Subgroups — #26121462

Order 12032.

1142 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	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
2	C₂×#15688886	C₂	1	4	yes	maximal
3	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₄₇	—	2	2	no	—
6	C₄×C₄⋊Dic₄₇	—	2	4	no	—
7	C₄×C₄⋊Dic₄₇	C₄	1	8	yes	complemented
8	C₂×C₄₇⋊#95	—	2	2	no	—
9	#15688886	C₄	1	16	yes	complemented
10	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#97×C₄₇	—	2	4	no	—
13	C₂×#97×C₄₇	C₄	1	4	yes	complemented
14	C₂³×Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	8	no	—
17	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	16	yes	—
19	C₂×C₄×Dic₄₇	—	2	8	no	—
20	C₂×C₄×Dic₄₇	—	4	8	no	—
21	C₄₇⋊#95	—	2	8	no	—
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	—	2	4	no	—
25	C₂²×C₈×C₄₇	—	2	4	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	4	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
32	#97×C₄₇	—	2	8	no	—
33	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	#41	1	4	yes	—
45	C₂²×Dic₄₇	—	2	8	no	—
46	C₂²×Dic₄₇	—	2	16	no	—
47	C₂²×Dic₄₇	—	4	8	no	—
48	C₂²×Dic₄₇	—	4	4	no	—
49	C₂²×Dic₄₇	—	4	4	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	—	2	16	no	—
53	C₂×C₈×C₄₇	C₄²	1	8	yes	complemented
54	C₄²×C₄₇	C₄²	1	4	yes	—
55	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₄⋊Dic₄₇	—	2	8	no	—
58	C₄⋊Dic₄₇	—	2	16	no	—
59	C₄⋊Dic₄₇	C₄²	1	16	yes	complemented
60	C₄×Dic₄₇	—	4	16	no	—
61	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
62	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂³×C₄₇	C₄×D₄	1	4	yes	—
64	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
65	C₂³×C₄₇	#137	1	4	yes	—
66	C₂×C₄×C₄₇	—	2	8	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	#95	1	8	yes	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₄₇	#117	1	4	yes	—
77	C₂×C₄×C₄₇	#117	1	2	yes	—
78	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
79	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
80	C₂×Dic₄₇	—	2	16	no	—
81	C₂×Dic₄₇	—	4	8	no	—
82	C₂×Dic₄₇	—	4	8	no	—
83	C₂×Dic₄₇	—	4	16	no	—
84	C₂×Dic₄₇	—	4	16	no	—
85	C₈×C₄₇	—	2	8	no	—
86	#10703	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
88	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
89	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
90	C₂²×C₄₇	#417	1	4	yes	—
91	C₂²×C₄₇	#418	1	2	yes	—
92	C₂²×C₄₇	#419	1	2	yes	—
93	C₂²×C₄₇	#428	1	4	yes	—
94	C₂²×C₄₇	#440	1	8	yes	—
95	C₂²×C₄₇	#441	1	4	yes	—
96	C₂²×C₄₇	#442	1	4	yes	—
97	C₄×C₄₇	—	2	8	no	—
98	C₄×C₄₇	C₄³	1	1	yes	characteristic
99	C₄×C₄₇	#376	1	1	yes	characteristic
100	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
101	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
102	Dic₄₇	—	4	16	no	—
103	C₂²×#97	D₄₇	1	1	yes	characteristic
104	C₂×C₄×C₄⋊C₄	—	47	2	no	—
105	C₂×#340	—	47	4	no	—
106	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
107	C₂×C₄₇	#1736	1	2	yes	—
108	C₂×C₄₇	#1762	1	4	yes	—
109	C₂×C₄₇	#1763	1	4	yes	—
110	C₂×C₄₇	#1776	1	4	yes	—
111	C₂³×C₈	C₂×D₄₇	1	2	yes	—
112	C₂²×C₄⋊C₄	—	47	4	no	—
113	C₂²×C₄²	—	94	2	no	—
114	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
115	C₂×#97	—	2	4	no	—
116	C₂×#97	Dic₄₇	1	4	yes	complemented
117	C₄×C₄⋊C₄	—	47	8	no	—
118	C₄×C₄⋊C₄	—	94	4	no	—
119	C₂×#95	—	94	2	no	—
120	#340	—	47	16	no	—
121	C₄₇	#10703	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	94	4	no	—
123	C₂³×C₄	D₁₈₈	1	2	yes	—
124	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₄×D₄₇	1	8	yes	—
128	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄⋊C₄	—	47	16	no	—
130	C₂×C₄⋊C₄	—	47	8	no	—
131	C₂×C₄⋊C₄	—	94	8	no	—
132	C₂×C₄⋊C₄	—	94	4	no	—
133	C₂×C₄²	—	94	8	no	—
134	C₂×C₄²	—	188	8	no	—
135	C₂×C₄²	Dic₉₄	1	2	yes	—
136	C₂×C₄²	C₄×D₄₇	1	4	yes	—
137	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
139	#97	—	2	8	no	—
140	#95	—	94	8	no	—
141	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	94	8	no	—
145	C₂²×C₄	—	94	16	no	—
146	C₂²×C₄	—	188	8	no	—
147	C₂²×C₄	—	188	4	no	—
148	C₂²×C₄	—	188	4	no	—
149	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
150	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
151	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
152	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
155	C₂²×C₄	#10625350	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	C₄×Dic₄₇	1	8	yes	complemented
162	C₄⋊C₄	—	47	16	no	—
163	C₄⋊C₄	—	94	8	no	—
164	C₄⋊C₄	—	94	16	no	—
165	C₄²	—	188	16	no	—
166	C₄²	C₄×Dic₄₇	1	4	yes	—
167	C₄²	Dic₄₇⋊C₄	1	8	yes	—
168	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
169	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
170	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
171	C₂³	C₄×D₁₈₈	1	4	yes	—
172	C₂³	#13094052	1	4	yes	—
173	C₂³	#13094057	1	4	yes	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	94	16	no	—
179	C₂×C₄	—	188	16	no	—
180	C₂×C₄	—	188	16	no	—
181	C₂×C₄	—	188	8	no	—
182	C₂×C₄	—	188	8	no	—
183	C₂×C₄	C₄₇⋊#95	1	8	yes	—
184	C₂×C₄	C₄²×D₄₇	1	4	yes	—
185	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
189	C₂×C₄	#13093981	1	4	yes	—
190	C₂×C₄	#13093983	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
192	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
195	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	4	yes	—
196	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
197	C₂²	#15689125	1	4	yes	—
198	C₂²	#15689129	1	2	yes	—
199	C₂²	#15689133	1	2	yes	—
200	C₂²	#15689165	1	4	yes	—
201	C₂²	#15689225	1	8	yes	—
202	C₂²	#15689231	1	4	yes	—
203	C₂²	#15689236	1	4	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	188	16	no	—
206	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
207	C₄	#15688969	1	1	yes	characteristic
208	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
209	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
210	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
211	C₂	#20243108	1	2	yes	—
212	C₂	#20243214	1	4	yes	—
213	C₂	#20243219	1	4	yes	—
214	C₂	#20243285	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#10703
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₄₇	C₄	semidirect
9. #15688886	C₄	semidirect
13. C₂×#97×C₄₇	C₄	semidirect
53. C₂×C₈×C₄₇	C₄²	semidirect
59. C₄⋊Dic₄₇	C₄²	semidirect
116. C₂×#97	Dic₄₇	semidirect
121. C₄₇	#10703	semidirect
161. C₂×C₈	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₇	2
C₂×#15688886	2
C₂²×#97×C₄₇	2
#10703	47