Subgroups — #20257299

Order 6016.

1104 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689846	C₂	1	2	yes	maximal, complemented
2	#15689838	C₂	1	2	yes	maximal, complemented
3	#15689833	C₂	1	2	yes	maximal, complemented
4	#15689819	C₂	1	4	yes	maximal, complemented
5	#15689782	C₂	1	4	yes	maximal, complemented
6	#15689778	C₂	1	2	yes	maximal, complemented
7	#15689744	C₂	1	4	yes	maximal, complemented
8	#15689701	C₂	1	2	yes	maximal, complemented
9	#15689680	C₂	1	2	yes	maximal, complemented
10	#15689673	C₂	1	1	yes	maximal, characteristic, complemented
11	#15689673	C₂	1	2	yes	maximal, complemented
12	C₂×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
13	#15689603	C₂	1	2	yes	maximal, complemented
14	C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#10625350	C₂²	1	2	yes	—
16	C₂×#10625350	C₂²	1	4	yes	—
17	C₂²×Dic₉₄	C₂²	1	2	yes	—
18	#13094024	C₂²	1	2	yes	—
19	#13094024	C₂²	1	2	yes	—
20	#13094024	C₂²	1	2	yes	—
21	#13094024	C₂²	1	2	yes	—
22	#13094024	C₂²	1	1	yes	characteristic
23	#13094021	C₂²	1	1	yes	characteristic
24	#13094019	C₂²	1	8	yes	—
25	#13094016	C₂²	1	4	yes	—
26	#13094012	C₂²	1	2	yes	—
27	#13094012	C₂²	1	4	yes	—
28	#13094010	C₂²	1	8	yes	—
29	#13094006	C₂²	1	8	yes	—
30	#13094003	C₂²	1	4	yes	—
31	#13094003	C₂²	1	4	yes	—
32	#13094002	C₂²	1	8	yes	—
33	#13094000	C₂²	1	4	yes	—
34	#13093998	C₂²	1	4	yes	—
35	#13093998	C₂²	1	2	yes	—
36	#13093998	C₂²	1	2	yes	—
37	#13093998	C₂²	1	4	yes	—
38	#13093996	C₂²	1	8	yes	—
39	#13093995	C₂²	1	4	yes	—
40	C₄×Dic₉₄	C₂²	1	4	yes	—
41	C₄×Dic₉₄	C₂²	1	4	yes	—
42	C₄×D₁₈₈	C₂²	1	4	yes	—
43	C₄×D₁₈₈	C₂²	1	2	yes	—
44	C₄×D₁₈₈	C₂²	1	4	yes	—
45	C₄×D₁₈₈	C₂²	1	2	yes	—
46	(C₂²⋊D₄₇)⋊C₄	C₂²	1	8	yes	—
47	#13093983	C₂²	1	4	yes	—
48	#13093982	C₂²	1	4	yes	—
49	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
50	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
51	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
52	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₄₇	C₂²	1	2	yes	—
57	C₄×D₄×C₄₇	C₂²	1	2	yes	—
58	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
59	#117×C₄₇	C₂²	1	2	yes	—
60	#117×C₄₇	C₂²	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
63	#41×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
74	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
75	C₂²×Dic₄₇	C₂³	1	4	yes	—
76	#10625350	—	2	8	no	—
77	#10625350	—	4	8	no	—
78	C₂×D₁₈₈	C₂³	1	2	yes	—
79	C₂×D₁₈₈	C₂³	1	2	yes	—
80	C₂×D₁₈₈	C₂³	1	2	yes	—
81	C₂×C₄×D₄₇	C₂³	1	4	yes	—
82	C₂×C₄×D₄₇	C₂³	1	8	yes	—
83	C₂×Dic₉₄	—	4	4	no	—
84	C₂×Dic₉₄	C₂³	1	2	yes	—
85	C₂×Dic₉₄	C₂³	1	4	yes	—
86	C₂×Dic₉₄	C₂³	1	4	yes	—
87	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
90	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
91	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
92	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
93	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
94	C₄²×C₄₇	C₂³	1	1	yes	characteristic
95	C₄²×C₄₇	C₂³	1	2	yes	—
96	C₄²×C₄₇	C₂³	1	1	yes	characteristic
97	C₄²×C₄₇	C₂³	1	2	yes	—
98	C₄²×C₄₇	C₂³	1	1	yes	characteristic
99	C₄²×C₄₇	C₂³	1	1	yes	characteristic
100	C₄²×C₄₇	C₂³	1	2	yes	—
101	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
102	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
103	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
104	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
