Subgroups — #26107512

Order 12032.

414 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×#13093958	C₂	1	1	yes	maximal, characteristic
2	#20243255	C₂	1	2	yes	maximal
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	#20241228	C₂	1	2	yes	maximal
5	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#13093934	—	2	2	no	—
9	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#15688886	C₄	1	2	yes	—
11	#15688877	—	2	2	no	—
12	#15688876	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₈²×C₄₇	C₄	1	4	yes	complemented
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₄₇	—	2	2	no	—
20	#13093958	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
23	#13093934	—	4	4	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₄₇	—	2	1	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	—	2	1	no	—
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
40	C₂²×Dic₄₇	—	2	4	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	C₄²	1	2	yes	—
47	C₂×C₈×C₄₇	C₄²	1	2	yes	—
48	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
51	C₂×C₈×C₄₇	D₈	1	2	yes	—
52	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
53	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₄₇	#41	1	1	yes	characteristic
57	C₄⋊Dic₄₇	—	2	4	no	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄×Dic₄₇	—	4	2	no	—
60	C₄×Dic₄₇	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂×C₄₇⋊C₈	—	4	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₄₇	#131	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
78	C₂×Dic₄₇	—	4	4	no	—
79	C₂×Dic₄₇	—	4	4	no	—
80	C₂×Dic₄₇	—	4	4	no	—
81	C₈×C₄₇	—	2	4	no	—
82	C₈×C₄₇	—	2	2	no	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	#97	1	4	yes	complemented
85	C₈×C₄₇	#107	1	4	yes	complemented
86	C₄₇⋊C₈	—	4	4	no	—
87	#8282	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄₇	#436	1	2	yes	—
90	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
91	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
92	C₂²×C₄₇	#443	1	2	yes	—
93	C₄×C₄₇	—	2	1	no	—
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
96	C₄×C₄₇	#405	1	1	yes	characteristic
97	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
98	C₄×C₄₇	#427	1	1	yes	characteristic
99	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
100	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₄₇	#446	1	2	yes	—
102	Dic₄₇	—	8	4	no	—
103	C₄×#107	—	47	1	no	—
104	#1770	—	47	2	no	—
105	C₂×#335	—	47	1	no	—
106	C₂×C₈²	D₄₇	1	1	yes	characteristic
107	#1279	—	47	2	no	—
108	C₂×C₄₇	#1589	1	2	yes	—
109	C₂×C₄₇	#1591	1	2	yes	—
110	C₂×C₄₇	#1770	1	1	yes	characteristic
111	C₂×C₄₇	#1772	1	1	yes	characteristic
112	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
113	C₂×#107	—	47	2	no	—
114	C₂×C₄⋊C₈	—	47	2	no	—
115	C₂×#97	—	94	2	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	47	2	no	—
120	#340	—	47	2	no	—
121	#336	—	94	2	no	—
122	#335	—	47	4	no	—
123	C₈²	Dic₄₇	1	4	yes	complemented
124	C₄₇	#8282	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	—	94	2	no	—
127	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
130	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	47	4	no	—
132	C₂×C₄²	—	94	2	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	#107	—	94	4	no	—
135	C₄⋊C₈	—	47	4	no	—
136	C₄⋊C₈	—	94	2	no	—
137	#97	—	188	4	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	Dic₉₄	1	2	yes	—
140	C₄×C₈	C₄×D₄₇	1	2	yes	—
141	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
142	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
143	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
144	#95	—	94	2	no	—
145	C₂²×C₄	—	94	4	no	—
146	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
148	C₂²×C₄	#10625350	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	94	4	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	—	188	2	no	—
157	C₂×C₈	#10625324	1	4	yes	—
158	C₂×C₈	D₃₇₆	1	2	yes	—
159	C₂×C₈	Dic₁₈₈	1	2	yes	—
160	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
161	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
163	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
164	C₄⋊C₄	—	94	4	no	—
165	C₄⋊C₄	—	94	4	no	—
166	C₄²	—	188	2	no	—
167	C₄²	—	188	2	no	—
168	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₄²	#10625350	1	1	yes	characteristic
171	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
172	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	188	4	no	—
175	C₂×C₄	—	188	4	no	—
176	C₂×C₄	—	188	4	no	—
177	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	#13093983	1	1	yes	characteristic
180	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
181	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
182	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
183	C₂×C₄	C₂×#10625324	1	2	yes	—
184	C₂×C₄	#13094030	1	2	yes	—
185	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
187	C₂×C₄	#13094046	1	1	yes	characteristic
188	C₂×C₄	#13094046	1	1	yes	characteristic
189	C₈	—	2	4	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	188	4	no	—
193	C₈	#13093934	1	4	yes	complemented
194	C₈	#13093958	1	4	yes	complemented
195	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
196	C₂²	#15689202	1	2	yes	—
197	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
198	C₂²	C₄×Dic₁₈₈	1	1	yes	characteristic
199	C₂²	#15689241	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	376	4	no	—
203	C₄	C₂×#13093934	1	1	yes	characteristic
204	C₄	#15689078	1	1	yes	characteristic
205	C₄	C₂×#13093958	1	1	yes	characteristic
206	C₄	#15689161	1	1	yes	characteristic
207	C₄	C₄×D₃₇₆	1	1	yes	characteristic
208	C₄	C₄×Dic₁₈₈	1	1	yes	characteristic
209	C₄	#15689253	1	2	yes	—
210	C₂	#20242274	1	2	yes	—
211	C₂	#20242286	1	2	yes	—
212	C₂	#20243255	1	1	yes	characteristic
213	C₂	#20243267	1	1	yes	characteristic
214	C₂	C₄×#13093958	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#8282
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #15688876	C₄	semidirect
16. C₈²×C₄₇	C₄	semidirect
84. C₈×C₄₇	#97	semidirect
85. C₈×C₄₇	#107	semidirect
123. C₈²	Dic₄₇	semidirect
124. C₄₇	#8282	semidirect
193. C₈	#13093934	semidirect
194. C₈	#13093958	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13093958	2
#20243255	2
C₂×#15688876	2
#20241228	2
C₂×C₈²×C₄₇	2
#8282	47