Subgroups — #26109364

Order 12032.

434 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20243390	C₂	1	2	yes	maximal
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	#20241271	C₂	1	2	yes	maximal
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	#15689253	—	2	2	no	—
8	#15689253	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₇⋊C₈	—	2	1	no	—
12	C₂×#13093947	—	2	2	no	—
13	C₂×#13093934	—	2	1	no	—
14	#15688885	—	2	2	no	—
15	#15688876	—	2	2	no	—
16	C₂×#106×C₄₇	—	2	1	no	—
17	C₂×#106×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄×Dic₉₄	—	2	1	no	—
24	C₄×Dic₉₄	—	2	1	no	—
25	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
26	C₄×Dic₉₄	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
28	C₂×Dic₄₇⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₄₇	—	2	1	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₄₇⋊C₈	—	2	4	no	—
35	#13093947	—	4	4	no	—
36	#13093934	—	2	4	no	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#106×C₄₇	—	4	2	no	—
44	#106×C₄₇	—	4	2	no	—
45	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	1	no	—
52	C₂²×Dic₄₇	—	2	1	no	—
53	C₂×Dic₉₄	—	2	2	no	—
54	C₂×Dic₉₄	—	2	2	no	—
55	C₂×Dic₉₄	—	2	1	no	—
56	C₂×Dic₉₄	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
62	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₄⋊C₄×C₄₇	—	4	2	no	—
65	C₄⋊C₄×C₄₇	—	4	2	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	8	2	no	—
68	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₄₇	#41	1	1	yes	characteristic
72	C₄⋊Dic₄₇	—	2	1	no	—
73	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
74	Dic₄₇⋊C₄	—	2	2	no	—
75	Dic₄₇⋊C₄	—	2	2	no	—
76	C₄×Dic₄₇	—	2	2	no	—
77	C₄×Dic₄₇	—	2	2	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	4	1	no	—
80	C₂×C₄₇⋊C₈	—	4	1	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂³×C₄₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	8	2	no	—
87	C₂×C₄×C₄₇	—	8	2	no	—
88	C₂×C₄×C₄₇	—	8	2	no	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₄₇	#137	1	2	yes	—
100	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	4	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	Dic₉₄	—	2	4	no	—
106	Dic₉₄	—	4	1	no	—
107	Dic₉₄	—	4	1	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	#103	1	4	yes	complemented
110	C₄₇⋊C₈	—	4	2	no	—
111	C₄₇⋊C₈	—	8	2	no	—
112	#8651	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	#380	1	1	yes	characteristic
114	C₂²×C₄₇	#460	1	1	yes	characteristic
115	C₂²×C₄₇	#462	1	1	yes	characteristic
116	C₂²×C₄₇	#468	1	2	yes	—
117	C₂²×C₄₇	#471	1	2	yes	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	16	2	no	—
121	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
122	C₄×C₄₇	#417	1	1	yes	characteristic
123	C₄×C₄₇	#428	1	2	yes	—
124	C₄×C₄₇	#469	1	1	yes	characteristic
125	C₄×C₄₇	#470	1	1	yes	characteristic
126	C₄×C₄₇	#475	1	1	yes	characteristic
127	C₄×C₄₇	#477	1	1	yes	characteristic
128	Dic₄₇	—	4	2	no	—
129	Dic₄₇	—	4	2	no	—
130	C₂×#446	—	47	1	no	—
131	#1844	D₄₇	1	1	yes	characteristic
132	#1803	—	47	2	no	—
133	C₂×#335	—	47	1	no	—
134	#1296	—	47	2	no	—
135	C₂×C₄₇	#1686	1	2	yes	—
136	C₂×C₄₇	#1692	1	2	yes	—
137	C₂×C₄₇	#1803	1	1	yes	characteristic
138	C₂×C₄₇	#1809	1	1	yes	characteristic
139	C₂×C₄₇	#1815	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	47	1	no	—
141	#446	—	47	4	no	—
142	#446	—	94	2	no	—
143	C₂×#106	—	2	1	no	—
144	C₂×#106	—	2	1	no	—
145	C₂×C₄⋊C₈	—	47	1	no	—
146	C₂×C₄⋊C₈	—	47	2	no	—
147	C₂×C₄⋊C₈	—	94	1	no	—
148	C₂×#103	—	94	2	no	—
