Subgroups — #20246237

Order 6016.

792 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×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689016	C₂	1	2	yes	maximal, complemented
4	#15689016	C₂	1	2	yes	maximal, complemented
5	#15688998	C₂	1	2	yes	maximal, complemented
6	#15688986	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
8	#15688979	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊C₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₄₇	C₂²	1	2	yes	—
13	#13093983	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	complemented
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
17	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂²	1	2	yes	complemented
25	C₂×C₄×Dic₄₇	C₂²	1	2	yes	complemented
26	C₄₇⋊#95	C₂²	1	2	yes	complemented
27	C₄₇⋊#95	C₂²	1	2	yes	complemented
28	C₄₇⋊#95	C₂²	1	2	yes	complemented
29	C₄₇⋊#95	C₂²	1	2	yes	complemented
30	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₄₇	C₄	1	8	yes	complemented
34	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₂⁴×C₄₇	—	2	2	no	—
39	C₂⁴×C₄₇	—	2	4	no	—
40	C₂⁴×C₄₇	—	2	2	no	—
41	C₂⁴×C₄₇	—	4	2	no	—
42	C₂⁴×C₄₇	C₂×C₄	1	4	yes	—
43	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
44	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
45	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
50	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	4	no	—
56	C₂²×Dic₄₇	—	2	4	no	—
57	C₂²×Dic₄₇	—	4	2	no	—
58	C₂²×Dic₄₇	C₂³	1	2	yes	—
59	C₂²×Dic₄₇	C₂³	1	2	yes	—
60	C₂²×Dic₄₇	C₂³	1	2	yes	—
61	C₂²×Dic₄₇	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
63	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
65	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
66	C₂²⋊Dic₄₇	—	2	2	no	—
67	C₂²⋊Dic₄₇	—	2	2	no	—
68	C₂²⋊Dic₄₇	—	2	4	no	—
69	C₂²⋊Dic₄₇	—	2	2	no	—
70	C₂²⋊Dic₄₇	—	2	4	no	—
71	C₂²⋊Dic₄₇	—	4	2	no	—
72	C₂²⋊Dic₄₇	D₄	1	4	yes	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄×Dic₄₇	—	2	4	no	—
76	C₄×Dic₄₇	—	4	2	no	—
77	C₂³×C₄₇	—	2	1	no	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	—	2	2	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	2	8	no	—
83	C₂³×C₄₇	—	2	8	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	—	4	8	no	—
90	C₂³×C₄₇	—	4	8	no	—
91	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂³×C₄₇	C₂×D₄	1	2	yes	—
95	C₂³×C₄₇	#41	1	4	yes	—
96	C₂³×C₄₇	#41	1	2	yes	—
97	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	4	16	no	—
135	C₂²×C₄₇	—	4	2	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	#117	1	4	yes	complemented
138	C₂²×C₄₇	C₄×D₄	1	4	yes	—
139	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
140	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#41	1	2	yes	—
143	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	Dic₄₇	—	2	4	no	—
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	4	2	no	—
152	#2342	—	47	1	no	maximal, Sylow
153	C₂×C₄₇	—	2	4	no	—
154	C₂×C₄₇	—	2	4	no	—
155	C₂×C₄₇	—	4	2	no	—
156	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
157	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×C₄₇	#518	1	1	yes	characteristic
159	C₂×C₄₇	#546	1	2	yes	—
160	C₂×C₄₇	#551	1	2	yes	—
161	C₂×#117	—	47	1	no	—
162	C₂²×C₂²⋊C₄	—	47	1	no	—
163	C₂²×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
164	#388	—	47	2	no	—
165	#388	—	47	2	no	—
166	#385	—	47	2	no	—
167	#381	—	47	1	no	—
168	#380	—	47	1	no	—
169	#379	—	47	1	no	—
170	C₄×C₂²⋊C₄	—	47	1	no	—
171	C₄×C₂²⋊C₄	—	47	2	no	—
172	C₄₇	#2342	1	1	yes	characteristic, complemented, Sylow
173	C₂⁵	C₂×D₄₇	1	1	yes	characteristic
174	C₂³×C₄	—	47	2	no	—
175	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
176	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
177	#117	—	94	4	no	—
178	C₂×C₄⋊C₄	—	47	2	no	—
179	C₂×C₄⋊C₄	—	47	2	no	—
180	C₂×C₂²⋊C₄	—	47	1	no	—
181	C₂×C₂²⋊C₄	—	47	1	no	—
182	C₂×C₂²⋊C₄	—	47	2	no	—
183	C₂×C₂²⋊C₄	—	47	1	no	—
184	C₂×C₂²⋊C₄	—	47	1	no	—
