Subgroups — #26105720

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20241221	C₂	1	4	yes	maximal, complemented
3	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	2	no	—
5	#15689180	C₄	1	4	yes	complemented
6	#15689176	—	2	2	no	—
7	#15689176	C₂²	1	4	yes	—
8	#15689147	C₄	1	4	yes	complemented
9	#15689087	—	2	2	no	—
10	C₄×#10625329	—	2	2	no	—
11	C₂²×C₄²×C₄₇	—	2	2	no	—
12	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄³×C₄₇	—	2	2	no	—
15	C₄³×C₄₇	C₄	1	2	yes	—
16	C₄³×C₄₇	C₂²	1	2	yes	—
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	#13093935	—	2	4	no	—
23	C₄×C₄₇⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₄₇	—	2	2	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	—	2	16	no	—
28	C₂×C₄²×C₄₇	—	2	4	no	—
29	C₂×C₄²×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	—	2	4	no	—
31	C₂×C₄²×C₄₇	—	2	8	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	16	no	—
42	C₂²×C₄×C₄₇	—	2	8	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₄²×C₄₇	—	2	16	no	—
57	C₄²×C₄₇	—	2	16	no	—
58	C₄²×C₄₇	—	2	8	no	—
59	C₄²×C₄₇	—	2	4	no	—
60	C₄²×C₄₇	C₄²	1	1	yes	characteristic
61	C₄²×C₄₇	C₄²	1	1	yes	characteristic
62	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₄₇	M₁₆	1	8	yes	complemented
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	#10625329	—	4	8	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	#95	1	2	yes	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	16	no	—
78	C₂×C₄×C₄₇	—	2	32	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	#95	1	2	yes	—
89	C₂×C₄×C₄₇	#95	1	2	yes	—
90	C₂×C₄×C₄₇	#95	1	2	yes	—
91	C₂×C₄×C₄₇	#104	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
97	C₄₇⋊C₈	—	4	8	no	—
98	#7867	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	8	no	—
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	#339	1	4	yes	—
104	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
105	C₂²×C₄₇	#407	1	2	yes	—
106	C₄×C₄₇	—	2	16	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	#339	1	4	yes	—
111	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
112	C₄×C₄₇	C₂×#104	1	4	yes	—
113	C₄×C₄₇	#423	1	2	yes	—
114	#2920	—	47	2	no	—
115	C₂×C₄³	D₄₇	1	1	yes	characteristic
116	#1276	—	47	4	no	—
117	C₂×C₄₇	—	2	2	no	—
118	C₂×C₄₇	—	2	4	no	—
119	C₂×C₄₇	#1728	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#339	1	2	yes	—
121	C₂²×C₄²	—	2	2	no	—
122	C₂²×C₄²	Dic₄₇	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	#432	—	47	4	no	—
125	#432	—	94	2	no	—
126	#431	—	47	4	no	—
127	#431	—	94	2	no	—
128	#423	—	47	4	no	—
129	#407	—	94	2	no	—
130	C₄×M₁₆	—	94	2	no	—
131	C₄³	—	2	2	no	—
132	C₄³	Dic₄₇	1	2	yes	—
133	C₄³	C₂×D₄₇	1	2	yes	—
134	C₄₇	#7867	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	—	2	2	no	—
136	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
138	C₂×M₁₆	—	94	4	no	—
139	C₂×C₄²	—	2	16	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	—	2	8	no	—
144	C₂×C₄²	Dic₉₄	1	2	yes	—
145	C₂×C₄²	C₄×D₄₇	1	2	yes	—
146	C₂×C₄²	C₄×D₄₇	1	2	yes	—
147	C₂×C₄²	D₁₈₈	1	2	yes	—
148	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
149	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
150	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
152	C₄⋊C₈	—	47	8	no	—
153	C₄⋊C₈	—	94	4	no	—
154	C₂²⋊C₈	—	94	4	no	—
155	C₂²⋊C₈	—	94	4	no	—
156	#97	—	94	4	no	—
157	C₄×C₈	—	94	4	no	—
158	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₂²×C₄	—	2	16	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
165	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
166	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
168	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
169	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
171	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	M₁₆	—	188	8	no	—
175	C₂×C₈	—	94	8	no	—
176	C₄²	—	2	16	no	—
177	C₄²	—	2	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	4	no	—
180	C₄²	#10625329	1	8	yes	complemented
181	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
182	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
183	C₄²	Dic₄₇⋊C₄	1	2	yes	—
184	C₄²	Dic₄₇⋊C₄	1	2	yes	—
185	C₄²	C₄⋊Dic₄₇	1	2	yes	—
186	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
187	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
188	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₄₇	1	2	yes	—
190	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₄₇⋊#95	1	2	yes	—
194	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	32	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	C₄₇⋊#95	1	2	yes	—
207	C₂×C₄	C₄₇⋊#95	1	2	yes	—
208	C₂×C₄	C₄₇⋊#95	1	2	yes	—
209	C₂×C₄	#13093952	1	8	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂×#10625329	1	4	yes	—
215	C₈	—	188	8	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	#15688883	1	4	yes	—
221	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
222	C₂²	#15689087	1	2	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	#15688883	1	4	yes	—
228	C₄	C₄×#10625329	1	2	yes	—
229	C₄	C₂×#13093952	1	4	yes	—
230	C₄	#15689147	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	#20243082	1	1	yes	characteristic
234	C₂	C₂×#15688883	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#7867
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20241221	C₂	semidirect
5. #15689180	C₄	semidirect
8. #15689147	C₄	semidirect
19. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
67. C₄²×C₄₇	M₁₆	semidirect
134. C₄₇	#7867	semidirect
180. C₄²	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20241221	2
C₂×C₄³×C₄₇	2
#7867	47