Subgroups — #26107429

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
5	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic
6	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15688878	C₄	1	2	yes	—
15	#15688878	C₄	1	2	yes	—
16	#15688845	—	2	2	no	—
17	#15688844	—	2	2	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#336	C₄	1	2	yes	—
22	C₄₇×#336	C₄	1	2	yes	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₄×C₄₇⋊C₈	—	2	1	no	—
30	C₄×C₄₇⋊C₈	—	2	1	no	—
31	C₄×C₄₇⋊C₈	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	1	no	—
33	C₄×C₄₇⋊C₈	—	2	1	no	—
34	C₄×C₄₇⋊C₈	—	2	1	no	—
35	C₄×C₄₇⋊C₈	—	2	1	no	—
36	C₄×C₄₇⋊C₈	—	2	1	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×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	—	2	1	no	—
48	C₄×C₈×C₄₇	D₄	1	2	yes	—
49	C₄×C₈×C₄₇	Q₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
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	—	2	2	no	—
68	C₄²×C₄₇	M₁₆	1	4	yes	—
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₄₇	#41	1	1	yes	characteristic
72	C₄²×C₄₇	#41	1	1	yes	characteristic
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	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	2	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
85	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
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	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	#97	1	2	yes	—
97	C₂×C₄×C₄₇	#97	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	#131	1	2	yes	—
107	C₂×C₄×C₄₇	#131	1	2	yes	—
108	C₂×C₄×C₄₇	#131	1	2	yes	—
109	C₂×C₄×C₄₇	#131	1	2	yes	—
110	C₈×C₄₇	—	2	4	no	—
111	C₈×C₄₇	—	4	2	no	—
112	C₄₇⋊C₈	—	2	4	no	—
113	C₄₇⋊C₈	—	4	2	no	—
114	C₄₇⋊C₈	—	4	2	no	—
115	C₄₇⋊C₈	—	4	2	no	—
116	#8265	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
119	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
120	C₂²×C₄₇	#405	1	1	yes	characteristic
121	C₂²×C₄₇	#405	1	1	yes	characteristic
122	C₂²×C₄₇	#405	1	1	yes	characteristic
123	C₂²×C₄₇	#405	1	1	yes	characteristic
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	4	1	no	—
127	C₄×C₄₇	—	4	1	no	—
128	C₄×C₄₇	#424	1	2	yes	—
129	C₄×C₄₇	#433	1	2	yes	—
130	C₄×C₄₇	#435	1	2	yes	—
131	C₄×C₄₇	#436	1	2	yes	—
132	C₄×C₄₇	#446	1	2	yes	—
133	C₄×C₄₇	#446	1	2	yes	—
134	C₄×C₄⋊C₈	—	47	1	no	—
135	C₄×C₄⋊C₈	—	47	1	no	—
136	#1744	D₄₇	1	1	yes	characteristic
137	C₂×#322	—	47	1	no	—
138	C₂×#322	—	47	1	no	—
139	#1273	—	47	1	no	—
140	#1273	—	47	1	no	—
141	C₂×C₄₇	#1451	1	1	yes	characteristic
142	C₂×C₄₇	#1452	1	1	yes	characteristic
143	C₂×C₄₇	#1453	1	1	yes	characteristic
144	C₂×C₄₇	#1454	1	1	yes	characteristic
145	C₂×C₄₇	#1768	1	1	yes	characteristic
146	C₂×C₄₇	#1770	1	1	yes	characteristic
147	C₂×C₄₇	#1771	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	47	2	no	—
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₄³	C₂×D₄₇	1	1	yes	characteristic
157	#336	—	47	2	no	—
158	#336	—	47	2	no	—
159	#336	Dic₄₇	1	2	yes	—
160	#336	Dic₄₇	1	2	yes	—
161	#322	—	94	2	no	—
162	#322	—	94	2	no	—
163	C₄₇	#8265	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₈	—	47	2	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	—	47	2	no	—
167	C₂²×C₈	—	47	2	no	—
168	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
169	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
170	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
177	C₄⋊C₈	—	94	4	no	—
178	C₄⋊C₈	—	94	4	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	94	1	no	—
182	C₄×C₈	—	94	1	no	—
183	C₄×C₈	—	94	1	no	—
184	C₄×C₈	—	94	1	no	—
185	C₄×C₈	—	94	1	no	—
186	C₄×C₈	—	94	1	no	—
187	C₄×C₈	—	94	1	no	—
188	C₄×C₈	—	94	1	no	—
189	C₄×C₈	Dic₉₄	1	2	yes	—
190	C₄×C₈	D₁₈₈	1	2	yes	—
191	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	94	2	no	—
205	C₂×C₈	—	94	2	no	—
206	C₂×C₈	—	94	2	no	—
207	C₂×C₈	—	94	2	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	94	2	no	—
214	C₂×C₈	—	94	2	no	—
215	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	#10625324	1	4	yes	—
222	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
223	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
224	C₄²	#10625350	1	1	yes	characteristic
225	C₄²	#10625350	1	1	yes	characteristic
226	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
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	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	#13093934	1	2	yes	—
238	C₂×C₄	#13093934	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
240	C₂×C₄	#13093983	1	1	yes	characteristic
241	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
242	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
243	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
244	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
245	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
246	C₂×C₄	C₂×#10625324	1	2	yes	—
247	C₂×C₄	#13094030	1	2	yes	—
248	C₂×C₄	#13094031	1	2	yes	—
249	C₂×C₄	#13094031	1	2	yes	—
250	C₂×C₄	#13094032	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	4	2	no	—
253	C₈	—	94	4	no	—
254	C₈	—	188	2	no	—
255	C₈	—	188	2	no	—
256	C₈	—	188	2	no	—
257	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
258	C₂²	C₂×#13093934	1	1	yes	characteristic
259	C₂²	M₁₆×Dic₄₇	1	1	yes	characteristic
260	C₂²	#15689078	1	1	yes	characteristic
261	C₂²	#15689079	1	1	yes	characteristic
262	C₂²	#15689079	1	1	yes	characteristic
263	C₂²	#15689081	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#15689150	1	2	yes	—
269	C₄	#15689184	1	2	yes	—
270	C₄	#15689196	1	2	yes	—
271	C₄	#15689202	1	2	yes	—
272	C₄	#15689251	1	2	yes	—
273	C₄	#15689253	1	2	yes	—
274	C₂	#20241678	1	1	yes	characteristic
275	C₂	#20241681	1	1	yes	characteristic
276	C₂	#20241683	1	1	yes	characteristic
277	C₂	#20241688	1	1	yes	characteristic
278	C₂	#20243245	1	1	yes	characteristic
279	C₂	#20243255	1	1	yes	characteristic
280	C₂	#20243260	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1744	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8265
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
163. C₄₇	#8265	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688845	2
C₂×#15688844	2
C₄₇⋊#1273	2
C₄₇×#1744	2
#8265	47