Subgroups — #26214351

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689163	C₂	1	2	yes	maximal
2	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
3	#20245745	C₂	1	2	yes	maximal
4	#20243284	C₂	1	4	yes	maximal, complemented
5	#20243280	C₂	1	2	yes	maximal, complemented
6	#20243276	C₂	1	2	yes	maximal, complemented
7	#20243105	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	2	yes	—
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	#15689163	C₄	1	16	yes	complemented
13	C₂×#13093957	C₂²	1	4	yes	—
14	C₂×#13093955	C₂²	1	4	yes	—
15	#15689078	C₄	1	8	yes	complemented
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₄₇	—	2	2	no	—
19	C₄⋊C₄×Dic₄₇	—	2	2	no	—
20	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	C₄×C₂²⋊Dic₄₇	—	2	4	no	—
23	#15688886	C₂²	1	1	yes	characteristic
24	#15688886	C₂²	1	1	yes	characteristic
25	#15688886	C₂²	1	2	yes	—
26	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄₇×#340	C₂²	1	2	yes	—
29	C₄₇×#340	C₂²	1	2	yes	—
30	C₂×#10625329	C₂×C₄	1	4	yes	—
31	C₂×#10625329	C₂×C₄	1	8	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	#13093983	—	2	4	no	—
34	#13093983	C₂×C₄	1	8	yes	—
35	#13093983	C₂×C₄	1	4	yes	—
36	#13093983	Q₈	1	4	yes	—
37	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
42	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	4	no	—
45	C₂×C₄×Dic₄₇	—	2	4	no	—
46	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
47	#13093957	C₂×C₄	1	16	yes	complemented
48	#13093955	C₂×C₄	1	16	yes	complemented
49	#13093934	C₂×C₄	1	8	yes	complemented
50	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
51	C₄₇⋊#95	—	2	4	no	—
52	C₄₇⋊#95	—	2	4	no	—
53	C₂³×C₄×C₄₇	—	2	2	no	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	—	2	4	no	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
62	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
63	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	8	2	no	—
67	C₂²×C₄×C₄₇	—	8	2	no	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₇	—	2	8	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	8	no	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
86	C₂²⋊Dic₄₇	—	2	8	no	—
87	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
91	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
92	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
93	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
94	C₄×Dic₄₇	—	2	8	no	—
95	C₄×Dic₄₇	—	8	4	no	—
96	C₄×Dic₄₇	—	8	4	no	—
97	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
98	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
99	#10625329	C₄²	1	16	yes	complemented
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄₇	C₄×D₄	1	4	yes	—
107	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	8	no	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	4	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
135	#20453	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
141	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
143	C₂²×C₄₇	#576	1	1	yes	characteristic
144	C₂²×C₄₇	#577	1	2	yes	—
145	C₂²×C₄₇	#578	1	1	yes	characteristic
146	C₄×C₄₇	—	4	8	no	—
147	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
148	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
150	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	4	4	no	—
154	Dic₄₇	—	4	4	no	—
155	C₂×#428	—	47	2	no	—
156	C₂×#405	—	47	1	no	—
157	#2274	—	47	2	no	—
158	#1776	—	47	4	no	—
159	C₄×#107	—	47	2	no	—
160	C₄×#106	—	47	2	no	—
161	#1735	—	47	1	no	—
162	C₂×#340	D₄₇	1	1	yes	characteristic
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄₇	#2912	1	1	yes	characteristic
166	C₂×C₄₇	#2913	1	1	yes	characteristic
167	C₂×C₄₇	#2946	1	2	yes	—
168	C₂×C₄₇	#2947	1	2	yes	—
169	C₂²×M₁₆	—	47	1	no	—
170	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#117	—	47	1	no	—
172	C₂×#117	—	47	2	no	—
173	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
174	#428	—	47	16	no	—
175	C₂×#107	—	47	4	no	—
176	C₂×#106	—	47	4	no	—
177	#405	—	47	8	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×C₄×C₈	—	47	2	no	—
