Subgroups — #26106406

Order 12032.

604 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	2	yes	maximal
2	#20241273	C₂	1	4	yes	maximal
3	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
4	#15688991	C₄	1	4	yes	complemented
5	#15688989	C₄	1	4	yes	complemented
6	#15688986	—	2	2	no	—
7	#15688983	—	2	2	no	—
8	C₄×C₄⋊Dic₄₇	—	2	2	no	—
9	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
10	#15688969	—	2	2	no	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#432	—	2	2	no	—
13	C₄₇×#431	—	2	2	no	—
14	C₄₇×#423	C₄	1	2	yes	complemented
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₄×M₁₆×C₄₇	C₄	1	2	yes	complemented
17	C₂×C₄⋊Dic₄₇	—	2	4	no	—
18	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
19	C₂×C₄×Dic₄₇	—	2	4	no	—
20	C₄₇⋊#95	—	2	4	no	—
21	C₄₇⋊#95	—	2	4	no	—
22	C₄₇⋊#95	—	2	4	no	—
23	C₄₇⋊#95	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₄₇	D₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
34	C₄⋊C₈×C₄₇	—	2	4	no	—
35	C₂²⋊C₈×C₄₇	—	2	4	no	—
36	#97×C₄₇	—	2	2	no	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	4	2	no	—
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	8	no	—
52	M₁₆×C₄₇	—	4	4	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₄²×C₄₇	C₄²	1	2	yes	—
55	C₄²×C₄₇	C₄²	1	1	yes	characteristic
56	C₄²×C₄₇	C₄²	1	1	yes	characteristic
57	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
61	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²⋊Dic₄₇	—	4	8	no	—
65	C₄⋊Dic₄₇	—	4	8	no	—
66	C₄×Dic₄₇	—	4	8	no	—
67	C₂³×C₄₇	—	2	4	no	—
68	C₂³×C₄₇	—	4	1	no	—
69	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
70	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	4	2	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	1	no	—
81	C₂×C₄×C₄₇	#95	1	2	yes	—
82	C₂×C₄×C₄₇	#95	1	2	yes	—
83	C₂×C₄×C₄₇	#95	1	2	yes	—
84	C₂×C₄×C₄₇	#95	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×Dic₄₇	—	4	8	no	—
89	C₂×Dic₄₇	—	4	8	no	—
90	C₂×Dic₄₇	—	4	8	no	—
91	C₈×C₄₇	—	4	4	no	—
92	#8050	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	—	4	4	no	—
98	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
99	C₂²×C₄₇	#421	1	4	yes	—
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	#410	1	2	yes	—
107	Dic₄₇	—	8	8	no	—
108	#2920	D₄₇	1	1	yes	characteristic
109	#2283	—	47	2	no	—
110	#1297	—	47	4	no	—
111	C₂×C₄₇	—	2	4	no	—
112	C₂×C₄₇	—	4	1	no	—
113	C₂×C₄₇	#1733	1	2	yes	—
114	C₂×C₄₇	#1740	1	1	yes	characteristic
115	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
116	#432	—	2	2	no	—
117	#431	—	2	2	no	—
118	#423	Dic₄₇	1	2	yes	complemented
119	#407	C₂×D₄₇	1	2	yes	—
120	C₄×M₁₆	Dic₄₇	1	2	yes	complemented
121	#383	—	47	4	no	—
122	#382	—	47	4	no	—
123	#381	—	94	2	no	—
124	#380	—	94	2	no	—
125	C₄×C₄⋊C₄	—	94	2	no	—
126	C₄×C₂²⋊C₄	—	47	4	no	—
127	#376	—	94	2	no	—
128	C₄₇	#8050	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	D₁₈₈	1	2	yes	—
130	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
132	C₂×C₄⋊C₄	—	94	4	no	—
133	C₂×C₂²⋊C₄	—	94	4	no	—
134	C₂×C₄²	—	94	4	no	—
135	C₂×C₄²	Dic₉₄	1	2	yes	—
136	C₂×C₄²	C₄×D₄₇	1	2	yes	—
137	C₂×C₄²	C₄×D₄₇	1	2	yes	—
138	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
141	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
142	C₄⋊C₈	—	2	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	#97	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	#95	—	47	8	no	—
147	#95	—	94	4	no	—
148	#95	—	94	4	no	—
149	#95	—	94	4	no	—
150	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	4	4	no	—
153	C₂²×C₄	—	94	8	no	—
154	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
155	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
156	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
157	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
158	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
159	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
160	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
164	M₁₆	—	4	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₄⋊C₄	—	188	8	no	—
167	C₂²⋊C₄	—	188	8	no	—
168	C₄²	—	188	8	no	—
169	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
170	C₄²	C₄×Dic₄₇	1	2	yes	—
171	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
172	C₄²	Dic₄₇⋊C₄	1	2	yes	—
173	C₄²	Dic₄₇⋊C₄	1	2	yes	—
174	C₄²	Dic₄₇⋊C₄	1	2	yes	—
175	C₄²	Dic₄₇⋊C₄	1	2	yes	—
176	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
177	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
178	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	4	1	no	—
181	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
182	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	188	8	no	—
194	C₂×C₄	—	188	8	no	—
195	C₂×C₄	—	188	8	no	—
196	C₂×C₄	C₄₇⋊#95	1	2	yes	—
197	C₂×C₄	C₄₇⋊#95	1	2	yes	—
198	C₂×C₄	C₄₇⋊#95	1	2	yes	—
199	C₂×C₄	C₄₇⋊#95	1	2	yes	—
200	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
203	C₈	—	4	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
210	C₂²	#15689139	1	4	yes	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	376	8	no	—
218	C₄	#15689097	1	2	yes	—
219	C₂	—	2	4	no	—
220	C₂	—	4	1	no	—
221	C₂	#20243099	1	2	yes	—
222	C₂	#20243121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2920	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2920	D₄₇
hypocenter	C₄₇	#8050
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
4. #15688991	C₄	semidirect
5. #15688989	C₄	semidirect
14. C₄₇×#423	C₄	semidirect
16. C₄×M₁₆×C₄₇	C₄	semidirect
118. #423	Dic₄₇	semidirect
120. C₄×M₁₆	Dic₄₇	semidirect
128. C₄₇	#8050	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
#20241273	2
C₄₇×#2920	2
#8050	47