Subgroups — #26114889

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
2	#20242141	C₂	1	2	yes	maximal
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
4	#20241237	C₂	1	2	yes	maximal
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093960	C₄	1	1	yes	characteristic
7	C₂×#13093960	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₄×#10625329	C₂²	1	2	yes	—
10	#15688888	C₄	1	2	yes	—
11	#15688875	—	2	2	no	—
12	#15688874	—	2	1	no	—
13	#15688874	—	2	1	no	—
14	#15688873	—	2	2	no	—
15	C₄₇×#431	C₂²	1	1	yes	characteristic
16	C₄₇×#431	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#322	—	2	1	no	—
19	C₈²×C₄₇	—	2	1	no	—
20	C₂×#10625329	C₂×C₄	1	2	yes	—
21	C₂×#10625329	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13093960	—	2	2	no	—
24	#13093960	C₈	1	4	yes	complemented
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	#13093935	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	1	no	—
33	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
34	C₂²⋊C₈×C₄₇	—	2	1	no	—
35	C₂²⋊C₈×C₄₇	Q₈	1	2	yes	—
36	C₄×C₈×C₄₇	—	2	1	no	—
37	C₄×C₈×C₄₇	—	2	1	no	—
38	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	D₄	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	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
55	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	1	yes	characteristic
59	#10625329	—	2	2	no	—
60	#10625329	—	2	2	no	—
61	#10625329	—	2	2	no	—
62	#10625329	C₂×C₈	1	4	yes	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	8	2	no	—
67	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	4	1	no	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
80	C₈×C₄₇	—	2	1	no	—
81	C₈×C₄₇	—	2	1	no	—
82	C₈×C₄₇	—	4	2	no	—
83	C₈×C₄₇	—	4	2	no	—
84	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
85	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
86	C₄₇⋊C₈	—	2	4	no	—
87	C₄₇⋊C₈	—	4	2	no	—
88	C₄₇⋊C₈	—	8	2	no	—
89	#9607	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	1	no	—
91	C₂²×C₄₇	—	2	1	no	—
92	C₂²×C₄₇	#389	1	1	yes	characteristic
93	C₂²×C₄₇	C₈×Q₈	1	1	yes	characteristic
94	C₂²×C₄₇	#446	1	1	yes	characteristic
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₄₇	#424	1	1	yes	characteristic
100	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
101	C₄×C₄₇	#436	1	1	yes	characteristic
102	C₄×C₄₇	#471	1	2	yes	—
103	C₄×#108	—	47	1	no	—
104	#1565	—	47	2	no	—
105	C₂×#334	—	47	1	no	—
106	#1450	D₄₇	1	1	yes	characteristic
107	#1283	—	47	2	no	—
108	C₂×C₄₇	—	2	1	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	#1939	1	1	yes	characteristic
111	C₂×C₄₇	#1951	1	1	yes	characteristic
112	C₂×C₄₇	#1953	1	1	yes	characteristic
113	#431	C₂×D₄₇	1	1	yes	characteristic
114	#431	C₂×D₄₇	1	1	yes	characteristic
115	C₂×#108	—	47	1	no	—
116	C₂×#108	—	47	1	no	—
117	C₂×C₄⋊C₈	—	47	2	no	—
118	C₄×M₁₆	—	47	2	no	—
119	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
120	#341	—	47	2	no	—
121	#335	—	94	2	no	—
122	#334	—	94	2	no	—
123	#334	—	94	1	no	—
124	#334	—	94	1	no	—
125	#322	—	2	1	no	—
126	C₈²	—	2	1	no	—
127	C₄₇	#9607	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	—	47	2	no	—
129	C₂×M₁₆	—	47	2	no	—
130	C₂²×C₈	—	94	2	no	—
131	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	#108	—	47	4	no	—
135	#108	—	94	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	94	2	no	—
138	C₄⋊C₈	—	94	2	no	—
139	C₄⋊C₈	D₁₈₈	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	C₂²⋊C₈	Dic₉₄	1	2	yes	—
142	#97	—	94	2	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	94	2	no	—
146	C₄×C₈	Dic₉₄	1	1	yes	characteristic
147	C₄×C₈	Dic₉₄	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂²×C₄	#10625350	1	1	yes	characteristic
153	M₁₆	—	47	4	no	—
154	M₁₆	—	94	2	no	—
155	M₁₆	—	94	2	no	—
156	M₁₆	—	94	2	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	94	2	no	—
162	C₂×C₈	—	94	2	no	—
163	C₂×C₈	—	376	2	no	—
164	C₂×C₈	—	188	2	no	—
165	C₂×C₈	C₈×D₄₇	1	2	yes	—
166	C₂×C₈	#10625324	1	2	yes	—
167	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
168	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
169	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
170	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
171	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
172	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
173	C₄²	#10625350	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
180	C₂×C₄	#13093998	1	1	yes	characteristic
181	C₂×C₄	#13094003	1	1	yes	characteristic
182	C₂×C₄	#13094016	1	1	yes	characteristic
183	C₂×C₄	#13094024	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
186	C₂×C₄	#13094030	1	1	yes	characteristic
187	C₂×C₄	#13094030	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	—	4	2	no	—
192	C₈	—	94	4	no	—
193	C₈	—	376	2	no	—
194	C₈	—	188	2	no	—
195	C₈	Dic₄₇⋊C₈	1	2	yes	—
196	C₈	Dic₄₇⋊C₈	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#15689021	1	1	yes	characteristic
200	C₂²	C₈×Dic₉₄	1	1	yes	characteristic
201	C₂²	#15689253	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
206	C₄	#15689149	1	1	yes	characteristic
207	C₄	C₈×D₁₈₈	1	1	yes	characteristic
208	C₄	#15689202	1	1	yes	characteristic
209	C₄	#15689386	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#20244106	1	1	yes	characteristic
213	C₂	#20244168	1	1	yes	characteristic
214	C₂	#20244178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₄×Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#9607
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
24. #13093960	C₈	semidirect
127. C₄₇	#9607	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093960	2
#20242141	2
C₂×#15688874	2
#20241237	2
C₄₇×#1450	2
#9607	47