Subgroups — #26134384

Order 12032.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688952	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20243560	C₂	1	1	yes	maximal, characteristic
4	#20241481	C₂	1	2	yes	maximal
5	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093960	—	2	1	no	—
8	C₂×#13093960	C₂²	1	2	yes	—
9	#15689147	—	2	1	no	—
10	#15688952	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
12	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
13	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₄₇	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#341	—	2	1	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	#13093960	—	2	2	no	—
23	#13093960	—	2	2	no	—
24	#13093960	—	4	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
26	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
27	C₂×M₁₆×C₄₇	—	2	1	no	—
28	C₂×M₁₆×C₄₇	—	2	1	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	#108×C₄₇	—	2	1	no	—
33	#108×C₄₇	—	2	1	no	—
34	#108×C₄₇	D₄	1	2	yes	—
35	#108×C₄₇	Q₈	1	2	yes	—
36	#97×C₄₇	—	4	1	no	—
37	C₄×C₈×C₄₇	—	4	1	no	—
38	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₄₇	—	2	2	no	—
46	M₁₆×C₄₇	—	2	2	no	—
47	M₁₆×C₄₇	—	4	1	no	—
48	M₁₆×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	—	2	1	no	—
50	C₂×C₈×C₄₇	—	2	1	no	—
51	C₂×C₈×C₄₇	—	4	1	no	—
52	C₂×C₈×C₄₇	—	4	1	no	—
53	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
60	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
61	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₄₇	D₈	1	1	yes	characteristic
64	C₄²×C₄₇	D₈	1	1	yes	characteristic
65	#10625329	—	4	2	no	—
66	#10625329	—	4	2	no	—
67	C₂×C₄₇⋊C₈	—	4	2	no	—
68	C₄₇⋊C₁₆	—	2	4	no	—
69	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	1	no	—
72	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	4	2	no	—
85	C₈×C₄₇	—	4	2	no	—
86	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₄₇	#125	1	1	yes	characteristic
89	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₄₇⋊C₈	—	8	2	no	—
91	#13179	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	—	2	1	no	—
94	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
95	C₂²×C₄₇	#462	1	1	yes	characteristic
96	C₂²×C₄₇	#477	1	1	yes	characteristic
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	1	no	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	#368	1	2	yes	complemented
101	C₄×C₄₇	#389	1	1	yes	characteristic
102	C₄×C₄₇	#428	1	1	yes	characteristic
103	C₄×C₄₇	#459	1	1	yes	characteristic
104	C₄×C₄₇	#474	1	1	yes	characteristic
105	C₂×#368	—	47	2	no	—
106	C₂×C₄⋊C₁₆	—	47	1	no	—
107	#1848	—	47	1	no	—
108	C₄×#108	D₄₇	1	1	yes	characteristic
109	#1381	—	47	2	no	—
110	C₂×C₄₇	—	2	1	no	—
111	C₂×C₄₇	—	2	1	no	—
112	C₂×C₄₇	#1947	1	1	yes	characteristic
113	C₂×C₄₇	C₂×#368	1	1	yes	characteristic
114	C₂×C₄₇	#2161	1	1	yes	characteristic
115	C₂²×C₁₆	—	47	2	no	—
116	C₂×#108	—	47	2	no	—
117	C₂×#108	—	94	1	no	—
118	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
119	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
120	#423	—	94	1	no	—
121	C₄×M₁₆	—	2	1	no	—
122	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
123	#368	—	94	4	no	—
124	C₄⋊C₁₆	—	47	2	no	—
125	C₄⋊C₁₆	—	47	2	no	—
126	#341	—	2	1	no	—
127	C₄₇	#13179	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	94	2	no	—
131	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₁₆	—	47	4	no	—
135	C₂×C₁₆	—	94	2	no	—
136	C₂×C₁₆	—	94	2	no	—
137	#108	—	2	1	no	—
138	#108	—	2	1	no	—
139	#108	—	94	2	no	—
140	#108	—	94	2	no	—
141	#108	—	188	2	no	—
142	#108	Dic₉₄	1	2	yes	—
143	#108	D₁₈₈	1	2	yes	—
144	C₄⋊C₈	—	188	1	no	—
145	C₄⋊C₈	—	188	1	no	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	Dic₉₄	1	1	yes	characteristic
149	C₄×C₈	Dic₉₄	1	1	yes	characteristic
150	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
151	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
152	C₂²×C₄	#10625337	1	1	yes	characteristic
153	C₂²×C₄	#10625337	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	4	1	no	—
159	M₁₆	—	188	2	no	—
160	M₁₆	—	188	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	188	2	no	—
166	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
167	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
172	C₂×C₈	#10625350	1	1	yes	characteristic
173	C₂×C₈	#10625350	1	1	yes	characteristic
174	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
175	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
176	C₄²	#10625334	1	1	yes	characteristic
177	C₄²	#10625334	1	1	yes	characteristic
178	C₁₆	—	94	4	no	—
179	C₂³	C₂×#10625337	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#13093957	1	1	yes	characteristic
183	C₂×C₄	#13093957	1	1	yes	characteristic
184	C₂×C₄	#13093957	1	1	yes	characteristic
185	C₂×C₄	#13093957	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#13093998	1	1	yes	characteristic
188	C₂×C₄	#13094003	1	1	yes	characteristic
189	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
190	C₂×C₄	#13094049	1	1	yes	characteristic
191	C₂×C₄	#13094054	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	376	2	no	—
197	C₈	C₄×D₁₈₈	1	1	yes	characteristic
198	C₈	C₄×Dic₉₄	1	1	yes	characteristic
199	C₈	#13094016	1	1	yes	characteristic
200	C₈	#13094024	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#13093957	1	1	yes	characteristic
204	C₂²	#15689333	1	1	yes	characteristic
205	C₂²	#15689418	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#15688952	1	2	yes	complemented
210	C₄	#15689021	1	1	yes	characteristic
211	C₄	#15689166	1	1	yes	characteristic
212	C₄	#15689318	1	1	yes	characteristic
213	C₄	#15689403	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#20244146	1	1	yes	characteristic
217	C₂	C₂×#15688952	1	1	yes	characteristic
218	C₂	#20245182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#13179
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
100. C₄×C₄₇	#368	semidirect
127. C₄₇	#13179	semidirect
209. C₄	#15688952	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688952	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20243560	2
#20241481	2
C₄×#108×C₄₇	2
#13179	47