Subgroups — #26105912

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 220 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	#20241233	C₂	1	4	yes	maximal, complemented
3	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	2	no	—
5	#15689180	C₄	1	4	yes	—
6	#15689176	—	2	2	no	—
7	#15689176	C₂²	1	4	yes	—
8	#15689147	—	2	2	no	—
9	#15689087	C₄	1	4	yes	—
10	C₄×#10625329	—	2	2	no	—
11	C₂³×C₈×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#97×C₄₇	—	2	2	no	—
14	C₂×#97×C₄₇	—	2	2	no	—
15	C₂×#97×C₄₇	C₄	1	2	yes	—
16	C₂×#97×C₄₇	C₂²	1	2	yes	—
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
22	#13093935	—	2	4	no	—
23	C₄×C₄₇⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₄₇	—	2	4	no	—
27	C₂²×C₈×C₄₇	—	2	8	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#97×C₄₇	—	2	2	no	—
37	#97×C₄₇	—	2	2	no	—
38	#97×C₄₇	—	2	4	no	—
39	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	4	4	no	—
42	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	—	2	16	no	—
53	C₂×C₈×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
56	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂×D₄	1	2	yes	—
65	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	#10625329	—	4	8	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂³×C₄₇	—	2	4	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	#95	1	2	yes	—
71	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
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	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	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	2	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	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
89	C₈×C₄₇	—	2	8	no	—
90	C₄₇⋊C₈	—	4	8	no	—
91	#7919	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	8	no	—
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	#344	1	4	yes	—
97	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
98	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
99	C₄×C₄₇	—	2	2	no	—
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	#407	1	2	yes	—
105	C₄×C₄₇	#423	1	2	yes	—
106	#2920	—	47	2	no	—
107	C₂²×#97	D₄₇	1	1	yes	characteristic
108	#1281	—	47	4	no	—
109	C₂×C₄₇	—	2	4	no	—
110	C₂×C₄₇	—	2	2	no	—
111	C₂×C₄₇	#1728	1	1	yes	characteristic
112	C₂×C₄₇	C₂×#344	1	2	yes	—
113	C₂³×C₈	Dic₄₇	1	2	yes	—
114	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
115	#432	—	47	4	no	—
116	#432	—	94	2	no	—
117	#431	—	47	4	no	—
118	#431	—	94	2	no	—
119	#423	—	94	2	no	—
120	#407	—	47	4	no	—
121	C₄×M₁₆	—	94	2	no	—
122	C₂×#97	—	2	2	no	—
123	C₂×#97	—	2	2	no	—
124	C₂×#97	Dic₄₇	1	2	yes	—
125	C₂×#97	C₂×D₄₇	1	2	yes	—
126	C₄₇	#7919	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	C₄×D₄₇	1	2	yes	—
128	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
129	C₂×M₁₆	—	94	4	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	2	8	no	—
132	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
133	C₂×C₄²	Dic₉₄	1	2	yes	—
134	C₂×C₄²	C₄×D₄₇	1	2	yes	—
135	C₂×C₄²	D₁₈₈	1	2	yes	—
136	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
139	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	C₄⋊C₈	—	94	4	no	—
141	C₄⋊C₈	—	94	4	no	—
142	C₂²⋊C₈	—	47	8	no	—
143	C₂²⋊C₈	—	94	4	no	—
144	#97	—	2	2	no	—
145	#97	—	2	2	no	—
146	#97	—	2	4	no	—
147	#97	—	94	4	no	—
148	C₄×C₈	—	94	4	no	—
149	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
153	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
154	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
155	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
156	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
157	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
158	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
159	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
162	M₁₆	—	188	8	no	—
163	C₂×C₈	—	2	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	94	8	no	—
167	C₂×C₈	#10625329	1	8	yes	complemented
168	C₄²	Dic₄₇⋊C₄	1	2	yes	—
169	C₄²	Dic₄₇⋊C₄	1	2	yes	—
170	C₄²	C₄⋊Dic₄₇	1	2	yes	—
171	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
172	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
173	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
174	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
176	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₄₇⋊#95	1	2	yes	—
181	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	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	C₄₇⋊#95	1	2	yes	—
192	C₂×C₄	C₄₇⋊#95	1	2	yes	—
193	C₂×C₄	C₄₇⋊#95	1	2	yes	—
194	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
196	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625329	1	4	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	188	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	#15688894	1	4	yes	—
206	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
207	C₂²	C₄×#10625329	1	2	yes	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	#15689087	1	2	yes	—
214	C₄	#15689147	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	4	no	—
217	C₂	#20243082	1	1	yes	characteristic
218	C₂	C₂×#15688894	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#7919
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #20241233	C₂	semidirect
55. C₂×C₈×C₄₇	M₁₆	semidirect
126. C₄₇	#7919	semidirect
167. C₂×C₈	#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
#20241233	2
C₂²×#97×C₄₇	2
#7919	47