Subgroups — #26114860

Order 12032.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688873	C₂	1	2	yes	maximal
4	#20241222	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688878	C₄	1	2	yes	—
13	#15688873	—	2	4	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	—	2	1	no	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₄⋊(C₄₇⋊C₈)	C₈	1	4	yes	complemented
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	1	no	—
31	C₂²×C₈×C₄₇	—	2	1	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	1	no	—
40	C₄⋊C₈×C₄₇	—	2	1	no	—
41	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
42	C₄⋊C₈×C₄₇	Q₈	1	2	yes	—
43	C₄×C₈×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₄⋊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×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	4	1	no	—
54	C₂×C₈×C₄₇	—	4	1	no	—
55	C₂×C₈×C₄₇	—	4	1	no	—
56	C₂×C₈×C₄₇	—	4	1	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	1	no	—
59	C₄²×C₄₇	—	2	1	no	—
60	C₄²×C₄₇	—	2	1	no	—
61	C₄²×C₄₇	—	2	1	no	—
62	C₄²×C₄₇	—	4	1	no	—
63	C₄²×C₄₇	C₂×C₈	1	2	yes	—
64	C₄²×C₄₇	M₁₆	1	2	yes	—
65	C₄²×C₄₇	SD₁₆	1	2	yes	—
66	C₄²×C₄₇	SD₁₆	1	2	yes	—
67	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₄₇	#41	1	1	yes	characteristic
70	C₄²×C₄₇	#41	1	1	yes	characteristic
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₂×C₄₇⋊C₈	—	2	2	no	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	—	2	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
80	C₂³×C₄₇	C₂×C₄⋊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	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₄₇	#106	1	2	yes	—
93	C₂×C₄×C₄₇	#106	1	2	yes	—
94	C₂×C₄×C₄₇	#108	1	2	yes	—
95	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
109	C₈×C₄₇	—	4	2	no	—
110	C₈×C₄₇	—	4	2	no	—
111	C₄₇⋊C₈	—	2	4	no	—
112	C₄₇⋊C₈	—	4	2	no	—
113	C₄₇⋊C₈	—	8	2	no	—
114	#9602	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	#389	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₄₇	#424	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
119	C₂²×C₄₇	#428	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
121	C₂²×C₄₇	#430	1	1	yes	characteristic
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	4	1	no	—
127	C₄×C₄₇	#334	1	4	yes	complemented
128	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
129	C₄×C₄₇	#436	1	1	yes	characteristic
130	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
131	C₄×C₄₇	#446	1	1	yes	characteristic
132	C₄×C₄₇	#460	1	1	yes	characteristic
133	C₄×C₄₇	#461	1	1	yes	characteristic
134	C₄×C₄₇	#475	1	1	yes	characteristic
135	C₄×C₄₇	#476	1	1	yes	characteristic
136	#1842	—	47	1	no	—
137	C₄×C₄⋊C₈	—	47	1	no	—
138	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
139	C₂×#334	—	47	2	no	—
140	#1277	—	47	2	no	—
141	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
142	C₂×C₄₇	#1565	1	1	yes	characteristic
143	C₂×C₄₇	#1570	1	1	yes	characteristic
144	C₂×C₄₇	#1574	1	1	yes	characteristic
145	C₂×C₄₇	#1939	1	1	yes	characteristic
146	C₂×C₄₇	#1948	1	1	yes	characteristic
147	C₂×C₄₇	#1950	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	47	1	no	—
149	C₂×C₄⋊C₈	—	47	1	no	—
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×C₄⋊C₈	—	94	1	no	—
152	C₂×C₄⋊C₈	—	94	1	no	—
153	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₄×C₈	—	2	1	no	—
156	C₂×C₄×C₈	—	47	2	no	—
157	C₄³	C₂×D₄₇	1	1	yes	characteristic
158	#336	—	2	1	no	—
159	#336	—	47	2	no	—
160	#334	—	94	4	no	—
161	C₄₇	#9602	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	47	2	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
168	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	47	4	no	—
177	C₄⋊C₈	—	94	2	no	—
178	C₄⋊C₈	—	94	2	no	—
179	C₄⋊C₈	—	94	2	no	—
180	C₄⋊C₈	—	188	2	no	—
181	C₄⋊C₈	—	188	2	no	—
182	C₄⋊C₈	Dic₉₄	1	2	yes	—
183	C₄⋊C₈	D₁₈₈	1	2	yes	—
184	C₄×C₈	—	4	2	no	—
185	C₄×C₈	—	94	2	no	—
186	C₄×C₈	—	94	2	no	—
187	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	94	2	no	—
202	C₂×C₈	—	94	2	no	—
203	C₂×C₈	—	94	2	no	—
204	C₂×C₈	—	94	2	no	—
205	C₂×C₈	—	94	2	no	—
206	C₂×C₈	—	188	2	no	—
207	C₂×C₈	—	188	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₈×D₄₇	1	2	yes	—
216	C₄²	#10625324	1	2	yes	—
217	C₄²	#10625335	1	2	yes	—
218	C₄²	#10625336	1	2	yes	—
219	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
220	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
221	C₄²	#10625350	1	1	yes	characteristic
222	C₄²	#10625350	1	1	yes	characteristic
223	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
234	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
235	C₂×C₄	#13093955	1	2	yes	—
236	C₂×C₄	#13093955	1	2	yes	—
237	C₂×C₄	#13093959	1	2	yes	—
238	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
239	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
240	C₂×C₄	#13093998	1	1	yes	characteristic
241	C₂×C₄	#13094003	1	1	yes	characteristic
242	C₂×C₄	#13094016	1	1	yes	characteristic
243	C₂×C₄	#13094024	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
246	C₂×C₄	#13094030	1	1	yes	characteristic
247	C₂×C₄	#13094030	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
250	C₂×C₄	#13094049	1	1	yes	characteristic
251	C₂×C₄	#13094054	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	94	4	no	—
255	C₈	—	376	2	no	—
256	C₈	—	188	2	no	—
257	C₂²	#15689021	1	1	yes	characteristic
258	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
259	C₂²	#15689149	1	1	yes	characteristic
260	C₂²	C₂×#13093955	1	1	yes	characteristic
261	C₂²	#15689166	1	1	yes	characteristic
262	C₂²	C₂×#13093959	1	1	yes	characteristic
263	C₂²	#15689171	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#15688873	1	4	yes	complemented
270	C₄	C₈×D₁₈₈	1	1	yes	characteristic
271	C₄	#15689202	1	1	yes	characteristic
272	C₄	C₈×Dic₉₄	1	1	yes	characteristic
273	C₄	#15689253	1	1	yes	characteristic
274	C₄	#15689323	1	1	yes	characteristic
275	C₄	#15689328	1	1	yes	characteristic
276	C₄	#15689408	1	1	yes	characteristic
277	C₄	#15689413	1	1	yes	characteristic
278	C₂	C₂×#15688873	1	1	yes	characteristic
279	C₂	#20242141	1	1	yes	characteristic
280	C₂	#20242158	1	1	yes	characteristic
281	C₂	#20242176	1	1	yes	characteristic
282	C₂	#20244106	1	1	yes	characteristic
283	C₂	#20244151	1	1	yes	characteristic
284	C₂	#20244163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9602
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₄⋊(C₄₇⋊C₈)	C₈	semidirect
127. C₄×C₄₇	#334	semidirect
161. C₄₇	#9602	semidirect
269. C₄	#15688873	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688873	2
#20241222	2
C₄×C₄⋊C₈×C₄₇	2
#9602	47