Subgroups — #26213624

Order 12032.

1470 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689078	C₂	1	4	yes	maximal
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
4	#20243076	C₂	1	4	yes	maximal, complemented
5	#97×Dic₄₇	C₂	1	4	yes	maximal, complemented
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689078	C₄	1	32	yes	complemented
10	C₄×#10625329	C₄	1	8	yes	complemented
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×#13093934	C₂²	1	8	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	8	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	—
16	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	—
17	#15688969	C₂²	1	2	yes	—
18	C₄²×Dic₄₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₄₇	C₄	1	8	yes	complemented
22	C₂×#97×C₄₇	C₂²	1	2	yes	—
23	C₂×#97×C₄₇	C₂²	1	2	yes	—
24	C₂×#10625329	C₂×C₄	1	8	yes	—
25	C₂×#10625329	C₂×C₄	1	16	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	C₂×C₄	1	16	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
32	#13093935	C₂×C₄	1	8	yes	—
33	#13093934	C₂×C₄	1	32	yes	complemented
34	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
35	C₈×Dic₄₇	C₂×C₄	1	32	yes	complemented
36	C₄₇⋊#95	C₂×C₄	1	8	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#97×C₄₇	C₂×C₄	1	8	yes	—
48	#97×C₄₇	C₂×C₄	1	8	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
60	C₂×C₈×C₄₇	C₄²	1	32	yes	complemented
61	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
64	C₄²×C₄₇	C₄²	1	8	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²⋊Dic₄₇	C₄²	1	16	yes	—
70	C₄⋊Dic₄₇	C₄²	1	16	yes	—
71	C₄×Dic₄₇	—	4	8	no	—
72	C₄×Dic₄₇	C₂²×C₄	1	16	yes	—
73	#10625329	C₄²	1	32	yes	complemented
74	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
78	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
79	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	4	8	no	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
85	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₄₇	—	2	16	no	—
93	C₂×Dic₄₇	C₂×C₄²	1	16	yes	—
94	C₈×C₄₇	C₂×C₄²	1	16	yes	—
95	C₄₇⋊C₈	C₂×C₄²	1	16	yes	—
96	#20331	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	—	2	8	no	—
99	C₂²×C₄₇	C₄³	1	4	yes	—
100	C₂²×C₄₇	C₂²×C₄²	1	2	yes	—
101	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₄₇	#568	1	4	yes	—
103	C₄×C₄₇	—	2	4	no	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	C₄³	1	4	yes	—
106	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
107	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
108	Dic₄₇	—	2	16	no	—
109	C₂×#405	—	47	4	no	—
110	C₂×C₄×M₁₆	—	47	1	no	—
111	C₂²×#97	D₄₇	1	1	yes	characteristic
112	C₄×#117	—	47	1	no	—
113	#1726	—	47	4	no	—
114	C₄×#97	—	47	4	no	—
115	C₂×C₄₇	—	2	4	no	—
116	C₂×C₄₇	C₂×C₄³	1	1	yes	characteristic
117	C₂×C₄₇	#2873	1	2	yes	—
118	C₂×C₄₇	#2875	1	4	yes	—
119	C₂²×M₁₆	—	47	2	no	—
120	C₂³×C₈	C₂×D₄₇	1	2	yes	—
121	C₂×#117	—	47	2	no	—
122	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	#405	—	47	32	no	—
124	C₄×M₁₆	—	47	8	no	—
125	C₂×#97	—	47	8	no	—
126	C₂×#97	—	47	2	no	—
127	C₂×#97	Dic₄₇	1	8	yes	complemented
128	C₂×#97	C₂×D₄₇	1	2	yes	—
129	C₂×#97	C₂×D₄₇	1	2	yes	—
130	C₂×C₄×C₈	—	47	8	no	—
131	C₂×C₄×C₈	—	47	2	no	—
132	C₄×C₄⋊C₄	—	47	4	no	—
133	C₄×C₂²⋊C₄	—	47	4	no	—
134	#376	—	47	2	no	—
135	C₄³	—	47	2	no	—
