Subgroups — #26116417

Order 12032.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251751	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	2	yes	maximal
4	#20241272	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15689021	—	2	1	no	—
9	#15689021	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	—	2	1	no	—
12	#15688993	C₂²	1	1	yes	characteristic
13	#15688993	C₂²	1	1	yes	characteristic
14	#15688989	C₄	1	2	yes	—
15	#15688878	—	2	1	no	—
16	#15688878	—	2	1	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	—	2	1	no	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#336	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094016	—	4	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	Q₈	1	4	yes	—
44	C₄×C₈×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	2	no	—
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	4	2	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	C₄²×C₄₇	—	2	2	no	—
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	1	no	—
63	C₄²×C₄₇	—	2	1	no	—
64	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
65	C₄²×C₄₇	#41	1	2	yes	—
66	C₄⋊Dic₄₇	—	4	2	no	—
67	C₄⋊Dic₄₇	—	4	2	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	C₄⋊Dic₄₇	—	4	1	no	—
70	C₄⋊Dic₄₇	—	4	1	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	Dic₄₇⋊C₄	—	4	4	no	—
74	Dic₄₇⋊C₄	—	4	4	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	#100	1	2	yes	—
92	C₂×C₄×C₄₇	#101	1	2	yes	—
93	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#135	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×C₄×C₄₇	#135	1	2	yes	—
100	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
102	C₂×Dic₄₇	—	4	2	no	—
103	C₂×Dic₄₇	—	4	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	2	no	—
107	C₂×Dic₄₇	—	4	2	no	—
108	C₈×C₄₇	—	4	4	no	—
109	C₄₇⋊C₈	—	8	2	no	—
110	#9901	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	#391	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
114	C₂²×C₄₇	#416	1	1	yes	characteristic
115	C₂²×C₄₇	#416	1	1	yes	characteristic
116	C₂²×C₄₇	#416	1	1	yes	characteristic
117	C₂²×C₄₇	#417	1	1	yes	characteristic
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	4	1	no	—
122	C₄×C₄₇	—	4	1	no	—
123	C₄×C₄₇	#478	1	2	yes	—
124	C₄×C₄₇	#479	1	2	yes	—
125	C₄×C₄₇	#488	1	2	yes	—
126	C₄×C₄₇	#499	1	2	yes	—
127	Dic₄₇	—	8	2	no	—
128	Dic₄₇	—	8	2	no	—
129	#2859	—	47	1	no	—
130	#1831	—	47	1	no	—
131	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
132	#1296	—	47	2	no	—
133	#1296	—	47	2	no	—
134	C₂×C₄₇	#1543	1	1	yes	characteristic
135	C₂×C₄₇	#1544	1	1	yes	characteristic
136	C₂×C₄₇	#1551	1	1	yes	characteristic
137	C₂×C₄₇	#1552	1	1	yes	characteristic
138	C₂×C₄₇	#1989	1	1	yes	characteristic
139	C₂×C₄₇	#1991	1	1	yes	characteristic
140	C₂×C₄₇	#1997	1	1	yes	characteristic
141	C₂×#125	—	47	2	no	—
142	C₂×C₄⋊C₈	—	47	2	no	—
143	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
144	C₂×C₄×C₈	—	2	1	no	—
145	#389	—	94	1	no	—
146	#389	—	94	1	no	—
147	#389	—	94	1	no	—
148	#389	—	94	1	no	—
149	#384	—	47	1	no	—
150	#384	—	47	1	no	—
151	#382	—	47	2	no	—
152	C₄³	C₂×D₄₇	1	1	yes	characteristic
153	#336	—	2	1	no	—
154	#336	—	94	1	no	—
155	#336	—	94	1	no	—
156	C₄₇	#9901	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	94	2	no	—
159	#125	—	188	4	no	—
160	C₂×C₄⋊C₄	—	47	2	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	94	2	no	—
163	C₂×C₄⋊C₄	—	94	2	no	—
164	C₂×C₄⋊C₄	—	94	1	no	—
165	C₂×C₄⋊C₄	—	94	1	no	—
166	C₂×C₄⋊C₄	—	94	1	no	—
167	C₂×C₄⋊C₄	—	94	1	no	—
168	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
171	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	94	4	no	—
177	C₄⋊C₈	Dic₉₄	1	4	yes	—
178	C₄×C₈	—	4	2	no	—
179	#95	—	94	2	no	—
180	#95	—	94	2	no	—
181	C₂²×C₄	—	94	2	no	—
182	C₂²×C₄	—	94	2	no	—
183	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	188	2	no	—
194	C₂×C₈	—	188	2	no	—
195	C₂×C₈	—	188	2	no	—
196	C₄⋊C₄	—	188	2	no	—
197	C₄⋊C₄	—	188	2	no	—
198	C₄⋊C₄	—	188	4	no	—
199	C₄⋊C₄	—	188	4	no	—
200	C₄⋊C₄	—	188	2	no	—
201	C₄⋊C₄	—	188	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₄⋊C₄	—	188	1	no	—
204	C₄⋊C₄	—	188	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂×Dic₉₄	1	2	yes	—
213	C₄²	#10625350	1	2	yes	—
214	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	188	2	no	—
228	C₂×C₄	—	188	2	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	—	188	2	no	—
232	C₂×C₄	—	188	2	no	—
233	C₂×C₄	#13093940	1	2	yes	—
234	C₂×C₄	#13093942	1	2	yes	—
235	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
236	C₂×C₄	#13094003	1	2	yes	—
237	C₂×C₄	#13094012	1	1	yes	characteristic
238	C₂×C₄	#13094024	1	1	yes	characteristic
239	C₂×C₄	#13094044	1	2	yes	—
240	C₂×C₄	#13094046	1	2	yes	—
241	C₂×C₄	#13094046	1	2	yes	—
242	C₂×C₄	#13094050	1	1	yes	characteristic
243	C₂×C₄	#13094055	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	—	376	2	no	—
246	C₂²	#15689029	1	1	yes	characteristic
247	C₂²	C₂×#13093940	1	1	yes	characteristic
248	C₂²	C₂×#13093942	1	1	yes	characteristic
249	C₂²	#15689117	1	1	yes	characteristic
250	C₂²	#15689120	1	1	yes	characteristic
251	C₂²	#15689120	1	1	yes	characteristic
252	C₂²	#15689122	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	376	2	no	—
259	C₄	—	376	2	no	—
260	C₄	#15689423	1	2	yes	—
261	C₄	#15689430	1	2	yes	—
262	C₄	#15689478	1	2	yes	—
263	C₄	#15689525	1	2	yes	—
264	C₂	#20242048	1	1	yes	characteristic
265	C₂	#20242052	1	1	yes	characteristic
266	C₂	#20242081	1	1	yes	characteristic
267	C₂	#20242086	1	1	yes	characteristic
268	C₂	#20244345	1	1	yes	characteristic
269	C₂	#20244353	1	1	yes	characteristic
270	C₂	#20244377	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9901
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
156. C₄₇	#9901	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
#20243505	2
#20241272	2
C₄×C₄⋊C₈×C₄₇	2
#9901	47