Subgroups — #26127876

Order 12032.

262 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241457	C₂	1	2	yes	maximal, complemented
4	#20241451	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
10	C₄×C₄₇⋊C₁₆	—	2	1	no	—
11	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₄₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₄×C₁₆×C₄₇	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	1	no	—
24	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
25	C₄×C₄₇⋊C₈	C₈	1	1	yes	characteristic
26	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
50	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	—	2	1	no	—
55	C₂×C₄₇⋊C₈	C₁₆	1	4	yes	complemented
56	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
60	C₁₆×C₄₇	—	4	2	no	—
61	C₄₇⋊C₁₆	—	4	2	no	—
62	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₄₇	—	2	1	no	—
75	C₈×C₄₇	—	2	1	no	—
76	C₈×C₄₇	#108	1	2	yes	—
77	C₈×C₄₇	#135	1	2	yes	—
78	C₄₇⋊C₈	—	2	2	no	—
79	C₄₇⋊C₈	—	2	2	no	—
80	C₄₇⋊C₈	—	2	1	no	—
81	C₄₇⋊C₈	—	2	1	no	—
82	C₄₇⋊C₈	C₂×C₁₆	1	2	yes	—
83	C₄₇⋊C₈	C₂×C₁₆	1	2	yes	—
84	#12012	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	1	no	—
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
88	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₄₇	—	2	1	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	C₄⋊C₁₆	1	2	yes	—
92	C₄×C₄₇	#364	1	2	yes	—
93	C₄×C₄₇	#427	1	1	yes	characteristic
94	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
95	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
96	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
97	#2162	—	47	1	no	—
98	#2162	D₄₇	1	1	yes	characteristic
99	C₂×C₈²	—	47	1	no	—
100	#1370	—	47	2	no	—
101	#1367	—	47	2	no	—
102	C₂×C₄₇	—	2	2	no	—
103	C₂×C₄₇	#1564	1	1	yes	characteristic
104	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
105	C₂×C₄₇	C₂×#364	1	1	yes	characteristic
106	C₂×C₄×C₈	—	47	1	no	—
107	C₂×C₄×C₈	—	47	1	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄⋊C₁₆	—	47	2	no	—
110	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
111	C₂²⋊C₁₆	—	2	1	no	—
112	C₂²⋊C₁₆	—	94	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	—	94	1	no	—
115	C₈²	—	47	1	no	—
116	C₈²	—	47	1	no	—
117	C₈²	—	47	1	no	—
118	C₈²	—	47	1	no	—
119	C₄₇	#12012	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	—	47	1	no	—
122	C₂²×C₈	—	94	1	no	—
123	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	94	2	no	—
128	C₄×C₈	—	47	1	no	—
129	C₄×C₈	—	47	1	no	—
130	C₄×C₈	—	47	1	no	—
131	C₄×C₈	—	47	1	no	—
132	C₄×C₈	—	47	1	no	—
133	C₄×C₈	—	47	1	no	—
134	C₄×C₈	—	94	1	no	—
135	C₄×C₈	Dic₉₄	1	1	yes	characteristic
136	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
140	C₂²×C₄	#10625324	1	1	yes	characteristic
141	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	47	4	no	—
145	C₂×C₈	—	47	1	no	—
146	C₂×C₈	—	47	1	no	—
147	C₂×C₈	—	47	1	no	—
148	C₂×C₈	—	47	1	no	—
149	C₂×C₈	—	94	4	no	—
150	C₂×C₈	—	94	2	no	—
151	C₂×C₈	—	94	1	no	—
152	C₂×C₈	—	94	1	no	—
153	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₄²	C₈×D₄₇	1	1	yes	characteristic
158	C₄²	#10625324	1	1	yes	characteristic
159	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	188	2	no	—
163	C₂³	Dic₄₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
168	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
169	C₂×C₄	Dic₄₇⋊C₈	1	1	yes	characteristic
170	C₂×C₄	C₁₆×D₄₇	1	2	yes	—
171	C₂×C₄	#13093963	1	2	yes	—
172	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
174	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	47	2	no	—
178	C₈	—	47	2	no	—
179	C₈	—	94	2	no	—
180	C₈	—	94	2	no	—
181	C₈	—	94	1	no	—
182	C₈	—	94	1	no	—
183	C₈	#13093959	1	2	yes	—
184	C₈	#13094044	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	Dic₄₇⋊C₁₆	1	2	yes	—
188	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	Dic₄₇⋊C₁₆	1	2	yes	—
192	C₄	#15688943	1	2	yes	—
193	C₄	#15689159	1	1	yes	characteristic
194	C₄	C₂×#13093959	1	1	yes	characteristic
195	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
196	C₄	C₂×#13093963	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#20242134	1	1	yes	characteristic
199	C₂	C₂×Dic₄₇⋊C₁₆	1	1	yes	characteristic
200	C₂	C₂×#15688943	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2162	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2162	D₄₇
hypocenter	C₄₇	#12012
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20241457	C₂	semidirect
4. #20241451	C₂	semidirect
55. C₂×C₄₇⋊C₈	C₁₆	semidirect
119. C₄₇	#12012	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
C₂×C₈×C₄₇⋊C₈	2
#20241457	2
#20241451	2
C₄₇×#2162	2
#12012	47