136	C₄₇	#20331	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	C₄×D₄₇	1	2	yes	—
138	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×M₁₆	—	47	8	no	—
140	C₂×M₁₆	—	47	16	no	—
141	C₂²×C₈	—	47	4	no	—
142	C₂²×C₈	C₄×D₄₇	1	16	yes	—
143	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
144	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
145	#117	—	47	16	no	—
146	C₂×C₄⋊C₄	—	47	4	no	—
147	C₂×C₂²⋊C₄	—	47	4	no	—
148	C₂×C₄²	—	47	4	no	—
149	C₂×C₄²	—	47	8	no	—
150	C₂×C₄²	C₄×D₄₇	1	2	yes	—
151	C₂×C₄²	C₄×D₄₇	1	4	yes	—
152	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
153	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	#97	—	47	32	no	—
156	#97	—	47	8	no	—
157	#97	C₂×Dic₄₇	1	8	yes	—
158	#97	C₂×Dic₄₇	1	8	yes	—
159	C₄×C₈	—	47	32	no	—
160	C₄×C₈	—	47	8	no	—
161	#95	—	47	8	no	—
162	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	—	47	8	no	—
165	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
167	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	M₁₆	—	47	32	no	—
174	C₂×C₈	—	47	8	no	—
175	C₂×C₈	—	47	16	no	—
176	C₂×C₈	C₄×Dic₄₇	1	32	yes	complemented
177	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
178	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
179	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
180	C₄⋊C₄	—	47	16	no	—
181	C₂²⋊C₄	—	47	16	no	—
182	C₄²	—	47	16	no	—
183	C₄²	—	188	8	no	—
184	C₄²	C₄×Dic₄₇	1	8	yes	—
185	C₄²	C₂×C₄×D₄₇	1	2	yes	—
186	C₄²	C₂×C₄×D₄₇	1	2	yes	—
187	C₄²	C₂²×Dic₄₇	1	2	yes	—
188	C₄²	C₂²×Dic₄₇	1	2	yes	—
189	C₂³	—	2	4	no	—
190	C₂³	C₄²×D₄₇	1	4	yes	—
191	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
192	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	47	16	no	—
197	C₂×C₄	—	94	16	no	—
198	C₂×C₄	C₄²×D₄₇	1	4	yes	—
199	C₂×C₄	C₄²×D₄₇	1	8	yes	—
200	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
201	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
202	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
204	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
206	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
207	C₈	—	47	16	no	—
208	C₈	C₂×C₄×Dic₄₇	1	16	yes	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	8	no	—
211	C₂²	C₄²×Dic₄₇	1	4	yes	—
212	C₂²	C₂×C₄²×D₄₇	1	2	yes	—
213	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
214	C₂²	#15689889	1	4	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	4	no	—
217	C₄	—	94	16	no	—
218	C₄	C₄²×Dic₄₇	1	4	yes	—
219	C₄	C₂×C₄²×D₄₇	1	2	yes	—
220	C₄	C₂²×C₄×Dic₄₇	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	C₂×C₄²×Dic₄₇	1	1	yes	characteristic
223	C₂	#20251806	1	2	yes	—
224	C₂	#20251813	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂×C₄³
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#20331
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #20243076	C₂	semidirect
5. #97×Dic₄₇	C₂	semidirect
9. #15689078	C₄	semidirect
10. C₄×#10625329	C₄	semidirect
21. C₂×#97×C₄₇	C₄	semidirect
33. #13093934	C₂×C₄	semidirect
35. C₈×Dic₄₇	C₂×C₄	semidirect
60. C₂×C₈×C₄₇	C₄²	semidirect
73. #10625329	C₄²	semidirect
127. C₂×#97	Dic₄₇	semidirect
136. C₄₇	#20331	semidirect
176. 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×#15689078	2
C₂×C₄×#10625329	2
C₄×#13093983	2
#20243076	2
#97×Dic₄₇	2
C₂²×#97×C₄₇	2
#20331	47