Subgroups — #26131128

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 230 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	#20241468	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	4	no	—
5	C₂×#13093964	—	2	4	no	—
6	C₂×#13093964	C₄	1	4	yes	complemented
7	C₂×#13093964	C₂²	1	4	yes	—
8	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	complemented
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	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
15	#13093964	—	2	8	no	—
16	#13093964	—	2	4	no	—
17	#13093964	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	—	2	8	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
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	2	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂²×C₈×C₄₇	D₄	1	4	yes	—
30	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	#117×C₄₇	—	2	2	no	—
33	#117×C₄₇	C₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	#97×C₄₇	—	2	4	no	—
38	C₄×C₈×C₄₇	—	2	4	no	—
39	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₄₇	—	2	8	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
60	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₂²⋊C₄×C₄₇	—	2	4	no	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	C₂×C₈	1	4	yes	—
69	C₄₇⋊C₁₆	—	2	8	no	—
70	C₂³×C₄₇	—	4	2	no	—
71	C₂³×C₄₇	#97	1	2	yes	—
72	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
73	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
74	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	4	4	no	—
77	C₂×C₄×C₄₇	#95	1	4	yes	—
78	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
79	C₂×C₄×C₄₇	#97	1	2	yes	—
80	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₄₇	—	2	4	no	—
91	C₈×C₄₇	#95	1	2	yes	—
92	C₈×C₄₇	#95	1	2	yes	—
93	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	#12547	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	—	4	2	no	—
99	C₂²×C₄₇	—	4	2	no	—
100	C₂²×C₄₇	#336	1	4	yes	—
101	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₄₇	—	4	4	no	—
105	C₄×C₄₇	#336	1	4	yes	—
106	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
107	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×M₃₂	—	47	2	no	—
111	C₂×#405	D₄₇	1	1	yes	characteristic
112	#1375	—	47	4	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	—	2	4	no	—
115	C₂×C₄₇	—	2	1	no	—
116	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
117	C₂×C₄₇	#2110	1	2	yes	—
118	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
119	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₂×#117	Dic₄₇	1	1	yes	characteristic
121	C₂×M₃₂	—	47	4	no	—
122	C₂×M₃₂	—	47	4	no	—
123	C₂×M₃₂	—	94	4	no	—
124	C₂×M₃₂	—	94	4	no	—
125	C₂²×C₁₆	—	47	4	no	—
126	#405	—	2	4	no	—
127	C₂×#97	C₂×D₄₇	1	2	yes	—
128	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
129	C₄₇	#12547	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	C₄₇⋊C₈	1	4	yes	—
134	C₂²×C₈	Dic₉₄	1	2	yes	—
135	C₂²×C₈	C₄×D₄₇	1	2	yes	—
136	C₂²×C₈	C₄×D₄₇	1	2	yes	—
137	C₂²×C₈	D₁₈₈	1	2	yes	—
138	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
139	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
141	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
142	#117	—	2	2	no	—
143	#117	C₄₇⋊C₈	1	4	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
147	M₃₂	—	94	8	no	—
148	M₃₂	—	94	4	no	—
149	M₃₂	—	94	4	no	—
150	C₂×C₁₆	—	47	8	no	—
151	C₂×C₁₆	—	94	8	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	#10625329	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
157	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
158	C₂²×C₄	#10625329	1	1	yes	characteristic
159	C₂²×C₄	#10625329	1	2	yes	—
160	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
161	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
162	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
168	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
169	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
170	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
171	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
172	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
173	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
174	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
175	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
176	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
178	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
184	C₁₆	—	94	8	no	—
185	C₂³	—	4	2	no	—
186	C₂³	#13093935	1	2	yes	—
187	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
188	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
189	C₂³	C₂×#10625329	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₄₇⋊#95	1	4	yes	—
193	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
194	C₂×C₄	#13093935	1	2	yes	—
195	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
196	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
197	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
198	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
199	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
204	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	C₄₇⋊#95	1	2	yes	—
207	C₈	C₄₇⋊#95	1	2	yes	—
208	C₈	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
209	C₈	C₂×Dic₄₇⋊C₄	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#15688878	1	4	yes	—
215	C₂²	C₂×#13093935	1	1	yes	characteristic
216	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
217	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
218	C₄	—	4	4	no	—
219	C₄	#15688878	1	4	yes	—
220	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
221	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
222	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
223	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	4	no	—
226	C₂	—	2	1	no	—
227	C₂	C₂×#15688878	1	1	yes	characteristic
228	C₂	#20244993	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#12547
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20241468	C₂	semidirect
6. C₂×#13093964	C₄	semidirect
8. C₂²×C₄₇⋊C₁₆	C₄	semidirect
129. C₄₇	#12547	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
#20241468	2
C₂×C₄₇×#405	2
#12547	47