Subgroups — #26115209

Order 12032.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	#20241782	C₂	1	2	yes	maximal
4	#20241711	C₂	1	2	yes	maximal
5	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
8	C₄×Dic₁₈₈	—	2	1	no	—
9	C₄×Dic₁₈₈	—	2	1	no	—
10	C₄×Dic₁₈₈	—	2	1	no	—
11	C₄×Dic₁₈₈	—	2	1	no	—
12	#15689176	C₂²	1	2	yes	—
13	C₂×#13093958	C₄	1	1	yes	characteristic
14	C₂×#13093949	C₄	1	2	yes	—
15	#15688876	—	2	1	no	—
16	#15688874	—	2	1	no	—
17	#15688853	—	2	4	no	—
18	#15688848	—	2	4	no	—
19	C₄₇×#431	C₂²	1	1	yes	characteristic
20	C₄₇×#431	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#322	—	2	1	no	—
23	C₈²×C₄₇	—	2	1	no	—
24	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₈₈	—	2	1	no	—
26	C₂×Dic₁₈₈	—	2	1	no	—
27	C₂×Dic₁₈₈	—	2	2	no	—
28	C₂×Dic₁₈₈	C₈	1	4	yes	complemented
29	C₄×Dic₉₄	—	2	2	no	—
30	C₄×Dic₉₄	—	2	2	no	—
31	C₄×Dic₉₄	—	2	2	no	—
32	C₄×Dic₉₄	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	#13093958	—	2	1	no	—
37	#13093958	—	2	1	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
39	#13093949	—	2	2	no	—
40	#13093949	—	2	2	no	—
41	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
42	C₄×C₄₇⋊C₈	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	1	no	—
47	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
48	C₂²⋊C₈×C₄₇	—	2	1	no	—
49	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
50	C₄×C₈×C₄₇	—	2	1	no	—
51	C₄×C₈×C₄₇	—	2	1	no	—
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	2	no	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂×Dic₉₄	—	2	2	no	—
62	C₂×Dic₉₄	—	2	2	no	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	2	2	no	—
65	C₂×Dic₉₄	—	2	2	no	—
66	C₂×Dic₉₄	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₄₇	—	2	1	no	—
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	#41	1	1	yes	characteristic
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	Dic₄₇⋊C₄	—	4	2	no	—
84	Dic₄₇⋊C₄	—	4	2	no	—
85	C₄×Dic₄₇	—	4	2	no	—
86	C₄×Dic₄₇	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	Dic₁₈₈	—	2	4	no	—
89	Dic₁₈₈	—	2	4	no	—
90	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	4	1	no	—
93	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	2	no	—
107	C₂×Dic₄₇	—	8	2	no	—
108	Dic₉₄	—	2	4	no	—
109	Dic₉₄	—	2	4	no	—
110	Dic₉₄	—	4	2	no	—
111	Dic₉₄	—	4	2	no	—
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	—	2	1	no	—
114	C₈×C₄₇	—	4	2	no	—
115	C₈×C₄₇	—	4	2	no	—
116	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
117	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
118	C₄₇⋊C₈	—	8	2	no	—
119	#9669	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	1	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	#390	1	1	yes	characteristic
123	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#436	1	1	yes	characteristic
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₄₇	#408	1	1	yes	characteristic
130	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
131	C₄×C₄₇	#436	1	1	yes	characteristic
132	C₄×C₄₇	#473	1	2	yes	—
133	Dic₄₇	—	4	4	no	—
134	Dic₄₇	—	8	2	no	—
135	C₂×C₄×Q₁₆	—	47	1	no	—
136	#1564	—	47	1	no	—
137	#1479	—	47	2	no	—
138	#1461	—	47	2	no	—
139	#1450	D₄₇	1	1	yes	characteristic
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	#1954	1	1	yes	characteristic
143	C₂×C₄₇	#1976	1	1	yes	characteristic
144	C₂×C₄₇	#1979	1	1	yes	characteristic
145	C₂²×Q₁₆	—	47	1	no	—
146	C₂×C₄×Q₈	—	47	2	no	—
147	C₄×Q₁₆	—	94	1	no	—
148	C₄×Q₁₆	—	94	1	no	—
149	C₄×Q₁₆	—	94	1	no	—
150	C₄×Q₁₆	—	94	1	no	—
151	#431	—	47	2	no	—
152	#431	C₂×D₄₇	1	1	yes	characteristic
