Subgroups — #26368857

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689519	C₂	1	2	yes	maximal
3	#20251765	C₂	1	2	yes	maximal
4	#20243551	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094024	C₂²	1	2	yes	—
8	#15689528	—	2	4	no	—
9	#15689519	—	2	8	no	—
10	C₂×#13093958	C₂²	1	4	yes	—
11	C₂×#13093955	C₂²	1	8	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
14	#15689030	—	2	4	no	—
15	#15689028	—	2	4	no	—
16	#15688988	C₂²	1	8	yes	—
17	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
19	C₄₇×#500	—	2	4	no	—
20	C₂×#107×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₉₄	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
24	#13094024	—	2	4	no	—
25	#13094024	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
28	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
29	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
31	#13093958	—	2	8	no	—
32	#13093955	—	2	16	no	—
33	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
34	C₈×Dic₄₇	Q₈	1	16	yes	complemented
35	C₄₇⋊#95	—	2	16	no	—
36	C₂²×Q₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₄₇	—	2	4	no	—
39	C₄⋊Q₈×C₄₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#107×C₄₇	—	2	8	no	—
44	C₄×C₈×C₄₇	D₄	1	4	yes	—
45	C₂×Q₈×C₄₇	—	4	4	no	—
46	C₂×Q₈×C₄₇	—	4	8	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	4	no	—
52	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
53	C₂×Dic₉₄	—	4	4	no	—
54	C₂×Dic₉₄	—	4	8	no	—
55	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
56	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
57	C₄⋊C₄×C₄₇	—	2	4	no	—
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	4	4	no	—
60	C₄²×C₄₇	C₂×D₄	1	2	yes	—
61	C₄²×C₄₇	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₄₇	—	2	4	no	—
63	C₄⋊Dic₄₇	—	2	4	no	—
64	C₄⋊Dic₄₇	—	4	4	no	—
65	C₄×Dic₄₇	—	2	4	no	—
66	C₄×Dic₄₇	—	2	4	no	—
67	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
68	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
69	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
70	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₄₇	—	8	8	no	—
72	C₂×C₄×C₄₇	—	4	4	no	—
73	C₂×C₄×C₄₇	—	4	8	no	—
74	C₂×C₄×C₄₇	#127	1	2	yes	—
75	C₂×C₄×C₄₇	#127	1	2	yes	—
76	C₂×C₄×C₄₇	#135	1	4	yes	—
77	C₂×C₄×C₄₇	#135	1	4	yes	—
78	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
81	C₂×Dic₄₇	—	2	4	no	—
82	C₂×Dic₄₇	—	2	4	no	—
83	C₂×Dic₄₇	—	4	4	no	—
84	C₂×Dic₄₇	—	4	8	no	—
85	C₂×Dic₄₇	C₄⋊Q₈	1	8	yes	—
86	Dic₉₄	—	8	8	no	—
87	C₈×C₄₇	C₄⋊Q₈	1	8	yes	—
88	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	—
89	#33313	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
92	C₂²×C₄₇	C₂×#135	1	2	yes	—
93	C₂²×C₄₇	C₂×#135	1	2	yes	—
94	C₄×C₄₇	—	8	4	no	—
95	C₄×C₄₇	#495	1	4	yes	—
96	C₄×C₄₇	#495	1	4	yes	—
97	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
98	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
99	Dic₄₇	—	2	8	no	—
100	Dic₄₇	—	8	4	no	—
101	C₂×#500	—	47	1	no	—
102	C₂×#500	D₄₇	1	1	yes	characteristic
103	C₂×#498	—	47	2	no	—
104	#2862	—	47	2	no	—
105	#1846	—	47	8	no	—
106	C₄²×C₈	—	47	1	no	—
107	C₂×C₄₇	#2868	1	1	yes	characteristic
108	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
110	C₂×C₄₇	#3160	1	4	yes	—
111	C₂×C₄⋊Q₈	—	47	2	no	—
112	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
113	#500	—	2	4	no	—
114	#500	—	94	4	no	—
115	#498	—	94	8	no	—
116	C₂×#107	—	47	4	no	—
117	C₂×#107	C₂×D₄₇	1	4	yes	—
118	C₂×#106	—	47	8	no	—
119	C₂×C₄×C₈	—	47	4	no	—
120	C₂×C₄×C₈	—	47	1	no	—
121	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
122	#391	—	94	4	no	—
123	#391	—	94	4	no	—
124	#382	—	47	8	no	—
125	C₄³	—	47	1	no	—
126	C₄₇	#33313	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×Q₈	—	94	2	no	—
129	C₂²×C₈	—	47	2	no	—
130	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	—	94	4	no	—
134	C₄⋊Q₈	—	94	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	47	4	no	—
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
139	C₂×C₄²	—	47	2	no	—
140	C₂×C₄²	—	47	2	no	—
141	C₂×C₄²	—	47	2	no	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	#107	—	2	8	no	—
144	#107	—	94	8	no	—
145	#106	—	94	16	no	—
146	C₄×C₈	—	47	16	no	—
147	C₄×C₈	—	47	4	no	—
148	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
149	#95	—	94	16	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	8	no	—
152	C₂×Q₈	—	188	4	no	—
153	C₂×Q₈	—	188	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	47	4	no	—
156	C₂²×C₄	—	94	4	no	—
157	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
158	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
159	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
160	C₂×C₈	—	47	4	no	—
161	C₂×C₈	—	47	8	no	—
162	C₂×C₈	Q₈×D₄₇	1	8	yes	—
163	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	94	4	no	—
168	C₄⋊C₄	—	94	4	no	—
169	C₄⋊C₄	—	188	4	no	—
170	C₄²	—	47	8	no	—
171	C₄²	—	94	4	no	—
172	C₄²	—	94	4	no	—
173	C₄²	D₄×D₄₇	1	2	yes	—
174	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
175	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
176	Q₈	—	8	8	no	—
177	Q₈	—	376	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	47	8	no	—
181	C₂×C₄	—	94	4	no	—
182	C₂×C₄	—	94	4	no	—
183	C₂×C₄	—	188	8	no	—
184	C₂×C₄	—	188	4	no	—
185	C₂×C₄	#13094020	1	2	yes	—
186	C₂×C₄	#13094020	1	2	yes	—
187	C₂×C₄	#13094043	1	4	yes	—
188	C₂×C₄	#13094047	1	4	yes	—
189	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
192	C₈	—	47	8	no	—
193	C₈	#13094022	1	8	yes	—
194	C₂²	C₂×#13094020	1	1	yes	characteristic
195	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
196	C₂²	C₂×#13094043	1	2	yes	—
197	C₂²	C₂×#13094047	1	2	yes	—
198	C₄	—	8	4	no	—
199	C₄	—	94	8	no	—
200	C₄	—	376	4	no	—
201	C₄	#15689506	1	4	yes	—
202	C₄	#15689508	1	4	yes	—
203	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
204	C₄	C₂×#13094022	1	2	yes	—
205	C₂	#20251784	1	1	yes	characteristic
206	C₂	C₂×#15689506	1	1	yes	characteristic
207	C₂	C₂×#15689508	1	1	yes	characteristic
208	C₂	#20253899	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#33313
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₄₇	Q₈	semidirect
126. C₄₇	#33313	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689528	2
C₂×#15689519	2
#20251765	2
#20243551	2
C₄×C₈×Dic₄₇	2
C₂×C₄₇×#500	2
#33313	47