185	C₂×C₂²⋊C₄	—	47	2	no	—
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	94	4	no	—
188	C₂×C₂²⋊C₄	Dic₄₇	1	8	yes	complemented
189	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
193	C₂×C₄²	—	47	2	no	—
194	C₂×C₄²	—	47	2	no	—
195	#95	—	47	2	no	—
196	#95	—	47	2	no	—
197	#95	—	47	2	no	—
198	#95	—	47	2	no	—
199	C₂⁴	—	2	4	no	—
200	C₂⁴	—	2	2	no	—
201	C₂⁴	—	2	2	no	—
202	C₂⁴	—	4	2	no	—
203	C₂⁴	C₂×Dic₄₇	1	4	yes	—
204	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
205	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
206	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	2	no	—
214	C₂²×C₄	—	94	4	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	—	188	2	no	—
217	C₂²×C₄	C₂×Dic₄₇	1	4	yes	complemented
218	C₂²×C₄	C₂×Dic₄₇	1	4	yes	complemented
219	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
223	C₄⋊C₄	—	94	4	no	—
224	C₄⋊C₄	—	188	2	no	—
225	C₂²⋊C₄	—	47	4	no	—
226	C₂²⋊C₄	—	94	4	no	—
227	C₂²⋊C₄	—	94	4	no	—
228	C₂²⋊C₄	—	94	2	no	—
229	C₂²⋊C₄	—	94	2	no	—
230	C₂²⋊C₄	—	94	2	no	—
231	C₂²⋊C₄	—	188	2	no	—
232	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
233	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
234	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
235	C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	complemented
236	C₄²	—	94	4	no	—
237	C₄²	—	188	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	2	8	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	8	no	—
251	C₂³	—	4	8	no	—
252	C₂³	#10625350	1	4	yes	—
253	C₂³	D₄×D₄₇	1	2	yes	—
254	C₂³	#10625352	1	2	yes	—
255	C₂³	C₂²×Dic₄₇	1	2	yes	—
256	C₂³	C₂²×Dic₄₇	1	2	yes	—
257	C₂³	C₂²×Dic₄₇	1	2	yes	—
258	C₂³	C₂³×D₄₇	1	1	yes	characteristic
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	47	4	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	2	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	188	8	no	—
280	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
281	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
282	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
283	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	16	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#13093983	1	4	yes	complemented
299	C₂²	D₄×Dic₄₇	1	4	yes	—
300	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
301	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₂²	C₂×#10625352	1	1	yes	characteristic
303	C₂²	C₂×#10625350	1	2	yes	—
304	C₂²	#13094069	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	94	4	no	—
310	C₄	—	188	2	no	—
311	C₄	—	188	2	no	—
312	C₄	—	188	2	no	—
313	C₂	—	2	4	no	—
314	C₂	—	2	4	no	—
315	C₂	—	4	2	no	—
316	C₂	C₂×#13093983	1	1	yes	characteristic
317	C₂	C₂×D₄×Dic₄₇	1	1	yes	characteristic
318	C₂	#15689598	1	1	yes	characteristic
319	C₂	#15689763	1	2	yes	—
320	C₂	#15689799	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂²×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#2342
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13093983	C₂	semidirect
3. #15689016	C₂	semidirect
4. #15689016	C₂	semidirect
5. #15688998	C₂	semidirect
6. #15688986	C₂	semidirect
7. #15688983	C₂	semidirect
9. C₂²⋊C₄×Dic₄₇	C₂	semidirect
10. C₄×C₂²⋊Dic₄₇	C₂	semidirect
14. C₂×C₄⋊Dic₄₇	C₂²	semidirect
15. C₂×C₄⋊Dic₄₇	C₂²	semidirect
24. C₂×C₄×Dic₄₇	C₂²	semidirect
25. C₂×C₄×Dic₄₇	C₂²	semidirect
26. C₄₇⋊#95	C₂²	semidirect
27. C₄₇⋊#95	C₂²	semidirect
28. C₄₇⋊#95	C₂²	semidirect
29. C₄₇⋊#95	C₂²	semidirect
33. C₂×C₂²⋊C₄×C₄₇	C₄	semidirect
49. C₂²×C₄×C₄₇	C₂×C₄	semidirect
50. C₂²×C₄×C₄₇	C₂×C₄	semidirect
64. C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
65. C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
137. C₂²×C₄₇	#117	semidirect
172. C₄₇	#2342	semidirect
188. C₂×C₂²⋊C₄	Dic₄₇	semidirect
217. C₂²×C₄	C₂×Dic₄₇	semidirect
218. C₂²×C₄	C₂×Dic₄₇	semidirect
234. C₂²⋊C₄	C₂×Dic₄₇	semidirect
235. C₂²⋊C₄	C₂×Dic₄₇	semidirect
298. C₂²	#13093983	semidirect