Subgroups — #26134394

Order 12032.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245177	C₂	1	2	yes	maximal
2	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
3	#20243548	C₂	1	1	yes	maximal, characteristic
4	#20241498	C₂	1	2	yes	maximal
5	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093958	—	2	1	no	—
8	C₂×#13093958	C₂²	1	2	yes	—
9	#15688993	—	2	1	no	—
10	#15688950	—	2	2	no	—
11	#15688948	—	2	2	no	—
12	#15688944	C₄	1	4	yes	complemented
13	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#427	C₂²	1	1	yes	characteristic
15	C₄₇×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#339	—	2	1	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	4	2	no	—
20	#13093964	—	2	2	no	—
21	#13093964	C₂×C₄	1	4	yes	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	#13093958	—	2	2	no	—
24	#13093958	—	2	2	no	—
25	#13093958	—	4	1	no	—
26	#13093958	—	4	1	no	—
27	C₂×M₁₆×C₄₇	—	2	1	no	—
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	#117×C₄₇	—	2	1	no	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	#108×C₄₇	—	2	1	no	—
33	#108×C₄₇	Q₈	1	2	yes	—
34	#107×C₄₇	D₄	1	1	yes	characteristic
35	#107×C₄₇	D₄	1	1	yes	characteristic
36	#106×C₄₇	—	2	1	no	—
37	#97×C₄₇	—	4	1	no	—
38	C₄×C₈×C₄₇	—	4	1	no	—
39	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₄₇	Q₈	1	2	yes	—
41	C₂²×C₄×C₄₇	—	2	1	no	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	4	2	no	—
44	M₁₆×C₄₇	—	2	2	no	—
45	M₁₆×C₄₇	—	4	1	no	—
46	C₂×C₈×C₄₇	—	4	1	no	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
49	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
50	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₄⋊C₄×C₄₇	—	2	1	no	—
59	C₂²⋊C₄×C₄₇	—	4	1	no	—
60	C₄²×C₄₇	—	2	1	no	—
61	C₄²×C₄₇	—	4	1	no	—
62	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	8	4	no	—
66	C₄₇⋊C₁₆	—	2	4	no	—
67	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	1	no	—
72	C₂×C₄×C₄₇	—	4	1	no	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
79	C₂×Dic₄₇	—	8	4	no	—
80	C₂×Dic₄₇	—	8	2	no	—
81	C₈×C₄₇	—	4	2	no	—
82	C₈×C₄₇	#107	1	2	yes	—
83	C₈×C₄₇	#107	1	2	yes	—
84	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
85	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₄₇	#125	1	1	yes	characteristic
87	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
88	#13181	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	1	no	—
90	C₂²×C₄₇	—	2	1	no	—
91	C₂²×C₄₇	#428	1	1	yes	characteristic
92	C₂²×C₄₇	#459	1	1	yes	characteristic
93	C₂²×C₄₇	#477	1	1	yes	characteristic
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	—	8	1	no	—
96	C₄×C₄₇	—	8	1	no	—
97	C₄×C₄₇	#389	1	1	yes	characteristic
98	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
99	C₄×C₄₇	#462	1	1	yes	characteristic
100	C₄×C₄₇	#474	1	1	yes	characteristic
101	Dic₄₇	—	16	2	no	—
102	#2159	—	47	2	no	—
103	C₂×#364	—	47	1	no	—
104	#1845	—	47	1	no	—
105	#1772	D₄₇	1	1	yes	characteristic
106	#1387	—	47	2	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	—	2	1	no	—
109	C₂×C₄₇	#1947	1	1	yes	characteristic
110	C₂×C₄₇	#2213	1	1	yes	characteristic
111	C₂×C₄₇	#2214	1	1	yes	characteristic
112	C₂×M₃₂	—	47	2	no	—
113	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
114	#427	C₂×D₄₇	1	1	yes	characteristic
115	C₂×#107	—	47	2	no	—
116	C₂×#107	—	94	1	no	—
117	#405	—	2	1	no	—
118	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
119	#384	—	94	1	no	—
120	#367	—	94	2	no	—
121	#366	—	94	2	no	—
122	#364	—	47	4	no	—
123	#339	—	2	1	no	—
124	C₄₇	#13181	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₁₆	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	94	2	no	—
130	C₂×C₄⋊C₄	—	188	2	no	—
131	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
132	M₃₂	—	47	4	no	—
133	M₃₂	—	94	2	no	—
134	C₂×C₁₆	—	94	2	no	—
135	#108	—	2	1	no	—
136	#108	Dic₉₄	1	2	yes	—
137	#107	—	94	2	no	—
138	#107	—	94	2	no	—
139	#107	—	188	1	no	—
140	#107	—	188	1	no	—
141	#107	D₁₈₈	1	1	yes	characteristic
142	#107	D₁₈₈	1	1	yes	characteristic
143	#106	—	2	1	no	—
144	#97	—	4	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	Dic₉₄	1	2	yes	—
147	C₄×C₈	C₄×D₄₇	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	188	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
155	C₂×C₈	#10625334	1	1	yes	characteristic
156	C₂×C₈	#10625334	1	1	yes	characteristic
157	C₂×C₈	#10625337	1	2	yes	—
158	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
162	C₂×C₈	#10625350	1	1	yes	characteristic
163	C₂×C₈	#10625350	1	1	yes	characteristic
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	376	4	no	—
167	C₄⋊C₄	—	188	2	no	—
168	C₄⋊C₄	—	188	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	Dic₄₇⋊C₄	1	2	yes	—
173	C₁₆	—	94	4	no	—
174	C₂³	#13094049	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	376	4	no	—
181	C₂×C₄	—	376	2	no	—
182	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
183	C₂×C₄	#13093998	1	1	yes	characteristic
184	C₂×C₄	#13094003	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
186	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
187	C₂×C₄	#13094054	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	#13093957	1	2	yes	—
190	C₈	#13093957	1	2	yes	—
191	C₈	C₄×D₁₈₈	1	1	yes	characteristic
192	C₈	C₄×Dic₉₄	1	1	yes	characteristic
193	C₈	#13094016	1	1	yes	characteristic
194	C₈	#13094024	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	#15689166	1	1	yes	characteristic
198	C₂²	#15689318	1	1	yes	characteristic
199	C₂²	#15689418	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	8	1	no	—
202	C₄	—	8	1	no	—
203	C₄	—	752	2	no	—
204	C₄	#15689021	1	1	yes	characteristic
205	C₄	C₂×#13093957	1	1	yes	characteristic
206	C₄	#15689333	1	1	yes	characteristic
207	C₄	#15689403	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#20244146	1	1	yes	characteristic
211	C₂	#20245464	1	1	yes	characteristic
212	C₂	#20245469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689318
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1772	C₂
socle	C₂²×C₄₇	#459
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1772	D₄₇
hypocenter	C₄₇	#13181
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #15688944	C₄	semidirect
124. C₄₇	#13181	semidirect

Maximal subgroups

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

subgroup	index
#20245177	2
C₂×#15688944	2
#20243548	2
#20241498	2
C₄₇×#1772	2
#13181	47