Subgroups — #26104740

Order 12032.

448 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	#20251766	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	4	yes	maximal
4	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic
6	C₂×#13094024	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15689029	—	2	1	no	—
9	#15689029	—	2	1	no	—
10	#15689029	C₄	1	2	yes	—
11	#15689029	C₂²	1	2	yes	—
12	#15688989	—	2	1	no	—
13	#15688989	—	2	1	no	—
14	#15688878	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×#97×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13094024	—	4	2	no	—
24	#13094024	—	4	2	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	C₄×C₄₇⋊C₈	—	2	4	no	—
30	C₄₇⋊#95	—	2	2	no	—
31	C₄₇⋊#95	—	2	2	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	4	no	—
43	#97×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂²×Dic₄₇	—	2	2	no	—
53	C₂×Dic₉₄	—	2	4	no	—
54	C₂×Dic₉₄	—	4	2	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×Dic₉₄	—	4	2	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₂×Dic₉₄	—	4	2	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	—	4	1	no	—
68	C₄²×C₄₇	SD₁₆	1	2	yes	—
69	C₄²×C₄₇	Q₁₆	1	2	yes	—
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	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	1	no	—
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	#103	1	2	yes	—
93	C₂×C₄×C₄₇	#103	1	2	yes	—
94	C₂×C₄×C₄₇	#104	1	2	yes	—
95	C₂×C₄×C₄₇	#104	1	2	yes	—
96	C₂×C₄×C₄₇	#131	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#137	1	2	yes	—
100	C₂×Dic₄₇	—	4	2	no	—
101	C₂×Dic₄₇	—	4	2	no	—
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	Dic₉₄	—	8	4	no	—
107	Dic₉₄	—	8	4	no	—
108	C₈×C₄₇	—	4	4	no	—
109	C₄₇⋊C₈	—	4	4	no	—
110	#7682	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	#408	1	1	yes	characteristic
112	C₂²×C₄₇	#410	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
115	C₂²×C₄₇	#419	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#104	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	—	2	1	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	4	1	no	—
125	C₄×C₄₇	#329	1	4	yes	—
126	Dic₄₇	—	8	2	no	—
127	Dic₄₇	—	8	2	no	—
128	#2862	—	47	1	no	—
129	#1767	D₄₇	1	1	yes	characteristic
130	#1744	—	47	1	no	—
131	#1272	—	47	4	no	—
132	C₂×C₄₇	#1469	1	1	yes	characteristic
133	C₂×C₄₇	#1485	1	1	yes	characteristic
134	C₂×C₄₇	#1493	1	1	yes	characteristic
135	C₂×C₄₇	#1503	1	1	yes	characteristic
136	C₂×C₄₇	#1506	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#329	1	1	yes	characteristic
138	C₂×C₄₇	#1548	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	—	47	1	no	—
140	C₂×C₄⋊Q₈	—	47	1	no	—
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×#97	—	2	1	no	—
143	C₂×C₄×C₈	—	94	1	no	—
144	#391	—	47	2	no	—
145	#391	—	47	2	no	—
146	#391	—	94	1	no	—
147	#391	—	94	1	no	—
148	#382	—	94	1	no	—
149	#382	—	94	1	no	—
150	C₄³	C₂×D₄₇	1	1	yes	characteristic
151	#336	—	47	2	no	—
152	#336	—	94	1	no	—
153	#336	C₂×D₄₇	1	2	yes	—
154	C₄₇	#7682	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	47	2	no	—
156	C₂²×Q₈	—	94	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	94	2	no	—
159	C₄⋊Q₈	—	188	2	no	—
160	C₄⋊Q₈	—	188	2	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	94	1	no	—
163	C₂×C₄⋊C₄	—	94	1	no	—
164	C₂×C₄⋊C₄	—	94	1	no	—
165	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
167	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
168	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
172	C₄⋊C₈	—	2	4	no	—
173	#97	—	4	2	no	—
174	C₄×C₈	—	94	4	no	—
175	#95	—	94	2	no	—
176	#95	—	94	2	no	—
177	#95	—	94	2	no	—
178	#95	—	94	2	no	—
179	C₂×Q₈	—	94	4	no	—
180	C₂×Q₈	—	188	2	no	—
181	C₂×Q₈	—	188	2	no	—
182	C₂×Q₈	—	188	2	no	—
183	C₂×Q₈	—	188	2	no	—
184	C₂×Q₈	—	188	2	no	—
185	C₂²×C₄	—	94	2	no	—
186	C₂²×C₄	—	94	2	no	—
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×D₄₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	94	4	no	—
198	C₂×C₈	—	188	2	no	—
199	C₂×C₈	—	188	2	no	—
200	C₄⋊C₄	—	188	2	no	—
201	C₄⋊C₄	—	188	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₄⋊C₄	—	188	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	#10625335	1	2	yes	—
211	C₄²	#10625337	1	2	yes	—
212	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	Q₈	—	376	4	no	—
214	Q₈	—	376	4	no	—
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	—	2	1	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	—	188	2	no	—
234	C₂×C₄	#13093940	1	2	yes	—
235	C₂×C₄	#13093947	1	2	yes	—
236	C₂×C₄	#13093947	1	2	yes	—
237	C₂×C₄	#13093950	1	2	yes	—
238	C₂×C₄	#13093952	1	2	yes	—
239	C₂×C₄	#13094032	1	2	yes	—
240	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
241	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
242	C₂×C₄	#13094057	1	2	yes	—
243	C₈	—	4	4	no	—
244	C₈	—	188	4	no	—
245	C₂²	#15689091	1	1	yes	characteristic
246	C₂²	#15689096	1	1	yes	characteristic
247	C₂²	C₂×#13093940	1	1	yes	characteristic
248	C₂²	C₂×#13093947	1	1	yes	characteristic
249	C₂²	#15689133	1	1	yes	characteristic
250	C₂²	C₂×#13093950	1	1	yes	characteristic
251	C₂²	C₂×#13093952	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	376	2	no	—
260	C₄	—	376	2	no	—
261	C₄	#15688862	1	4	yes	—
262	C₂	#20241741	1	1	yes	characteristic
263	C₂	#20241810	1	1	yes	characteristic
264	C₂	#20241842	1	1	yes	characteristic
265	C₂	#20241886	1	1	yes	characteristic
266	C₂	#20241901	1	1	yes	characteristic
267	C₂	C₂×#15688862	1	1	yes	characteristic
268	C₂	#20242070	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	#7682
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
154. C₄₇	#7682	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20243134	2
#20241213	2
C₄₇×#1767	2
#7682	47