Subgroups — #26131134

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	2	yes	maximal
2	#20241470	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	2	no	—
5	C₂×#13093964	—	2	4	no	—
6	C₂×#13093964	C₄	1	8	yes	complemented
7	C₂×#13093964	C₂²	1	4	yes	—
8	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
9	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
12	C₄₇×#405	—	2	4	no	—
13	C₂×#97×C₄₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
15	#13093964	—	2	8	no	—
16	#13093964	—	2	4	no	—
17	#13093964	C₂×C₄	1	8	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	8	no	—
19	C₂×C₄₇⋊C₁₆	—	2	4	no	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₄₇	—	2	2	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	C₈	1	4	yes	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂²×C₈×C₄₇	D₄	1	4	yes	—
29	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	#117×C₄₇	—	2	2	no	—
32	#117×C₄₇	C₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	#97×C₄₇	—	2	4	no	—
37	C₄×C₈×C₄₇	—	2	4	no	—
38	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₄₇	—	2	8	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	C₄²	1	2	yes	—
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₂²⋊C₄×C₄₇	—	2	4	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	C₂×C₈	1	4	yes	—
66	C₄₇⋊C₁₆	—	2	8	no	—
67	C₂³×C₄₇	—	4	2	no	—
68	C₂³×C₄₇	#97	1	2	yes	—
69	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
70	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
71	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	4	4	no	—
74	C₂×C₄×C₄₇	#95	1	4	yes	—
75	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₄₇	#97	1	2	yes	—
77	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₄₇	—	2	4	no	—
88	C₈×C₄₇	#95	1	4	yes	—
89	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
90	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	#12549	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	—	2	4	no	—
94	C₂²×C₄₇	—	4	2	no	—
95	C₂²×C₄₇	—	4	2	no	—
96	C₂²×C₄₇	#336	1	4	yes	—
97	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
98	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₄₇	—	4	4	no	—
101	C₄×C₄₇	#336	1	4	yes	—
102	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
103	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
104	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×M₃₂	—	47	2	no	—
107	C₂×#405	D₄₇	1	1	yes	characteristic
108	#1376	—	47	4	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	2	4	no	—
111	C₂×C₄₇	—	2	1	no	—
112	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
113	C₂×C₄₇	#2153	1	2	yes	—
114	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
115	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
116	C₂×#117	Dic₄₇	1	1	yes	characteristic
117	C₂×M₃₂	—	47	8	no	—
118	C₂×M₃₂	—	47	4	no	—
119	C₂×M₃₂	—	94	2	no	—
120	C₂×M₃₂	—	94	4	no	—
121	C₂²×C₁₆	—	94	2	no	—
122	#405	—	2	4	no	—
123	C₂×#97	Dic₄₇	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
125	C₄₇	#12549	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	C₄₇⋊C₈	1	4	yes	—
130	C₂²×C₈	Dic₉₄	1	2	yes	—
131	C₂²×C₈	C₄×D₄₇	1	4	yes	—
132	C₂²×C₈	D₁₈₈	1	2	yes	—
133	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
134	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
136	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
137	#117	—	2	2	no	—
138	#117	C₄₇⋊C₈	1	4	yes	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
142	M₃₂	—	47	8	no	—
143	M₃₂	—	94	8	no	—
144	M₃₂	—	94	4	no	—
145	C₂×C₁₆	—	94	8	no	—
146	C₂×C₁₆	—	94	4	no	—
147	#97	—	2	4	no	—
148	C₄×C₈	—	2	4	no	—
149	C₂⁴	#10625329	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
152	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
153	C₂²×C₄	#10625329	1	1	yes	characteristic
154	C₂²×C₄	#10625329	1	2	yes	—
155	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
156	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
157	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
159	M₁₆	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
163	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
164	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
165	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
166	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
167	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
168	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
169	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
171	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
177	C₁₆	—	94	8	no	—
178	C₂³	—	4	2	no	—
179	C₂³	#13093935	1	2	yes	—
180	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
181	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
182	C₂³	C₂×#10625329	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₄₇⋊#95	1	4	yes	—
186	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
187	C₂×C₄	#13093935	1	2	yes	—
188	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
189	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
190	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
191	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
192	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₄₇⋊#95	1	4	yes	—
200	C₈	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
201	C₈	C₂×Dic₄₇⋊C₄	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	#15688878	1	4	yes	—
207	C₂²	C₂×#13093935	1	1	yes	characteristic
208	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
209	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	#15688878	1	4	yes	—
212	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
213	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
215	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	4	no	—
219	C₂	C₂×#15688878	1	1	yes	characteristic
220	C₂	#20245154	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#12549
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20241470	C₂	semidirect
6. C₂×#13093964	C₄	semidirect
125. C₄₇	#12549	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093964	2
#20241470	2
C₂×C₄₇×#405	2
#12549	47