Subgroups — #26131497

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688927	C₂	1	2	yes	maximal
2	C₂×#15688899	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093958	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal
5	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₄₇	—	2	1	no	—
9	#15688927	C₄	1	8	yes	complemented
10	#15688899	—	2	2	no	—
11	#15688886	—	2	1	no	—
12	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
14	C₄×Q₁₆×C₄₇	—	2	2	no	—
15	C₄×Q₁₆×C₄₇	C₄	1	4	yes	complemented
16	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#103×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄×Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
24	#13093958	—	2	2	no	—
25	#13093958	C₂×C₄	1	4	yes	—
26	C₄₇⋊#95	—	4	1	no	—
27	C₂²×Q₈×C₄₇	—	2	1	no	—
28	C₂×Q₁₆×C₄₇	—	2	1	no	—
29	C₂×Q₁₆×C₄₇	—	2	2	no	—
30	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	—	2	4	no	—
35	C₄×Q₈×C₄₇	—	4	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
38	C₂×C₄²×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	1	no	—
41	#107×C₄₇	C₂×C₄	1	2	yes	—
42	#103×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×Q₈×C₄₇	—	2	2	no	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	—	4	1	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	—	4	1	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	4	2	no	—
55	Q₁₆×C₄₇	—	2	2	no	—
56	Q₁₆×C₄₇	—	2	4	no	—
57	Q₁₆×C₄₇	C₄²	1	4	yes	complemented
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	#41	1	2	yes	—
64	C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	—	4	4	no	—
68	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₄₇	SD₁₆	1	2	yes	—
70	C₄⋊Dic₄₇	—	2	4	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄×Dic₄₇	—	8	2	no	—
73	C₄₇⋊C₁₆	—	2	4	no	—
74	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₄₇	—	2	4	no	—
76	Q₈×C₄₇	—	4	2	no	—
77	Q₈×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	—	8	1	no	—
85	C₂×C₄×C₄₇	—	8	1	no	—
86	C₂×C₄×C₄₇	#102	1	2	yes	—
87	C₂×C₄×C₄₇	#102	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#135	1	2	yes	—
92	C₂×Dic₄₇	—	4	4	no	—
93	C₂×Dic₄₇	—	8	2	no	—
94	C₂×Dic₄₇	—	8	2	no	—
95	C₈×C₄₇	—	2	2	no	—
96	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₄₇	#117	1	1	yes	characteristic
98	C₈×C₄₇	C₄×D₄	1	2	yes	—
99	#12638	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
101	C₂²×C₄₇	C₄×D₈	1	2	yes	—
102	C₂²×C₄₇	#509	1	1	yes	characteristic
103	C₂²×C₄₇	#510	1	1	yes	characteristic
104	C₂²×C₄₇	#510	1	2	yes	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	4	4	no	—
108	C₄×C₄₇	—	8	2	no	—
109	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₄₇	#416	1	1	yes	characteristic
111	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
112	Dic₄₇	—	8	4	no	—
113	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
114	C₂×#358	—	47	2	no	—
115	C₂×#346	—	47	1	no	—
116	C₄×#107	—	47	1	no	—
117	#1380	—	47	2	no	—
118	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
119	C₂×C₄₇	#2139	1	1	yes	characteristic
120	C₂×C₄₇	#2141	1	1	yes	characteristic
121	C₂×C₄₇	#2175	1	2	yes	—
122	C₂×C₄₇	#2176	1	2	yes	—
123	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	2	1	no	—
125	C₂²×C₁₆	—	47	2	no	—
126	C₄×Q₁₆	—	2	2	no	—
127	C₄×Q₁₆	Dic₄₇	1	4	yes	complemented
128	C₂×#107	—	47	2	no	—
129	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
130	C₂×#103	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	94	1	no	—
133	#358	—	47	8	no	—
134	#346	—	94	2	no	—
135	#340	—	94	1	no	—
136	C₄₇	#12638	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	2	1	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₄×D₄₇	1	4	yes	—
141	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
142	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
144	C₄×Q₈	—	2	4	no	—
145	C₄×Q₈	—	4	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	94	2	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	188	1	no	—
151	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂×C₁₆	—	47	4	no	—
153	C₂×C₁₆	—	94	2	no	—
154	C₂×C₁₆	—	94	2	no	—
155	#107	—	2	1	no	—
156	#107	—	47	4	no	—
157	#107	—	94	2	no	—
158	#107	C₂×Dic₄₇	1	2	yes	—
159	#103	—	2	4	no	—
160	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
161	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
162	#95	—	188	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	188	2	no	—
170	C₂²×C₄	#10625334	1	1	yes	characteristic
171	C₂²×C₄	#10625334	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
173	Q₁₆	—	2	2	no	—
174	Q₁₆	—	2	4	no	—
175	Q₁₆	C₄×Dic₄₇	1	4	yes	complemented
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
179	C₂×C₈	#10625350	1	2	yes	—
180	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	94	4	no	—
186	C₄⋊C₄	—	188	2	no	—
187	C₄²	—	4	4	no	—
188	C₄²	—	376	2	no	—
189	C₄²	#10625335	1	2	yes	—
190	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
191	C₁₆	—	94	4	no	—
192	C₂³	C₂×#10625334	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	376	2	no	—
205	C₂×C₄	—	376	2	no	—
206	C₂×C₄	—	188	4	no	—
207	C₂×C₄	#13093945	1	2	yes	—
208	C₂×C₄	#13093945	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
211	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
212	C₂×C₄	#13094044	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
215	C₈	#13093983	1	1	yes	characteristic
216	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
217	C₂²	C₂×#13093945	1	1	yes	characteristic
218	C₂²	C₄×#10625334	1	2	yes	—
219	C₂²	#15689558	1	1	yes	characteristic
220	C₂²	#15689563	1	1	yes	characteristic
221	C₂²	#15689564	1	2	yes	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	—	376	4	no	—
227	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₄	#15689119	1	1	yes	characteristic
229	C₄	C₄×#10625335	1	2	yes	—
230	C₂	C₄×#13093945	1	1	yes	characteristic
231	C₂	#20245100	1	1	yes	characteristic
232	C₂	#20245108	1	1	yes	characteristic
233	C₂	#20245250	1	2	yes	—
234	C₂	#20245257	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#12638
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. #15688927	C₄	semidirect
15. C₄×Q₁₆×C₄₇	C₄	semidirect
57. Q₁₆×C₄₇	C₄²	semidirect
127. C₄×Q₁₆	Dic₄₇	semidirect
136. C₄₇	#12638	semidirect
175. Q₁₆	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×#15688927	2
C₂×#15688899	2
C₄×#13093958	2
#20241479	2
C₂×C₄×Q₁₆×C₄₇	2
#12638	47