149	C₂×#97	—	94	1	no	—
150	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
151	#384	C₂×D₄₇	1	2	yes	—
152	#340	—	94	2	no	—
153	#335	—	94	2	no	—
154	C₄₇	#8651	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	47	1	no	—
156	C₂²×C₈	—	94	1	no	—
157	C₂²×C₈	—	94	2	no	—
158	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₄×Q₈	—	47	2	no	—
161	C₄×Q₈	—	47	2	no	—
162	C₄×Q₈	—	94	1	no	—
163	C₄×Q₈	—	94	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	47	1	no	—
168	C₂×C₄⋊C₄	—	94	1	no	—
169	C₂×C₄²	—	94	1	no	—
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	#106	—	4	2	no	—
172	#106	—	4	2	no	—
173	C₄⋊C₈	—	47	2	no	—
174	C₄⋊C₈	—	94	4	no	—
175	C₄⋊C₈	—	94	1	no	—
176	C₄⋊C₈	—	94	2	no	—
177	C₄⋊C₈	—	94	2	no	—
178	#103	—	188	4	no	—
179	#97	—	94	4	no	—
180	C₄×C₈	C₄×D₄₇	1	2	yes	—
181	C₄×C₈	D₁₈₈	1	2	yes	—
182	C₂×Q₈	—	47	2	no	—
183	C₂×Q₈	—	94	1	no	—
184	C₂×Q₈	—	94	2	no	—
185	C₂×Q₈	—	94	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	94	1	no	—
188	C₂²×C₄	—	94	1	no	—
189	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	#10625352	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	94	2	no	—
195	C₂×C₈	—	188	2	no	—
196	C₂×C₈	—	188	2	no	—
197	C₂×C₈	—	188	2	no	—
198	C₂×C₈	—	188	1	no	—
199	C₂×C₈	—	188	1	no	—
200	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
201	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
202	C₂×C₈	#10625335	1	2	yes	—
203	C₂×C₈	#10625337	1	2	yes	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	—	47	2	no	—
210	C₄⋊C₄	—	94	2	no	—
211	C₄⋊C₄	—	94	1	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄²	—	94	2	no	—
214	C₄²	—	94	2	no	—
215	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
217	C₄²	Q₈×D₄₇	1	1	yes	characteristic
218	C₄²	#10625354	1	1	yes	characteristic
219	C₂³	#13093995	1	1	yes	characteristic
220	Q₈	—	94	4	no	—
221	Q₈	—	188	1	no	—
222	Q₈	—	188	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	2	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	#13093982	1	1	yes	characteristic
234	C₂×C₄	#13093999	1	1	yes	characteristic
235	C₂×C₄	#13094000	1	1	yes	characteristic
236	C₂×C₄	#13094004	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
239	C₂×C₄	#13094050	1	1	yes	characteristic
240	C₂×C₄	#13094051	1	2	yes	—
241	C₂×C₄	#13094053	1	2	yes	—
242	C₂×C₄	#13094055	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	376	2	no	—
245	C₈	—	188	2	no	—
246	C₈	#13093947	1	4	yes	complemented
247	C₂²	#15688980	1	1	yes	characteristic
248	C₂²	#15689320	1	1	yes	characteristic
249	C₂²	#15689330	1	1	yes	characteristic
250	C₂²	#15689366	1	2	yes	—
251	C₂²	#15689383	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	16	2	no	—
255	C₄	—	188	2	no	—
256	C₄	—	188	2	no	—
257	C₄	C₂×#13093947	1	1	yes	characteristic
258	C₄	#15689122	1	1	yes	characteristic
259	C₄	#15689162	1	2	yes	—
260	C₄	#15689377	1	1	yes	characteristic
261	C₄	#15689382	1	1	yes	characteristic
262	C₄	#15689405	1	1	yes	characteristic
263	C₄	#15689415	1	1	yes	characteristic
264	C₂	#20242857	1	2	yes	—
265	C₂	#20242891	1	2	yes	—
266	C₂	#20243390	1	1	yes	characteristic
267	C₂	#20243418	1	1	yes	characteristic
268	C₂	#20243448	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#8651
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
8. #15689253	C₄	semidirect
109. C₈×C₄₇	#103	semidirect
154. C₄₇	#8651	semidirect
246. C₈	#13093947	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20243390	2
C₂×#15688876	2
#20241271	2
C₄₇×#1844	2
#8651	47