Subgroups — #20252606

Order 6016.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094044	C₂	1	2	yes	maximal, complemented
2	#15689634	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689285	C₂	1	2	yes	maximal, complemented
4	#15689279	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689274	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689268	C₂	1	2	yes	maximal, complemented
7	#15689262	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689257	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689119	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	1	yes	characteristic, complemented
13	#13094044	—	2	8	no	—
14	C₂×#10625337	C₂²	1	2	yes	complemented
15	C₂×#10625335	C₂²	1	2	yes	complemented
16	C₂×#10625336	C₂²	1	2	yes	complemented
17	C₂×#10625334	C₂²	1	2	yes	complemented
18	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	complemented
19	#13094007	—	2	1	no	—
20	#13094003	C₂²	1	1	yes	characteristic
21	#13094003	C₂²	1	1	yes	characteristic, complemented
22	#13093998	C₂²	1	1	yes	characteristic
23	#13093998	C₂²	1	1	yes	characteristic, complemented
24	#13093993	—	2	1	no	—
25	C₄×C₂²⋊D₄₇	—	2	1	no	—
26	C₄×C₂²⋊D₄₇	—	2	1	no	—
27	#13093983	C₂²	1	1	yes	characteristic, complemented
28	#13093948	C₂²	1	2	yes	complemented
29	#13093948	C₂²	1	2	yes	complemented
30	#13093945	C₂²	1	2	yes	complemented
31	#13093945	C₂²	1	2	yes	complemented
32	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
33	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
35	C₂×#41×C₄₇	—	2	8	no	—
36	C₂×#41×C₄₇	—	2	1	no	—
37	C₂×#41×C₄₇	—	2	1	no	—
38	C₂×#41×C₄₇	C₂²	1	2	yes	—
39	C₂×#41×C₄₇	C₂²	1	2	yes	—
40	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₄₇	—	2	1	no	—
42	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	—	2	1	no	—
44	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
45	C₂⁴×C₄₇	—	2	1	no	—
46	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
47	#41×C₄₇	—	2	8	no	—
48	#41×C₄₇	—	2	8	no	—
49	#41×C₄₇	—	2	4	no	—
50	#41×C₄₇	—	2	4	no	—
51	#41×C₄₇	—	2	4	no	—
52	#41×C₄₇	D₄	1	8	yes	complemented
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	—	2	1	no	—
55	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
56	C₂×D₄×C₄₇	—	2	8	no	—
57	C₂×D₄×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	—	2	4	no	—
60	C₂×D₄×C₄₇	—	2	1	no	—
61	C₂×D₄×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	—	2	1	no	—
74	C₂×C₂²⋊D₄₇	—	2	1	no	—
75	C₂×C₂²⋊D₄₇	—	2	1	no	—
76	#10625350	—	2	4	no	—
77	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₄₇	—	2	1	no	—
79	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	—	2	1	no	—
81	C₂²⋊Dic₄₇	—	2	1	no	—
82	C₂²⋊Dic₄₇	—	2	1	no	—
83	#10625337	—	2	4	no	—
84	#10625336	—	2	4	no	—
85	#10625335	—	2	4	no	—
86	#10625334	—	2	4	no	—
87	(C₂×D₄₇)⋊C₄	—	2	1	no	—
88	(C₂×D₄₇)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
91	Dic₄₇⋊C₄	—	2	1	no	—
92	Dic₄₇⋊C₄	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₄₇	—	2	1	no	—
108	Q₈×C₄₇	—	2	4	no	—
109	Q₈×C₄₇	—	2	2	no	—
110	Q₈×C₄₇	C₂×D₄	1	4	yes	—
111	D₄×C₄₇	—	2	8	no	—
112	D₄×C₄₇	—	2	4	no	—
113	D₄×C₄₇	—	2	8	no	—
114	D₄×C₄₇	—	2	2	no	—
115	D₄×C₄₇	C₂×D₄	1	4	yes	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₄₇	—	4	2	no	—
131	C₂²⋊D₄₇	—	4	2	no	—
132	C₂²⋊D₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	2	1	no	—
134	C₂×Dic₄₇	—	2	1	no	—
135	C₂×Dic₄₇	—	2	1	no	—
136	D₁₈₈	—	2	2	no	—
137	C₄×D₄₇	—	4	2	no	—
138	Dic₉₄	—	2	2	no	—
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	4	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	4	4	no	—
150	C₂²×C₄₇	—	4	8	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	#120	1	2	yes	—
154	C₂²×C₄₇	#135	1	2	yes	complemented
155	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₄₇	—	4	1	no	—
157	C₂×D₄₇	—	4	1	no	—
158	C₂×D₄₇	—	4	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	4	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	#120	1	2	yes	—
163	C₄×C₄₇	C₂²×D₄	1	2	yes	—
164	Dic₄₇	—	4	1	no	—
165	Dic₄₇	—	4	1	no	—
166	Dic₄₇	—	4	1	no	—
167	#3002	—	47	1	no	maximal, Sylow
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	—	4	1	no	—
171	C₂×C₄₇	—	4	4	no	—
172	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
174	C₂×C₄₇	#575	1	1	yes	characteristic
175	D₄₇	—	8	1	no	—
