Subgroups — #26127712

Order 12032.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 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	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20241446	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
5	#15689147	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	—	2	1	no	—
9	#15688876	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₄₇×#500	—	2	1	no	—
13	C₄₇×#500	—	2	1	no	—
14	C₄₇×#500	C₄	1	2	yes	—
15	C₄₇×#500	C₂²	1	2	yes	—
16	C₂×#107×C₄₇	—	2	1	no	—
17	C₂×#107×C₄₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₄₇	—	4	1	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₄₇	—	2	2	no	—
27	C₄⋊Q₈×C₄₇	—	2	2	no	—
28	C₄⋊Q₈×C₄₇	—	4	1	no	—
29	C₄⋊Q₈×C₄₇	—	4	1	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	#107×C₄₇	—	2	2	no	—
34	#107×C₄₇	—	2	2	no	—
35	#107×C₄₇	—	2	2	no	—
36	#107×C₄₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	—	4	2	no	—
42	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂×Q₈×C₄₇	—	8	4	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₄₇	—	2	4	no	—
55	C₄⋊C₄×C₄₇	—	4	2	no	—
56	C₄⋊C₄×C₄₇	—	4	2	no	—
57	C₄⋊C₄×C₄₇	—	4	2	no	—
58	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₄₇	D₈	1	1	yes	characteristic
60	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
62	#10625329	—	8	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₄₇⋊C₁₆	—	4	2	no	—
65	C₂³×C₄₇	#102	1	1	yes	characteristic
66	Q₈×C₄₇	—	8	4	no	—
67	Q₈×C₄₇	—	8	4	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	4	4	no	—
72	C₂×C₄×C₄₇	—	8	4	no	—
73	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
77	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
81	C₈×C₄₇	—	2	1	no	—
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	#101	1	2	yes	—
84	C₈×C₄₇	#137	1	2	yes	—
85	C₄₇⋊C₈	—	8	2	no	—
86	#11972	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	—	2	1	no	—
89	C₂²×C₄₇	#358	1	2	yes	—
90	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	1	no	—
93	C₄×C₄₇	—	8	4	no	—
94	C₄×C₄₇	#358	1	2	yes	—
95	C₄×C₄₇	#361	1	2	yes	—
96	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
97	C₄×C₄₇	#419	1	1	yes	characteristic
98	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
99	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
100	C₂×#500	D₄₇	1	1	yes	characteristic
101	#2162	—	47	1	no	—
102	#1567	—	47	1	no	—
103	#1365	—	47	4	no	—
104	C₂×C₄₇	—	2	2	no	—
105	C₂×C₄₇	#1546	1	1	yes	characteristic
106	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
107	C₂×C₄₇	C₂×#361	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	1	no	—
111	#500	Dic₄₇	1	2	yes	—
112	#500	C₂×D₄₇	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	Dic₄₇	1	2	yes	—
115	#423	—	94	1	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	47	2	no	—
118	C₂²⋊C₁₆	—	94	1	no	—
119	C₄×C₁₆	—	94	1	no	—
120	#335	—	47	2	no	—
121	#335	—	94	1	no	—
122	C₄₇	#11972	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	4	1	no	—
124	C₂×M₁₆	—	188	1	no	—
125	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₄⋊Q₈	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₁₆	—	94	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	C₂×Dic₄₇	1	4	yes	—
139	C₄⋊C₈	—	94	2	no	—
140	C₄⋊C₈	—	188	1	no	—
141	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
144	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	#10625334	1	1	yes	characteristic
150	C₂²×C₄	#10625335	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
152	M₁₆	—	376	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	#10625334	1	1	yes	characteristic
165	C₄²	#10625335	1	1	yes	characteristic
166	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₁₆	—	188	2	no	—
169	C₂³	#13093945	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#13093945	1	1	yes	characteristic
178	C₂×C₄	#13093945	1	1	yes	characteristic
179	C₂×C₄	#13093945	1	1	yes	characteristic
180	C₂×C₄	#13093967	1	2	yes	—
181	C₂×C₄	#13093970	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
184	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	376	2	no	—
188	C₈	#13093943	1	2	yes	—
189	C₈	#13094055	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#15688927	1	2	yes	—
193	C₂²	C₂×#13093945	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#15688927	1	2	yes	—
198	C₄	#15688936	1	2	yes	—
199	C₄	C₂×#13093943	1	1	yes	characteristic
200	C₄	#15689132	1	1	yes	characteristic
201	C₄	C₂×#13093967	1	1	yes	characteristic
202	C₄	C₂×#13093970	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#20242060	1	1	yes	characteristic
205	C₂	C₂×#15688927	1	1	yes	characteristic
206	C₂	C₂×#15688936	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#11972
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #20241446	C₂	semidirect
122. C₄₇	#11972	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
#20242147	2
#20241446	2
C₂×C₄₇×#500	2
#11972	47