153	#431	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#107	—	47	1	no	—
155	C₂×#103	—	47	2	no	—
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#335	—	94	1	no	—
158	#334	—	94	1	no	—
159	Q₈⋊C₈	—	94	4	no	—
160	#324	—	94	4	no	—
161	#322	—	2	1	no	—
162	C₈²	—	2	1	no	—
163	C₄₇	#9669	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	—	47	2	no	—
165	C₂×Q₁₆	—	47	4	no	—
166	C₂×Q₁₆	—	94	2	no	—
167	C₂×Q₁₆	—	94	1	no	—
168	C₂×Q₁₆	—	94	1	no	—
169	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
171	C₄×Q₈	—	94	2	no	—
172	C₄×Q₈	—	94	2	no	—
173	C₄×Q₈	—	94	2	no	—
174	C₄×Q₈	—	94	2	no	—
175	C₂×C₄⋊C₄	—	47	2	no	—
176	C₂×C₄⋊C₄	—	94	2	no	—
177	C₂×C₄²	—	94	2	no	—
178	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
179	#107	—	94	1	no	—
180	#107	—	94	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	94	2	no	—
183	C₄⋊C₈	D₁₈₈	1	2	yes	—
184	#103	—	94	2	no	—
185	#103	—	94	2	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	94	2	no	—
188	C₂²⋊C₈	D₁₈₈	1	2	yes	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	94	2	no	—
192	C₄×C₈	D₁₈₈	1	1	yes	characteristic
193	C₄×C₈	D₁₈₈	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×Q₈	—	47	4	no	—
197	C₂×Q₈	—	94	2	no	—
198	C₂×Q₈	—	94	2	no	—
199	C₂×Q₈	—	94	2	no	—
200	C₂×Q₈	—	94	2	no	—
201	C₂×Q₈	—	94	2	no	—
202	C₂²×C₄	—	94	2	no	—
203	C₂²×C₄	—	94	2	no	—
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
206	C₂²×C₄	#10625350	1	1	yes	characteristic
207	Q₁₆	—	94	4	no	—
208	Q₁₆	—	94	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	188	2	no	—
214	C₂×C₈	C₈×D₄₇	1	2	yes	—
215	C₂×C₈	#10625324	1	2	yes	—
216	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
217	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₄⋊C₄	—	94	2	no	—
221	C₄⋊C₄	—	94	2	no	—
222	C₄⋊C₄	—	188	2	no	—
223	C₄⋊C₄	—	188	2	no	—
224	C₄²	—	188	2	no	—
225	C₄²	—	188	2	no	—
226	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
227	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
228	C₄²	#10625350	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
231	Q₈	—	94	4	no	—
232	Q₈	—	94	4	no	—
233	Q₈	—	188	2	no	—
234	Q₈	—	188	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	94	4	no	—
238	C₂×C₄	—	376	2	no	—
239	C₂×C₄	—	188	2	no	—
240	C₂×C₄	—	188	2	no	—
241	C₂×C₄	—	188	2	no	—
242	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
244	C₂×C₄	#13093998	1	1	yes	characteristic
245	C₂×C₄	#13093998	1	1	yes	characteristic
246	C₂×C₄	#13094012	1	1	yes	characteristic
247	C₂×C₄	#13094021	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
250	C₂×C₄	#13094030	1	1	yes	characteristic
251	C₂×C₄	#13094030	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	376	2	no	—
257	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
258	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#15689025	1	1	yes	characteristic
262	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
263	C₂²	#15689202	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	376	2	no	—
268	C₄	—	188	4	no	—
269	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
270	C₄	#15689089	1	1	yes	characteristic
271	C₄	C₈×D₁₈₈	1	1	yes	characteristic
272	C₄	#15689202	1	1	yes	characteristic
273	C₄	#15689398	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#20244182	1	1	yes	characteristic
277	C₂	#20244286	1	1	yes	characteristic
278	C₂	#20244303	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#9669
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
28. C₂×Dic₁₈₈	C₈	semidirect
163. C₄₇	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
#20242136	2
#20241782	2
#20241711	2
C₄₇×#1450	2
#9669	47