180	C₄×C₄⋊C₄	—	47	4	no	—
181	C₄×C₄⋊C₄	—	47	4	no	—
182	C₄×C₄⋊C₄	—	94	2	no	—
183	C₄×C₄⋊C₄	—	94	2	no	—
184	C₄×C₂²⋊C₄	—	94	4	no	—
185	#340	—	47	1	no	—
186	#340	—	47	1	no	—
187	#340	—	47	2	no	—
188	#340	C₂×D₄₇	1	2	yes	—
189	#340	C₂×D₄₇	1	2	yes	—
190	C₄₇	#20453	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	47	4	no	—
194	C₂×M₁₆	—	47	8	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	C₄×D₄₇	1	4	yes	—
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	#117	—	47	4	no	—
201	#117	—	47	4	no	—
202	#117	—	47	8	no	—
203	#117	—	94	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	47	1	no	—
207	C₂×C₄⋊C₄	—	47	1	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	47	2	no	—
210	C₂×C₄⋊C₄	C₄×D₄₇	1	8	yes	—
211	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
212	C₂×C₂²⋊C₄	—	47	2	no	—
213	C₂×C₂²⋊C₄	—	94	2	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	94	2	no	—
216	C₂×C₄²	—	94	4	no	—
217	C₂×C₄²	—	94	4	no	—
218	#107	—	47	16	no	—
219	#106	—	47	16	no	—
220	#97	—	47	8	no	—
221	C₄×C₈	—	47	8	no	—
222	#95	—	94	4	no	—
223	#95	—	94	4	no	—
224	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	8	2	no	—
228	C₂²×C₄	—	8	2	no	—
229	C₂²×C₄	—	47	2	no	—
230	C₂²×C₄	—	47	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
234	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
235	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	#10625352	1	4	yes	—
237	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
239	M₁₆	—	47	16	no	—
240	C₂×C₈	—	2	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	47	4	no	—
244	C₂×C₈	—	47	8	no	—
245	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	47	4	no	—
250	C₄⋊C₄	—	47	4	no	—
251	C₄⋊C₄	—	47	4	no	—
252	C₄⋊C₄	—	47	4	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	C₂×C₄×D₄₇	1	8	yes	—
256	C₂²⋊C₄	—	47	8	no	—
257	C₂²⋊C₄	—	94	8	no	—
258	C₄²	—	47	4	no	—
259	C₄²	—	47	4	no	—
260	C₄²	—	94	8	no	—
261	C₄²	—	376	4	no	—
262	C₄²	—	376	4	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	1	no	—
266	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
267	C₂³	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
268	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	—	47	4	no	—
274	C₂×C₄	—	47	4	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	4	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	C₄²×D₄₇	1	4	yes	—
285	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
286	C₂×C₄	#13093980	1	4	yes	—
287	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
288	C₂×C₄	#13093989	1	8	yes	—
289	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
291	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#10625352	1	2	yes	—
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	47	8	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#15688974	1	4	yes	—
301	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
302	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
303	C₂²	#15689924	1	1	yes	characteristic
304	C₂²	#15689929	1	2	yes	—
305	C₂²	#15689936	1	1	yes	characteristic
306	C₄	—	4	8	no	—
307	C₄	—	94	4	no	—
308	C₄	—	94	4	no	—
309	C₄	—	188	4	no	—
310	C₄	—	188	4	no	—
311	C₄	#15688974	1	4	yes	—
312	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
313	C₄	C₂×#13093980	1	1	yes	characteristic
314	C₄	C₂×#13093989	1	2	yes	—
315	C₂	—	2	4	no	—
316	C₂	C₂×#15688974	1	1	yes	characteristic
317	C₂	#20251979	1	1	yes	characteristic
318	C₂	#20251984	1	1	yes	characteristic
319	C₂	#20252214	1	2	yes	—
320	C₂	#20252225	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	#20453
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
4. #20243284	C₂	semidirect
5. #20243280	C₂	semidirect
6. #20243276	C₂	semidirect
12. #15689163	C₄	semidirect
15. #15689078	C₄	semidirect
47. #13093957	C₂×C₄	semidirect
48. #13093955	C₂×C₄	semidirect
49. #13093934	C₂×C₄	semidirect
50. C₈×Dic₄₇	C₂×C₄	semidirect
99. #10625329	C₄²	semidirect
190. C₄₇	#20453	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689163	2
C₂×#15689078	2
#20245745	2
#20243284	2
#20243280	2
#20243276	2
#20243105	2
C₂×C₄₇×#340	2
#20453	47