Subgroups — #26109076

Order 12032.

516 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	#20243540	C₂	1	2	yes	maximal
2	C₂×#15688872	C₂	1	2	yes	maximal
3	#20241271	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	#15688872	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
11	C₄₇×#391	C₄	1	2	yes	—
12	C₄₇×#389	—	2	2	no	—
13	C₄₇×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
21	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₄⋊Q₈×C₄₇	—	2	2	no	—
23	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₄₇	Q₈	1	2	yes	—
25	C₄×Q₈×C₄₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	#95×C₄₇	—	2	4	no	—
37	C₂×Q₈×C₄₇	—	2	2	no	—
38	C₂×Q₈×C₄₇	—	4	2	no	—
39	C₂×Q₈×C₄₇	—	4	2	no	—
40	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₄⋊C₄×C₄₇	—	4	4	no	—
51	C₄⋊C₄×C₄₇	—	4	2	no	—
52	C₄⋊C₄×C₄₇	—	4	2	no	—
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄²×C₄₇	—	2	2	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	2	no	—
57	C₄²×C₄₇	—	4	1	no	—
58	C₄²×C₄₇	—	4	4	no	—
59	C₄²×C₄₇	SD₁₆	1	4	yes	—
60	C₄²×C₄₇	Q₁₆	1	4	yes	—
61	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₄₇	#41	1	2	yes	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂×C₄₇⋊C₈	—	4	4	no	—
66	C₂×C₄₇⋊C₈	—	4	4	no	—
67	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₄₇	—	4	4	no	—
69	Q₈×C₄₇	—	4	4	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	#100	1	2	yes	—
82	C₂×C₄×C₄₇	#103	1	4	yes	—
83	C₂×C₄×C₄₇	#103	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#121	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₄₇	#137	1	2	yes	—
92	C₄₇⋊C₈	—	8	4	no	—
93	#8587	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	#380	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
96	C₂²×C₄₇	#413	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#103	1	2	yes	—
98	C₂²×C₄₇	#417	1	2	yes	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	4	no	—
102	C₄×C₄₇	—	4	4	no	—
103	C₄×C₄₇	—	4	1	no	—
104	C₄×C₄₇	—	8	2	no	—
105	C₄×C₄₇	#333	1	4	yes	complemented
106	C₄×C₄₇	#460	1	2	yes	—
107	C₄×C₄₇	#462	1	2	yes	—
108	C₄×C₄₇	#475	1	2	yes	—
109	C₄×C₄₇	#477	1	2	yes	—
110	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
111	#1842	—	47	2	no	—
112	C₂×#333	—	47	2	no	—
113	#1296	—	47	2	no	—
114	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
115	C₂×C₄₇	#1550	1	2	yes	—
116	C₂×C₄₇	#1560	1	1	yes	characteristic
117	C₂×C₄₇	#1800	1	1	yes	characteristic
118	C₂×C₄₇	#1803	1	2	yes	—
119	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
120	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
121	C₂×C₄⋊C₈	—	47	4	no	—
122	C₂×C₄⋊C₈	—	94	2	no	—
123	C₂×C₄⋊C₈	—	94	2	no	—
124	#391	Dic₄₇	1	2	yes	—
125	#389	—	2	2	no	—
126	#384	C₂×D₄₇	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄³	C₂×D₄₇	1	1	yes	characteristic
129	#333	—	94	4	no	—
130	C₄₇	#8587	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
132	C₂²×C₈	—	94	4	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	Dic₉₄	1	2	yes	—
135	C₄⋊Q₈	D₁₈₈	1	2	yes	—
136	C₄×Q₈	—	4	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
145	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊C₈	—	94	4	no	—
148	C₄⋊C₈	—	94	4	no	—
149	C₄⋊C₈	—	188	4	no	—
150	C₄⋊C₈	—	188	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₈	—	188	4	no	—
164	C₂×C₈	—	188	4	no	—
165	C₂×C₈	—	188	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	#10625336	1	4	yes	—
178	C₄²	#10625337	1	4	yes	—
179	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
180	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
181	C₄²	#10625350	1	2	yes	—
182	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#13093941	1	2	yes	—
197	C₂×C₄	#13093948	1	4	yes	—
198	C₂×C₄	#13093948	1	4	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
200	C₂×C₄	#13093983	1	1	yes	characteristic
201	C₂×C₄	#13093998	1	2	yes	—
202	C₂×C₄	#13094003	1	2	yes	—
203	C₂×C₄	C₂×#10625336	1	2	yes	—
204	C₂×C₄	C₂×#10625337	1	2	yes	—
205	C₂×C₄	#13094049	1	2	yes	—
206	C₂×C₄	#13094054	1	2	yes	—
207	C₈	—	376	4	no	—
208	C₂²	#15688983	1	1	yes	characteristic
209	C₂²	C₂×#13093941	1	1	yes	characteristic
210	C₂²	#15689107	1	1	yes	characteristic
211	C₂²	C₂×#13093948	1	2	yes	—
212	C₂²	#15689124	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	#15688872	1	4	yes	complemented
220	C₄	#15689323	1	2	yes	—
221	C₄	#15689333	1	2	yes	—
222	C₄	#15689408	1	2	yes	—
223	C₄	#15689418	1	2	yes	—
224	C₂	C₂×#15688872	1	1	yes	characteristic
225	C₂	#20242078	1	2	yes	—
226	C₂	#20242123	1	1	yes	characteristic
227	C₂	#20243380	1	1	yes	characteristic
228	C₂	#20243393	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#8587
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
105. C₄×C₄₇	#333	semidirect
130. C₄₇	#8587	semidirect
219. C₄	#15688872	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₂×#15688872	2
#20241271	2
C₄×C₄⋊Q₈×C₄₇	2
#8587	47