176	C₂²×#41	D₄₇	1	1	yes	characteristic, complemented
177	C₂×#135	—	47	2	no	—
178	#525	—	47	1	no	—
179	#452	—	47	2	no	—
180	#451	—	47	1	no	—
181	#450	—	47	1	no	—
182	#449	—	47	2	no	—
183	#448	—	47	1	no	—
184	#447	—	47	1	no	—
185	#416	—	47	2	no	—
186	C₂×C₂²⋊C₈	—	47	1	no	—
187	C₄₇	#3002	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	2	8	no	—
189	C₂×#41	—	2	1	no	—
190	C₂×#41	—	2	1	no	—
191	C₂×#41	—	47	1	no	—
192	C₂×#41	C₂×D₄₇	1	2	yes	—
193	C₂×#41	C₂×D₄₇	1	2	yes	—
194	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
199	#135	—	94	8	no	—
200	C₂×Q₁₆	—	47	2	no	—
201	C₂×SD₁₆	—	47	2	no	—
202	C₂×SD₁₆	—	47	2	no	—
203	C₂×D₈	—	47	2	no	—
204	C₂²×C₈	—	47	2	no	—
205	#123	—	94	1	no	—
206	C₂²⋊Q₈	—	47	1	no	—
207	C₂²⋊Q₈	—	47	1	no	—
208	#121	—	47	1	no	—
209	#121	—	47	1	no	—
210	#120	—	94	1	no	—
211	C₄×D₄	—	94	1	no	—
212	C₄×D₄	—	94	1	no	—
213	#117	—	47	1	no	—
214	#103	—	47	2	no	—
215	#103	—	47	2	no	—
216	#102	—	47	2	no	—
217	#102	—	47	2	no	—
218	C₂²⋊C₈	—	47	1	no	—
219	C₂²⋊C₈	—	47	1	no	—
220	C₂²⋊C₈	—	47	2	no	—
221	C₂⁴	—	2	1	no	—
222	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
223	#41	—	2	8	no	—
224	#41	—	2	8	no	—
225	#41	—	2	4	no	—
226	#41	—	2	4	no	—
227	#41	—	2	4	no	—
228	#41	—	94	4	no	—
229	#41	C₂²⋊D₄₇	1	8	yes	complemented
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	47	1	no	—
233	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	47	1	no	—
240	C₂×D₄	—	94	1	no	—
241	C₂×D₄	—	94	1	no	—
242	C₂×D₄	—	94	1	no	—
243	C₂×D₄	C₂²×D₄₇	1	2	yes	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	94	1	no	—
250	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
256	Q₁₆	—	94	4	no	—
257	SD₁₆	—	94	4	no	—
258	SD₁₆	—	94	4	no	—
259	D₈	—	94	4	no	—
260	C₂×C₈	—	47	2	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	94	4	no	—
263	C₄⋊C₄	—	47	1	no	—
264	C₄⋊C₄	—	47	1	no	—
265	C₄⋊C₄	—	94	1	no	—
266	C₄⋊C₄	—	94	1	no	—
267	C₂²⋊C₄	—	94	1	no	—
268	C₂²⋊C₄	—	94	1	no	—
269	C₂²⋊C₄	—	94	1	no	—
270	C₂²⋊C₄	—	94	1	no	—
271	C₂²⋊C₄	—	94	1	no	—
272	C₄²	—	94	1	no	—
273	C₂³	—	2	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	94	1	no	—
281	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	94	2	no	—
287	Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	8	no	—
291	D₄	—	2	2	no	—
292	D₄	—	94	2	no	—
293	D₄	—	188	2	no	—
294	D₄	—	188	2	no	—
295	D₄	—	188	2	no	—
296	D₄	C₂×C₂²⋊D₄₇	1	4	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	94	1	no	—
307	C₂×C₄	—	94	1	no	—
308	C₂×C₄	—	94	1	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
315	C₈	—	94	2	no	—
316	C₈	—	94	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	8	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	188	1	no	—
330	C₂²	—	188	1	no	—
331	C₂²	—	188	1	no	—
332	C₂²	#13093993	1	2	yes	—
333	C₂²	#13094044	1	2	yes	complemented
334	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	1	no	—
338	C₄	—	188	1	no	—
339	C₄	—	188	1	no	—
340	C₄	—	188	1	no	—
341	C₄	#13093993	1	2	yes	—
342	C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	4	4	no	—
347	C₂	—	376	1	no	—
348	C₂	C₂×#13093993	1	1	yes	characteristic
349	C₂	C₂×#13094044	1	1	yes	characteristic
350	C₂	#15689921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂²×#41×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂²×#41	D₄₇
hypocenter	C₄₇	#3002
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13094044	C₂	semidirect
2. #15689634	C₂	semidirect
3. #15689285	C₂	semidirect
4. #15689279	C₂	semidirect
5. #15689274	C₂	semidirect
6. #15689268	C₂	semidirect
7. #15689262	C₂	semidirect
8. #15689257	C₂	semidirect
9. #15689119	C₂	semidirect
10. C₂×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
11. C₂²×#41×C₄₇	C₂	semidirect
12. C₂×#10625350	C₂²	semidirect
14. C₂×#10625337	C₂²	semidirect
15. C₂×#10625335	C₂²	semidirect
16. C₂×#10625336	C₂²	semidirect
17. C₂×#10625334	C₂²	semidirect
18. C₂²×C₄₇⋊C₈	C₂²	semidirect
21. #13094003	C₂²	semidirect
23. #13093998	C₂²	semidirect
27. #13093983	C₂²	semidirect
28. #13093948	C₂²	semidirect
29. #13093948	C₂²	semidirect
30. #13093945	C₂²	semidirect
31. #13093945	C₂²	semidirect
34. C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
52. #41×C₄₇	D₄	semidirect
154. C₂²×C₄₇	#135	semidirect
176. C₂²×#41	D₄₇	semidirect
187. C₄₇	#3002	semidirect
229. #41	C₂²⋊D₄₇	semidirect
333. C₂²	#13094044	semidirect