Subgroups — #26110015

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688873	C₂	1	4	yes	maximal
4	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15688873	—	2	8	no	—
12	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	—	2	4	no	—
22	C₄×C₄₇⋊C₈	C₈	1	4	yes	complemented
23	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
24	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	4	no	—
32	C₄⋊C₈×C₄₇	—	4	4	no	—
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	4	no	—
40	C₄²×C₄₇	—	2	4	no	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	C₂×C₈	1	2	yes	—
43	C₄²×C₄₇	M₁₆	1	2	yes	—
44	C₄²×C₄₇	SD₁₆	1	4	yes	—
45	C₄²×C₄₇	SD₁₆	1	4	yes	—
46	C₄²×C₄₇	C₂×D₄	1	2	yes	—
47	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄₇⋊C₈	—	2	2	no	—
49	C₂×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₄₇⋊C₈	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
54	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
55	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
56	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
61	C₂×C₄×C₄₇	#106	1	4	yes	—
62	C₂×C₄×C₄₇	#106	1	4	yes	—
63	C₂×C₄×C₄₇	#108	1	4	yes	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
73	C₈×C₄₇	—	8	2	no	—
74	C₄₇⋊C₈	—	2	4	no	—
75	C₄₇⋊C₈	—	8	2	no	—
76	C₄₇⋊C₈	C₄⋊C₈	1	8	yes	complemented
77	#8805	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	#384	1	1	yes	characteristic
79	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₄₇	#423	1	1	yes	characteristic
81	C₂²×C₄₇	C₂×#106	1	2	yes	—
82	C₂²×C₄₇	C₂×#108	1	2	yes	—
83	C₄×C₄₇	—	2	4	no	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	#334	1	8	yes	complemented
86	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₄₇	#423	1	2	yes	—
88	C₄×C₄₇	#492	1	1	yes	characteristic
89	C₄×C₄₇	#492	1	1	yes	characteristic
90	C₄×C₄₇	#498	1	2	yes	—
91	#1842	—	47	1	no	—
92	#1842	D₄₇	1	1	yes	characteristic
93	C₄²×C₈	—	47	1	no	—
94	C₂×#334	—	47	4	no	—
95	C₂×C₄₇	C₂×#334	1	2	yes	—
96	C₂×C₄₇	#1566	1	2	yes	—
97	C₂×C₄₇	#1842	1	1	yes	characteristic
98	C₂×C₄₇	#1844	1	1	yes	characteristic
99	C₂×C₄₇	#1848	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	—	47	2	no	—
102	C₂×C₄⋊C₈	—	94	2	no	—
103	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
104	C₂×C₄×C₈	—	47	1	no	—
105	C₂×C₄×C₈	—	47	2	no	—
106	C₂×C₄×C₈	—	47	2	no	—
107	C₂×C₄×C₈	—	47	1	no	—
108	C₄³	C₂×D₄₇	1	1	yes	characteristic
109	#334	—	94	8	no	—
110	C₄₇	#8805	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	47	2	no	—
113	C₂²×C₈	—	47	2	no	—
114	C₂²×C₈	—	94	2	no	—
115	C₂×C₄²	Dic₉₄	1	2	yes	—
116	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	94	4	no	—
123	C₄⋊C₈	—	188	4	no	—
124	C₄×C₈	—	47	4	no	—
125	C₄×C₈	—	47	4	no	—
126	C₄×C₈	—	47	4	no	—
127	C₄×C₈	—	94	4	no	—
128	C₄×C₈	—	94	2	no	—
129	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
130	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	47	4	no	—
137	C₂×C₈	—	47	4	no	—
138	C₂×C₈	—	47	4	no	—
139	C₂×C₈	—	47	4	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	188	2	no	—
143	C₂×C₈	—	188	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₈×D₄₇	1	2	yes	—
147	C₄²	#10625324	1	2	yes	—
148	C₄²	#10625335	1	4	yes	—
149	C₄²	#10625336	1	4	yes	—
150	C₄²	D₄×D₄₇	1	2	yes	—
151	C₄²	Q₈×D₄₇	1	2	yes	—
152	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	Dic₄₇⋊C₈	1	4	yes	—
157	C₂×C₄	#13093955	1	4	yes	—
158	C₂×C₄	#13093955	1	4	yes	—
159	C₂×C₄	#13093959	1	4	yes	—
160	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
161	C₂×C₄	#13094020	1	1	yes	characteristic
162	C₂×C₄	#13094022	1	1	yes	characteristic
163	C₂×C₄	#13094023	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
166	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
167	C₂×C₄	C₂×#10625336	1	2	yes	—
168	C₂×C₄	C₂×#10625335	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	47	8	no	—
171	C₈	—	94	4	no	—
172	C₈	—	376	2	no	—
173	C₂²	#15688992	1	1	yes	characteristic
174	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
175	C₂²	#15689146	1	1	yes	characteristic
176	C₂²	C₂×#13093955	1	2	yes	—
177	C₂²	C₂×#13093959	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	4	no	—
180	C₄	#15688873	1	8	yes	complemented
181	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
182	C₄	#15689145	1	2	yes	—
183	C₄	#15689495	1	1	yes	characteristic
184	C₄	#15689497	1	1	yes	characteristic
185	C₄	#15689519	1	2	yes	—
186	C₂	C₂×#15688873	1	2	yes	—
187	C₂	#20242143	1	2	yes	—
188	C₂	#20243539	1	1	yes	characteristic
189	C₂	#20243544	1	1	yes	characteristic
190	C₂	#20243559	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8805
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
22. C₄×C₄₇⋊C₈	C₈	semidirect
76. C₄₇⋊C₈	C₄⋊C₈	semidirect
85. C₄×C₄₇	#334	semidirect
110. C₄₇	#8805	semidirect
180. C₄	#15688873	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄²×C₄₇⋊C₈	2
C₂×#15688873	2
C₄₇×#1842	2
#8805	47