Subgroups — #26105737

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20241223	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689176	—	2	4	no	—
5	#15689176	C₄	1	8	yes	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
9	C₂³×C₈×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
18	C₄×C₄₇⋊C₈	—	2	8	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	—	2	4	no	—
24	C₂²×C₈×C₄₇	C₈	1	8	yes	complemented
25	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	4	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₇	—	2	4	no	—
33	C₄×C₈×C₄₇	—	2	4	no	—
34	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	—	2	4	no	—
37	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₄₇	—	2	16	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	—
53	C₄²×C₄₇	C₄²	1	2	yes	—
54	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
58	C₄²×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄₇⋊C₈	—	2	8	no	—
60	C₂×C₄₇⋊C₈	—	4	8	no	—
61	C₂³×C₄₇	—	2	2	no	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	#95	1	2	yes	—
64	C₂³×C₄₇	C₄⋊C₈	1	4	yes	—
65	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	#95	1	2	yes	—
71	C₂×C₄×C₄₇	#95	1	2	yes	—
72	C₂×C₄×C₄₇	#95	1	2	yes	—
73	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
74	C₂×C₄×C₄₇	#97	1	4	yes	—
75	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
76	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₄₇	#108	1	4	yes	—
78	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₄₇	#135	1	4	yes	—
85	C₈×C₄₇	—	2	8	no	—
86	C₄₇⋊C₈	—	4	8	no	—
87	#7872	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	—	2	8	no	—
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	#336	1	4	yes	—
92	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
94	C₂²×C₄₇	#427	1	2	yes	—
95	C₂²×C₄₇	C₂×#108	1	2	yes	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	#336	1	4	yes	—
99	C₄×C₄₇	#341	1	4	yes	—
100	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
101	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
102	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
103	C₄×C₄₇	#416	1	4	yes	—
104	C₂×#431	—	47	2	no	—
105	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
106	#1277	—	47	4	no	—
107	C₂×C₄₇	—	2	4	no	—
108	C₂×C₄₇	#1564	1	4	yes	—
109	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
110	C₂×C₄₇	#1735	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
112	C₂³×C₈	Dic₄₇	1	2	yes	—
113	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
114	#431	—	47	8	no	—
115	#431	—	94	4	no	—
116	C₂×C₄⋊C₈	—	47	4	no	—
117	C₂×C₂²⋊C₈	—	94	2	no	—
118	C₂×C₄×C₈	—	2	4	no	—
119	C₂×C₄×C₈	—	94	2	no	—
120	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
121	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
122	C₄₇	#7872	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
124	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
125	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	—	94	4	no	—
129	C₂²×C₈	C₄₇⋊C₈	1	8	yes	complemented
130	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
131	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₄₇	1	4	yes	—
133	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊C₈	—	47	8	no	—
138	C₄⋊C₈	—	94	4	no	—
139	C₂²⋊C₈	—	94	8	no	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	—	94	8	no	—
143	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
147	C₂²×C₄	#10625329	1	4	yes	—
148	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
149	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
150	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
153	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
154	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
155	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
158	C₂×C₈	—	2	16	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	94	8	no	—
162	C₂×C₈	—	188	8	no	—
163	C₂×C₈	C₂×C₄₇⋊C₈	1	8	yes	—
164	C₄²	C₄×Dic₄₇	1	2	yes	—
165	C₄²	Dic₄₇⋊C₄	1	4	yes	—
166	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
167	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
168	C₄²	C₂×C₄×D₄₇	1	2	yes	—
169	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₄₇⋊#95	1	2	yes	—
173	C₂³	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
174	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₄₇⋊#95	1	2	yes	—
180	C₂×C₄	C₄₇⋊#95	1	2	yes	—
181	C₂×C₄	C₄₇⋊#95	1	2	yes	—
182	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
183	C₂×C₄	#13093935	1	4	yes	—
184	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
185	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
186	C₂×C₄	#13093960	1	4	yes	—
187	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
188	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
189	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
191	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
192	C₂×C₄	C₂×#10625329	1	2	yes	—
193	C₂×C₄	#13094046	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₈	—	188	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	#15688878	1	4	yes	—
200	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
201	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
202	C₂²	#15689161	1	2	yes	—
203	C₂²	C₂×#13093960	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	#15688878	1	4	yes	—
207	C₄	#15688888	1	4	yes	—
208	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
209	C₄	C₂×#13093935	1	1	yes	characteristic
210	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
211	C₄	#15689120	1	4	yes	—
212	C₂	—	2	4	no	—
213	C₂	#20242136	1	4	yes	—
214	C₂	C₂×#15688878	1	1	yes	characteristic
215	C₂	#20243105	1	1	yes	characteristic
216	C₂	C₂×#15688888	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#7872
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20241223	C₂	semidirect
24. C₂²×C₈×C₄₇	C₈	semidirect
122. C₄₇	#7872	semidirect
129. C₂²×C₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20241223	2
C₂²×C₄×C₈×C₄₇	2
#7872	47