Subgroups — #26115285

Order 12032.

500 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	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688864	C₂	1	4	yes	maximal
4	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	C₂²	1	2	yes	—
6	C₂×#13093935	—	2	1	no	—
7	C₂×#13093935	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15689021	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	—	2	1	no	—
12	#15689021	—	2	1	no	—
13	#15688993	C₄	1	1	yes	characteristic
14	#15688993	C₄	1	1	yes	characteristic
15	#15688989	C₄	1	2	yes	—
16	#15688864	—	2	8	no	—
17	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₂×#97×C₄₇	C₂²	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#336	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094016	—	2	4	no	—
23	#13094016	—	2	4	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	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×Dic₄₇⋊C₄	—	2	2	no	—
32	#13093935	—	2	4	no	—
33	#13093935	—	2	2	no	—
34	#13093935	D₄	1	4	yes	—
35	C₄×C₄₇⋊C₈	—	2	4	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	4	no	—
47	#97×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	2	no	—
56	C₂²×Dic₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₄²×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	4	1	no	—
66	C₄²×C₄₇	C₂×D₄	1	2	yes	—
67	C₄²×C₄₇	#41	1	2	yes	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄⋊Dic₄₇	—	2	4	no	—
70	C₄⋊Dic₄₇	—	4	2	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	Dic₄₇⋊C₄	—	4	4	no	—
75	Dic₄₇⋊C₄	—	4	2	no	—
76	Dic₄₇⋊C₄	—	4	2	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	#101	1	2	yes	—
95	C₂×C₄×C₄₇	#101	1	2	yes	—
96	C₂×C₄×C₄₇	#104	1	2	yes	—
97	C₂×C₄×C₄₇	#104	1	2	yes	—
98	C₂×C₄×C₄₇	#104	1	2	yes	—
99	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	#121	1	2	yes	—
101	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
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	C₂×Dic₄₇	—	4	2	no	—
109	C₈×C₄₇	—	4	4	no	—
110	C₄₇⋊C₈	—	2	8	no	—
111	#9686	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	#390	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
118	C₂²×C₄₇	#421	1	1	yes	characteristic
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	4	1	no	—
126	C₄×C₄₇	#330	1	4	yes	complemented
127	Dic₄₇	—	8	2	no	—
128	Dic₄₇	—	8	2	no	—
129	#2859	—	47	1	no	—
130	#1767	D₄₇	1	1	yes	characteristic
131	C₄×#97	—	47	1	no	—
132	C₂×#330	—	47	4	no	—
133	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
134	C₂×C₄₇	#1526	1	1	yes	characteristic
135	C₂×C₄₇	#1528	1	1	yes	characteristic
136	C₂×C₄₇	#1531	1	1	yes	characteristic
137	C₂×C₄₇	#1962	1	1	yes	characteristic
138	C₂×C₄₇	#1966	1	1	yes	characteristic
139	C₂×C₄₇	#1980	1	1	yes	characteristic
140	C₂×#125	—	47	2	no	—
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×#97	—	47	2	no	—
143	C₂×#97	—	94	1	no	—
144	C₂×#97	C₂×D₄₇	1	2	yes	—
145	C₂×C₄×C₈	—	94	1	no	—
146	#389	—	94	1	no	—
147	#389	—	94	1	no	—
148	#389	—	94	1	no	—
149	#389	—	94	1	no	—
150	#384	—	47	1	no	—
151	#384	—	47	1	no	—
152	#382	—	47	2	no	—
153	C₄³	C₂×D₄₇	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#330	—	94	8	no	—
156	C₄₇	#9686	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	94	2	no	—
159	#125	—	94	4	no	—
160	#125	—	94	4	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	—	94	1	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	2	no	—
168	C₂×C₄⋊C₄	—	94	2	no	—
169	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
172	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	#97	—	2	4	no	—
178	#97	—	47	4	no	—
179	#97	—	94	4	no	—
180	#97	—	94	2	no	—
181	C₄×C₈	—	94	4	no	—
182	#95	—	94	2	no	—
183	#95	—	94	2	no	—
184	C₂²×C₄	—	94	2	no	—
185	C₂²×C₄	—	94	2	no	—
186	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	94	4	no	—
197	C₂×C₈	—	94	4	no	—
198	C₂×C₈	—	94	4	no	—
199	C₄⋊C₄	—	94	4	no	—
200	C₄⋊C₄	—	94	4	no	—
201	C₄⋊C₄	—	188	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₄⋊C₄	—	188	2	no	—
204	C₄⋊C₄	—	188	2	no	—
205	C₄⋊C₄	—	188	4	no	—
206	C₄⋊C₄	—	188	2	no	—
207	C₄⋊C₄	—	188	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	D₄×D₄₇	1	2	yes	—
215	C₄²	#10625354	1	2	yes	—
216	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	—	188	2	no	—
232	C₂×C₄	—	188	2	no	—
233	C₂×C₄	—	188	2	no	—
234	C₂×C₄	—	188	2	no	—
235	C₂×C₄	—	188	2	no	—
236	C₂×C₄	#13093942	1	2	yes	—
237	C₂×C₄	#13093942	1	2	yes	—
238	C₂×C₄	#13093950	1	2	yes	—
239	C₂×C₄	#13093950	1	2	yes	—
240	C₂×C₄	#13093952	1	2	yes	—
241	C₂×C₄	#13093986	1	2	yes	—
242	C₂×C₄	#13093999	1	2	yes	—
243	C₂×C₄	#13094013	1	1	yes	characteristic
244	C₂×C₄	#13094020	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₈	—	94	8	no	—
247	C₂²	#15689024	1	1	yes	characteristic
248	C₂²	C₂×#13093942	1	1	yes	characteristic
249	C₂²	C₂×#13093942	1	1	yes	characteristic
250	C₂²	C₂×#13093950	1	1	yes	characteristic
251	C₂²	C₂×#13093950	1	1	yes	characteristic
252	C₂²	C₂×#13093952	1	1	yes	characteristic
253	C₂²	#15689141	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	376	2	no	—
262	C₄	—	376	2	no	—
263	C₄	#15688864	1	4	yes	complemented
264	C₂	C₂×#15688864	1	1	yes	characteristic
265	C₂	#20241982	1	1	yes	characteristic
266	C₂	#20241991	1	1	yes	characteristic
267	C₂	#20242004	1	1	yes	characteristic
268	C₂	#20244215	1	1	yes	characteristic
269	C₂	#20244231	1	1	yes	characteristic
270	C₂	#20244306	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	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#9686
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
126. C₄×C₄₇	#330	semidirect
156. C₄₇	#9686	semidirect
263. C₄	#15688864	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
C₄×#13093935	2
C₂×#15688864	2
C₄₇×#1767	2
#9686	47