Subgroups — #26110036

Order 12032.

362 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688875	C₂	1	2	yes	maximal
4	C₂×#15688873	C₂	1	2	yes	maximal
5	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688878	—	2	2	no	—
10	#15688878	C₄	1	2	yes	—
11	#15688878	C₄	1	2	yes	—
12	#15688875	—	2	4	no	—
13	#15688873	—	2	4	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
25	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₄⋊C₈×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	4	4	no	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	4	no	—
44	C₄²×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄₇⋊C₈	—	2	2	no	—
50	C₂×C₄₇⋊C₈	—	2	2	no	—
51	C₂×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₄₇⋊C₈	—	2	2	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	4	no	—
55	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
56	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
57	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#131	1	2	yes	—
72	C₂×C₄×C₄₇	#131	1	2	yes	—
73	C₂×C₄×C₄₇	#135	1	4	yes	—
74	C₈×C₄₇	—	8	2	no	—
75	C₄₇⋊C₈	—	2	4	no	—
76	C₄₇⋊C₈	—	2	4	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	#8811	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	#384	1	1	yes	characteristic
80	C₂²×C₄₇	#424	1	1	yes	characteristic
81	C₂²×C₄₇	#424	1	1	yes	characteristic
82	C₂²×C₄₇	#427	1	2	yes	—
83	C₂²×C₄₇	#430	1	2	yes	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	—	4	2	no	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	#424	1	2	yes	—
89	C₄×C₄₇	#424	1	2	yes	—
90	C₄×C₄₇	#495	1	2	yes	—
91	C₄×C₄₇	#499	1	2	yes	—
92	#1831	—	47	1	no	—
93	#1831	D₄₇	1	1	yes	characteristic
94	#1744	—	47	1	no	—
95	C₂×#335	—	47	2	no	—
96	C₂×#334	—	47	2	no	—
97	C₂×C₄₇	#1572	1	2	yes	—
98	C₂×C₄₇	#1572	1	2	yes	—
99	C₂×C₄₇	#1843	1	1	yes	characteristic
100	C₂×C₄₇	#1846	1	1	yes	characteristic
101	C₂×C₄₇	#1849	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	47	2	no	—
103	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
104	C₂×C₄×C₈	—	47	1	no	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₄³	C₂×D₄₇	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	#336	—	47	2	no	—
109	#336	—	47	2	no	—
110	#336	—	94	2	no	—
111	#335	—	94	4	no	—
112	#334	—	94	4	no	—
113	C₄₇	#8811	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	47	2	no	—
116	C₂²×C₈	—	47	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	C₂×C₄²	Dic₉₄	1	2	yes	—
119	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄⋊C₈	—	94	2	no	—
127	C₄×C₈	—	47	2	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	—	47	2	no	—
130	C₄×C₈	—	47	2	no	—
131	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
132	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	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	—	94	2	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₂×C₈	—	188	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₄×D₄₇	1	1	yes	characteristic
151	C₄²	D₄×D₄₇	1	1	yes	characteristic
152	C₄²	Q₈×D₄₇	1	1	yes	characteristic
153	C₄²	Q₈×D₄₇	1	1	yes	characteristic
154	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
163	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₄	#13094020	1	1	yes	characteristic
165	C₂×C₄	#13094022	1	1	yes	characteristic
166	C₂×C₄	#13094023	1	1	yes	characteristic
167	C₂×C₄	#13094023	1	1	yes	characteristic
168	C₂×C₄	#13094030	1	2	yes	—
169	C₂×C₄	#13094032	1	2	yes	—
170	C₂×C₄	#13094044	1	4	yes	—
171	C₈	—	8	2	no	—
172	C₈	—	94	4	no	—
173	C₈	—	94	4	no	—
174	C₈	—	376	2	no	—
175	C₂²	#15688992	1	1	yes	characteristic
176	C₂²	#15689149	1	1	yes	characteristic
177	C₂²	#15689151	1	1	yes	characteristic
178	C₂²	#15689159	1	2	yes	—
179	C₂²	#15689171	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#15689148	1	2	yes	—
185	C₄	#15689148	1	2	yes	—
186	C₄	#15689505	1	2	yes	—
187	C₄	#15689522	1	2	yes	—
188	C₂	#20242166	1	2	yes	—
189	C₂	#20242169	1	2	yes	—
190	C₂	#20243541	1	1	yes	characteristic
191	C₂	#20243550	1	1	yes	characteristic
192	C₂	#20243562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#8811
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
113. C₄₇	#8811	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
#20243134	2
C₂×#15688875	2
C₂×#15688873	2
C₄₇×#1831	2
#8811	47