105	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
106	C₄⋊Dic₄₇	C₂³	1	4	yes	—
107	C₄⋊Dic₄₇	C₂³	1	4	yes	—
108	C₄⋊Dic₄₇	C₂³	1	4	yes	—
109	C₄⋊Dic₄₇	C₂³	1	4	yes	—
110	C₄⋊Dic₄₇	C₂³	1	2	yes	—
111	C₄⋊Dic₄₇	C₂³	1	2	yes	—
112	Dic₄₇⋊C₄	C₂³	1	8	yes	—
113	Dic₄₇⋊C₄	C₂³	1	8	yes	—
114	C₄×Dic₄₇	C₂³	1	8	yes	—
115	C₂³×C₄₇	C₂⁴	1	2	yes	—
116	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
117	C₂²×D₄₇	C₂⁴	1	4	yes	—
118	Q₈×C₄₇	—	2	2	no	—
119	D₄×C₄₇	—	2	4	no	—
120	D₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	#41	1	4	yes	complemented
127	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
128	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
129	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
130	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₄₇	—	2	8	no	—
137	C₂²⋊D₄₇	—	4	8	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	C₂⁴	1	4	yes	—
140	C₂×Dic₄₇	C₂⁴	1	8	yes	—
141	D₁₈₈	—	2	4	no	—
142	D₁₈₈	—	2	4	no	—
143	D₁₈₈	—	2	4	no	—
144	C₄×D₄₇	—	2	8	no	—
145	C₄×D₄₇	—	4	8	no	—
146	Dic₉₄	—	2	4	no	—
147	Dic₉₄	—	2	8	no	—
148	Dic₉₄	—	4	4	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	C₂×#41	1	2	yes	—
155	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
156	C₂×D₄₇	—	2	4	no	—
157	C₂×D₄₇	—	2	4	no	—
158	C₂×D₄₇	—	2	4	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	C₂×#41	1	2	yes	—
165	C₄×C₄₇	C₂×#41	1	2	yes	—
166	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
167	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
168	Dic₄₇	—	2	4	no	—
169	Dic₄₇	—	2	8	no	—
170	#3511	—	47	1	no	maximal, Sylow
171	C₂×C₄₇	—	2	2	no	—
172	C₂×C₄₇	—	4	2	no	—
173	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
174	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
176	D₄₇	—	4	4	no	—
177	#556	—	47	2	no	—
178	#555	—	47	2	no	—
179	#554	—	47	2	no	—
180	#553	—	47	4	no	—
181	#548	—	47	4	no	—
182	#547	—	47	2	no	—
183	#542	—	47	4	no	—
184	#537	—	47	2	no	—
185	#533	—	47	2	no	—
186	#532	—	47	1	no	—
187	#532	—	47	2	no	—
188	C₂×C₄⋊Q₈	—	47	1	no	—
189	#520	—	47	2	no	—
190	C₄×#41	D₄₇	1	1	yes	characteristic, complemented
191	C₄₇	#3511	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	47	2	no	—
193	C₂×#41	—	47	4	no	—
194	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
195	C₂²×Q₈	—	47	2	no	—
196	C₄⋊Q₈	—	47	2	no	—
197	C₄⋊Q₈	—	47	2	no	—
198	C₄⋊Q₈	—	47	2	no	—
199	C₄⋊Q₈	—	47	2	no	—
200	C₄⋊Q₈	—	47	1	no	—
201	#127	—	47	1	no	—
202	#126	—	47	8	no	—
203	#125	—	47	4	no	—
204	#124	—	47	8	no	—
205	#124	—	47	2	no	—
206	#124	—	47	4	no	—
207	#123	—	47	8	no	—
208	C₂²⋊Q₈	—	47	4	no	—
209	C₂²⋊Q₈	—	47	4	no	—
210	C₂²⋊Q₈	—	47	4	no	—
211	C₂²⋊Q₈	—	47	8	no	—
212	#121	—	47	4	no	—
213	#121	—	47	8	no	—
214	#121	—	47	4	no	—
215	#121	—	47	2	no	—
216	#121	—	47	2	no	—
217	#121	—	47	4	no	—
218	C₄×Q₈	—	47	4	no	—
219	C₄×Q₈	—	47	4	no	—
220	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
221	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
222	C₄×D₄	—	47	8	no	—
223	C₄×D₄	—	47	4	no	—
224	C₄×D₄	—	47	2	no	—
225	C₄×D₄	—	47	4	no	—
226	C₄×D₄	—	47	2	no	—
227	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
228	C₄×D₄	C₂×D₄₇	1	2	yes	—
229	C₄×D₄	C₂×D₄₇	1	2	yes	—
230	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
231	#117	—	47	4	no	—
232	#117	—	47	4	no	—
233	#117	C₂×D₄₇	1	1	yes	characteristic
234	#117	C₂×D₄₇	1	2	yes	—
235	C₂×C₄⋊C₄	—	47	2	no	—
236	C₂×C₄⋊C₄	—	47	2	no	—
237	C₂×C₄⋊C₄	—	47	4	no	—
