Subgroups — #26220425

Order 12032.

838 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	2	yes	maximal, complemented
2	#20244183	C₂	1	2	yes	maximal
3	#20244107	C₂	1	2	yes	maximal, complemented
4	#20244003	C₂	1	4	yes	maximal, complemented
5	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243154	C₂	1	2	yes	maximal, complemented
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#2293	C₂	1	1	yes	maximal, characteristic
9	#15689184	—	2	8	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
19	#15688878	C₂²	1	2	yes	—
20	#15688878	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#390	C₄	1	2	yes	—
24	C₄₇×#389	C₄	1	2	yes	—
25	C₄₇×#388	C₄	1	4	yes	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄³×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
36	#13093935	—	2	2	no	—
37	#13093935	—	2	4	no	—
38	#13093935	D₄	1	4	yes	complemented
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₄×C₄₇⋊C₈	—	2	4	no	—
41	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
43	C₄×D₄×C₄₇	—	2	4	no	—
44	#117×C₄₇	—	2	4	no	—
45	#117×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	#95×C₄₇	C₂×C₄	1	2	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂×D₄×C₄₇	—	2	2	no	—
62	C₂×D₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	8	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	2	4	no	—
78	C₂²⋊C₄×C₄₇	—	2	8	no	—
79	C₂²⋊C₄×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	4	2	no	—
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄²×C₄₇	#41	1	4	yes	—
87	C₄²×C₄₇	#41	1	2	yes	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
96	C₂×C₄₇⋊C₈	#41	1	4	yes	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	D₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	#117	1	2	yes	—
119	C₂×C₄×C₄₇	#117	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
121	C₂×C₄×C₄₇	#131	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
126	C₄₇⋊C₈	—	2	4	no	—
127	C₄₇⋊C₈	—	2	4	no	—
128	C₄₇⋊C₈	—	4	4	no	—
129	#21268	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	4	2	no	—
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	4	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	—	4	2	no	—
135	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#131	1	2	yes	—
139	C₂²×C₄₇	#568	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	4	no	—
142	C₄×C₄₇	—	4	4	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	#433	1	4	yes	complemented
146	C₄×C₄₇	#529	1	2	yes	—
147	C₄×C₄₇	#532	1	2	yes	—
148	C₂×#433	—	47	2	no	—
149	#2293	D₄₇	1	1	yes	characteristic
150	#1954	—	47	2	no	—
151	#1939	—	47	2	no	—
152	#1924	—	47	4	no	—
153	C₄×C₄⋊C₈	—	47	1	no	—
154	#1750	—	47	2	no	—
155	C₄×#97	—	47	1	no	—
156	C₂×C₄₇	—	8	2	no	—
157	C₂×C₄₇	#2293	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#433	1	1	yes	characteristic
159	C₂×C₄₇	#2923	1	1	yes	characteristic
160	C₂×C₄₇	#2935	1	2	yes	—
161	C₂×C₄₇	#2969	1	2	yes	—
162	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#117	C₂×D₄₇	1	2	yes	—
164	#433	—	94	8	no	—
165	C₂×C₄⋊C₈	—	47	4	no	—
166	C₂×C₄⋊C₈	—	47	2	no	—
167	C₂×C₂²⋊C₈	—	47	4	no	—
168	C₂×C₂²⋊C₈	—	47	4	no	—
169	C₂×#97	—	47	1	no	—
170	C₂×#97	—	47	2	no	—
171	C₂×#97	—	47	1	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	—	47	2	no	—
174	#390	Dic₄₇	1	2	yes	—
175	#389	Dic₄₇	1	2	yes	—
176	#388	Dic₄₇	1	4	yes	—
177	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
178	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
179	C₄³	C₂×D₄₇	1	1	yes	characteristic
180	#336	—	47	2	no	—
181	#336	—	47	4	no	—
182	C₄₇	#21268	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	4	no	—
188	C₄×D₄	—	2	4	no	—
189	#117	—	2	4	no	—
190	#117	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
198	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
199	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	94	8	no	—
204	C₄⋊C₈	—	94	4	no	—
205	C₂²⋊C₈	—	94	8	no	—
206	C₂²⋊C₈	—	94	8	no	—
207	#97	—	47	4	no	—
208	#97	—	94	4	no	—
209	#97	—	94	2	no	—
210	C₄×C₈	—	94	4	no	—
211	C₄×C₈	—	94	4	no	—
212	#95	C₂×Dic₄₇	1	4	yes	—
213	#95	C₂×Dic₄₇	1	2	yes	—
214	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	#10625350	1	4	yes	—
249	C₄²	D₄×D₄₇	1	2	yes	—
250	C₄²	#10625352	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#13093983	1	2	yes	—
273	C₂×C₄	#13093983	1	2	yes	—
274	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
275	C₂×C₄	#13094031	1	4	yes	—
276	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	#41×D₄₇	1	2	yes	—
278	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
279	C₂×C₄	C₂×#10625350	1	2	yes	—
280	C₈	—	94	4	no	—
281	C₈	—	94	4	no	—
282	C₈	—	188	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂×#13093983	1	1	yes	characteristic
289	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
290	C₂²	#41×Dic₄₇	1	1	yes	characteristic
291	C₂²	C₂×#13094031	1	2	yes	—
292	C₂²	#15689890	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	4	2	no	—
298	C₄	#15689184	1	4	yes	complemented
299	C₄	#15689655	1	2	yes	—
300	C₄	#15689670	1	2	yes	—
301	C₂	—	8	2	no	—
302	C₂	#20245853	1	1	yes	characteristic
303	C₂	C₂×#15689184	1	1	yes	characteristic
304	C₂	#20252039	1	1	yes	characteristic
305	C₂	#20252132	1	2	yes	—
306	C₂	#20252393	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2293	C₂
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2293	D₄₇
hypocenter	C₄₇	#21268
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2293

Complements

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

N	H ≅ G/N	type
1. C₂×#15689184	C₂	semidirect
3. #20244107	C₂	semidirect
4. #20244003	C₂	semidirect
5. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
6. #20243154	C₂	semidirect
7. C₄×#13093935	C₂	semidirect
38. #13093935	D₄	semidirect
145. C₄×C₄₇	#433	semidirect
182. C₄₇	#21268	semidirect
298. C₄	#15689184	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#20244183	2
#20244107	2
#20244003	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243154	2
C₄×#13093935	2
C₄₇×#2293	2
#21268	47