Subgroups — #26127861

Order 12032.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	#20241461	C₂	1	2	yes	maximal, complemented
4	#20241459	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
8	#15689176	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#15688876	C₂²	1	2	yes	—
14	#15688874	—	2	1	no	—
15	C₄₇×#431	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#335	C₂²	1	2	yes	—
18	C₄₇×#334	—	2	1	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	C₈	1	4	yes	complemented
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
27	C₄×C₄₇⋊C₈	—	4	1	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₂²⋊C₈×C₄₇	—	4	1	no	—
33	C₄×C₈×C₄₇	—	4	1	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	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	4	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×C₈	1	2	yes	—
46	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₄₇	D₈	1	1	yes	characteristic
51	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₄₇⋊C₁₆	—	2	1	no	—
55	C₄₇⋊C₁₆	—	2	1	no	—
56	C₄₇⋊C₁₆	C₂×C₈	1	2	yes	—
57	C₄₇⋊C₁₆	C₂×C₈	1	2	yes	—
58	C₂³×C₄₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#108	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	—	2	1	no	—
71	C₈×C₄₇	—	8	2	no	—
72	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₄₇⋊C₈	—	8	2	no	—
76	#12008	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	#335	1	2	yes	—
80	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	#335	1	2	yes	—
84	C₄×C₄₇	#368	1	2	yes	—
85	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
87	C₄×C₄₇	#508	1	2	yes	—
88	C₂×C₄×C₁₆	—	47	1	no	—
89	#1564	—	47	1	no	—
90	#1564	D₄₇	1	1	yes	characteristic
91	#1372	—	47	2	no	—
92	#1371	—	47	2	no	—
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
95	C₂×C₄₇	#2158	1	1	yes	characteristic
96	C₂×C₄₇	C₂×#368	1	1	yes	characteristic
97	C₂²×C₁₆	—	47	1	no	—
98	C₂²×C₁₆	—	47	1	no	—
99	#431	—	2	1	no	—
100	#431	—	94	1	no	—
101	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₄×C₁₆	—	47	1	no	—
103	C₄×C₁₆	—	47	1	no	—
104	C₄×C₁₆	—	47	1	no	—
105	C₄×C₁₆	—	47	1	no	—
106	#335	—	47	2	no	—
107	#335	C₂×D₄₇	1	2	yes	—
108	#334	—	2	1	no	—
109	#334	—	94	1	no	—
110	C₄₇	#12008	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
112	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
113	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂×C₁₆	—	47	4	no	—
115	C₂×C₁₆	—	47	1	no	—
116	C₂×C₁₆	—	47	1	no	—
117	C₂×C₁₆	—	47	1	no	—
118	C₂×C₁₆	—	47	1	no	—
119	C₂×C₁₆	—	94	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	94	2	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₂²⋊C₈	—	188	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	188	1	no	—
126	C₄×C₈	Dic₉₄	1	1	yes	characteristic
127	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
129	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	#10625334	1	1	yes	characteristic
132	C₂²×C₄	#10625337	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	188	2	no	—
136	C₂×C₈	C₈×D₄₇	1	2	yes	—
137	C₂×C₈	#10625324	1	2	yes	—
138	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
143	C₄²	#10625334	1	1	yes	characteristic
144	C₄²	#10625337	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₁₆	—	47	2	no	—
147	C₁₆	—	47	2	no	—
148	C₁₆	—	94	1	no	—
149	C₁₆	—	94	1	no	—
150	C₂³	#13093957	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
154	C₂×C₄	#13093957	1	1	yes	characteristic
155	C₂×C₄	#13093957	1	1	yes	characteristic
156	C₂×C₄	#13093957	1	1	yes	characteristic
157	C₂×C₄	#13093959	1	2	yes	—
158	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
160	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	376	2	no	—
165	C₈	Dic₄₇⋊C₈	1	2	yes	—
166	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
167	C₈	C₂×#10625324	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#15688875	1	2	yes	—
171	C₂²	C₂×#13093957	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#15688875	1	2	yes	—
175	C₄	#15688952	1	2	yes	—
176	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
177	C₄	C₂×#13093959	1	1	yes	characteristic
178	C₄	#15689553	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₂×#15688875	1	1	yes	characteristic
181	C₂	#20245170	1	1	yes	characteristic
182	C₂	C₂×#15688952	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#10625334
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	#107
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#12008
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #20241461	C₂	semidirect
4. #20241459	C₂	semidirect
20. C₂×C₄₇⋊C₁₆	C₈	semidirect
110. C₄₇	#12008	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
#20242136	2
#20241461	2
#20241459	2
C₄₇×#1564	2
#12008	47