Subgroups — #26230169

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252106	C₂	1	2	yes	maximal, complemented
2	#20244221	C₂	1	2	yes	maximal, complemented
3	#20243717	C₂	1	8	yes	maximal, complemented
4	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243139	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094031	C₂²	1	4	yes	—
8	#15689195	—	2	8	no	—
9	#15689147	C₂²	1	2	yes	—
10	C₂×#13093951	C₂²	1	8	yes	—
11	#15689090	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	1	yes	characteristic
13	#15688969	C₂²	1	1	yes	characteristic
14	#15688888	C₂²	1	4	yes	—
15	#15688877	C₂²	1	4	yes	—
16	C₂×#135×C₄₇	C₂²	1	4	yes	—
17	C₄₇×#532	C₂²	1	2	yes	—
18	C₄₇×#495	C₄	1	4	yes	complemented
19	C₄₇×#494	C₄	1	1	yes	characteristic, complemented
20	C₄₇×#493	C₄	1	1	yes	characteristic, complemented
21	C₄₇×#492	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	#13094031	—	4	8	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	#13093951	—	4	8	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
32	#13093935	D₄	1	4	yes	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₂×#41×C₄₇	C₂³	1	4	yes	—
35	#135×C₄₇	—	4	8	no	—
36	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×SD₁₆×C₄₇	C₂×C₄	1	8	yes	—
38	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
41	#127×C₄₇	C₂×C₄	1	2	yes	—
42	#124×C₄₇	C₂×C₄	1	4	yes	—
43	#121×C₄₇	—	2	4	no	—
44	C₄×D₄×C₄₇	—	2	4	no	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	#41×C₄₇	—	2	8	no	—
50	#41×C₄₇	—	2	8	no	—
51	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₄₇	—	2	4	no	—
53	C₂×D₄×C₄₇	—	2	4	no	—
54	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	#41	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	2	no	—
59	Q₁₆×C₄₇	—	4	4	no	—
60	SD₁₆×C₄₇	—	4	8	no	—
61	D₈×C₄₇	—	4	4	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₂²⋊C₄×C₄₇	—	2	8	no	—
67	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂×D₄	1	2	yes	—
70	C₄×Dic₄₇	—	2	4	no	—
71	#10625329	—	2	8	no	—
72	#10625329	—	4	8	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	4	8	no	—
76	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
79	Q₈×C₄₇	—	2	8	no	—
80	D₄×C₄₇	—	2	8	no	—
81	D₄×C₄₇	—	4	8	no	—
82	D₄×C₄₇	—	8	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
92	C₂×Dic₄₇	—	4	4	no	—
93	C₂×Dic₄₇	—	4	2	no	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	—	2	2	no	—
96	C₄₇⋊C₈	—	2	4	no	—
97	C₄₇⋊C₈	—	4	4	no	—
98	#22282	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	4	4	no	—
101	C₂²×C₄₇	—	4	8	no	—
102	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₄₇	#540	1	2	yes	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	4	4	no	—
107	C₄×C₄₇	#518	1	2	yes	—
108	C₄×C₄₇	#535	1	2	yes	—
109	Dic₄₇	—	8	2	no	—
110	#3147	D₄₇	1	1	yes	characteristic
111	#2932	—	47	2	no	—
112	#1963	—	47	2	no	—
113	#1876	—	47	8	no	—
114	#1837	—	47	1	no	—
115	#1746	—	47	1	no	—
116	C₂×C₄₇	—	2	2	no	—
117	C₂×C₄₇	—	8	4	no	—
118	C₂×C₄₇	#2328	1	1	yes	characteristic
119	C₂×C₄₇	#2957	1	2	yes	—
120	C₂×#135	C₂×D₄₇	1	4	yes	—
121	C₂×#131	—	47	4	no	—
122	#532	C₂×D₄₇	1	2	yes	—
123	#495	Dic₄₇	1	4	yes	complemented
124	#494	Dic₄₇	1	1	yes	characteristic, complemented
125	#493	Dic₄₇	1	1	yes	characteristic, complemented
126	#492	Dic₄₇	1	2	yes	complemented
127	#435	—	94	8	no	—
128	#423	—	47	2	no	—
129	C₂×#104	—	47	8	no	—
130	#408	—	47	4	no	—
131	C₂×#97	—	47	1	no	—
132	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
133	#376	—	47	1	no	—
134	#341	—	47	4	no	—
135	#336	—	47	4	no	—
136	C₄₇	#22282	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	C₂²×D₄₇	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
140	C₂×SD₁₆	C₂×Dic₄₇	1	8	yes	—
141	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
142	#131	—	188	8	no	—
143	C₂×M₁₆	—	47	4	no	—
144	C₂×M₁₆	—	94	4	no	—
145	C₂²×C₈	—	47	2	no	—
146	C₂²×C₈	—	94	4	no	—
147	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
149	#127	C₂×Dic₄₇	1	2	yes	—
150	#124	C₂×Dic₄₇	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	—	47	2	no	—
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	C₄⋊C₈	—	94	4	no	—
156	#104	—	188	8	no	—
157	C₂²⋊C₈	—	94	8	no	—
158	#97	—	47	4	no	—
159	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
160	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
161	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
162	#95	—	94	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	94	2	no	—
171	C₂²×C₄	#10625352	1	2	yes	—
172	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	94	8	no	—
177	M₁₆	—	188	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	47	4	no	—
180	C₂×C₈	—	94	4	no	—
181	C₂×C₈	—	94	4	no	—
182	C₂×C₈	—	188	8	no	—
183	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
184	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	94	4	no	—
188	C₄²	D₄×D₄₇	1	2	yes	—
189	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#10625352	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	188	4	no	—
202	C₂×C₄	—	188	2	no	—
203	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
204	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
205	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄	#13094069	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	94	4	no	—
211	C₈	—	188	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
216	C₂²	#15689727	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	376	2	no	—
221	C₄	#15689598	1	2	yes	—
222	C₄	#15689686	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#20246114	1	1	yes	characteristic
226	C₂	#20252305	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3147	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#22282
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20252106	C₂	semidirect
2. #20244221	C₂	semidirect
3. #20243717	C₂	semidirect
4. #20243526	C₂	semidirect
5. #20243139	C₂	semidirect
18. C₄₇×#495	C₄	semidirect
19. C₄₇×#494	C₄	semidirect
20. C₄₇×#493	C₄	semidirect
21. C₄₇×#492	C₄	semidirect
123. #495	Dic₄₇	semidirect
124. #494	Dic₄₇	semidirect
125. #493	Dic₄₇	semidirect
126. #492	Dic₄₇	semidirect
136. C₄₇	#22282	semidirect

Maximal subgroups

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

subgroup	index
#20252106	2
#20244221	2
#20243717	2
#20243526	2
#20243139	2
C₄₇×#3147	2
#22282	47