238	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
239	C₂×C₄²	C₂×D₄₇	1	2	yes	—
240	#41	—	2	4	no	—
241	#41	—	94	8	no	—
242	#41	—	188	8	no	—
243	C₂×Q₈	—	47	2	no	—
244	C₂×Q₈	—	47	4	no	—
245	C₂×Q₈	—	47	4	no	—
246	C₂×Q₈	—	188	4	no	—
247	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₂×D₄	—	47	4	no	—
249	C₂×D₄	—	47	8	no	—
250	C₂×D₄	—	47	2	no	—
251	C₂×D₄	—	47	2	no	—
252	C₂×D₄	—	47	2	no	—
253	C₂×D₄	C₂²×D₄₇	1	2	yes	—
254	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	—	47	4	no	—
256	C₂²×C₄	—	47	4	no	—
257	C₂²×C₄	—	47	8	no	—
258	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
260	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
261	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
263	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	—	47	8	no	—
265	C₄⋊C₄	—	47	8	no	—
266	C₄⋊C₄	—	47	4	no	—
267	C₄⋊C₄	—	47	4	no	—
268	C₄⋊C₄	—	47	4	no	—
269	C₄⋊C₄	—	47	4	no	—
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	47	2	no	—
272	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
275	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
276	C₂²⋊C₄	—	47	8	no	—
277	C₂²⋊C₄	—	47	8	no	—
278	C₂²⋊C₄	—	47	8	no	—
279	C₂²⋊C₄	—	47	4	no	—
280	C₂²⋊C₄	—	47	4	no	—
281	C₂²⋊C₄	C₂²×D₄₇	1	2	yes	—
282	C₂²⋊C₄	C₂²×D₄₇	1	2	yes	—
283	C₂²⋊C₄	C₂²×D₄₇	1	2	yes	—
284	C₄²	—	47	8	no	—
285	C₄²	C₂²×D₄₇	1	1	yes	characteristic
286	C₄²	C₂²×D₄₇	1	2	yes	—
287	C₄²	C₂²×D₄₇	1	1	yes	characteristic
288	C₄²	C₂²×D₄₇	1	2	yes	—
289	C₄²	C₂²×D₄₇	1	1	yes	characteristic
290	C₄²	C₂²×D₄₇	1	1	yes	characteristic
291	C₄²	C₂²×D₄₇	1	2	yes	—
292	C₂³	—	47	4	no	—
293	C₂³	C₂³×D₄₇	1	2	yes	—
294	C₂³	C₂³×D₄₇	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	Q₈	—	94	4	no	—
297	Q₈	—	94	8	no	—
298	Q₈	—	188	4	no	—
299	D₄	—	2	4	no	—
300	D₄	—	2	2	no	—
301	D₄	—	94	8	no	—
302	D₄	—	94	4	no	—
303	D₄	—	94	4	no	—
304	D₄	—	94	4	no	—
305	D₄	—	188	8	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	47	4	no	—
312	C₂×C₄	—	47	8	no	—
313	C₂×C₄	—	94	8	no	—
314	C₂×C₄	—	188	8	no	—
315	C₂×C₄	—	188	4	no	—
316	C₂×C₄	#10625350	1	4	yes	complemented
317	C₂×C₄	C₂³×D₄₇	1	2	yes	—
318	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₄₇	1	2	yes	—
321	C₂×C₄	C₂³×D₄₇	1	2	yes	—
322	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	94	4	no	—
332	C₂²	—	94	4	no	—
333	C₂²	—	94	4	no	—
334	C₂²	C₂×#10625350	1	2	yes	—
335	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	94	4	no	—
342	C₄	—	94	8	no	—
343	C₄	C₂×#10625350	1	2	yes	—
344	C₄	C₂×#10625350	1	2	yes	—
345	C₄	#13094070	1	2	yes	—
346	C₄	#13094071	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	4	2	no	—
349	C₂	—	188	4	no	—
350	C₂	C₂²×#10625350	1	1	yes	characteristic
351	C₂	C₂×#13094070	1	1	yes	characteristic
352	C₂	C₂×#13094071	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄×#41×C₄₇	C₂
hypercenter	C₄×#41	D₄₇
hypocenter	C₄₇	#3511
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #15689846	C₂	semidirect
2. #15689838	C₂	semidirect
3. #15689833	C₂	semidirect
4. #15689819	C₂	semidirect
5. #15689782	C₂	semidirect
6. #15689778	C₂	semidirect
7. #15689744	C₂	semidirect
8. #15689701	C₂	semidirect
9. #15689680	C₂	semidirect
10. #15689673	C₂	semidirect
11. #15689673	C₂	semidirect
12. C₂×#13094024	C₂	semidirect
13. #15689603	C₂	semidirect
14. C₄×#41×C₄₇	C₂	semidirect
126. C₂×C₄×C₄₇	#41	semidirect
190. C₄×#41	D₄₇	semidirect
191. C₄₇	#3511	semidirect
316. C₂×C₄	#10625350	semidirect