093943	1	2	yes	—
183	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
185	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
187	C₂×C₄	#13094044	1	8	yes	—
188	C₈	—	188	8	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
194	C₂²	C₂×#13093941	1	1	yes	characteristic
195	C₂²	C₂×#13093943	1	1	yes	characteristic
196	C₂²	#15689119	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#15688892	1	4	yes	—
202	C₄	#15689117	1	4	yes	—
203	C₄	#15689159	1	4	yes	—
204	C₂	—	4	2	no	—
205	C₂	#20242050	1	2	yes	—
206	C₂	#20242053	1	2	yes	—
207	C₂	#20243104	1	2	yes	—
208	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#8038
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #20241271	C₂	semidirect
5. #15689176	C₄	semidirect
119. C₄₇	#8038	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
#20241271	2
C₄₇×#2285	2
#8038	47