Subgroups — #20246839

Order 6016.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689056	C₂	1	2	yes	maximal, complemented
5	#15689046	C₂	1	2	yes	maximal, complemented
6	#15689029	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689003	C₂	1	2	yes	maximal
9	#15688983	C₂	1	1	yes	maximal, characteristic, complemented
10	#15688981	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂³×Dic₄₇	C₂²	1	2	yes	—
14	#13094003	—	2	4	no	—
15	#13094003	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	4	no	—
17	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
27	C₄₇⋊#95	C₂²	1	2	yes	—
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₄₇	—	2	4	no	—
35	C₂×C₄⋊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	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
39	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₄₇	—	2	2	no	—
42	C₂×D₄×C₄₇	—	2	2	no	—
43	C₂×D₄×C₄₇	Q₈	1	4	yes	complemented
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	D₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
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	C₂×Dic₉₄	—	2	2	no	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₂²⋊C₄×C₄₇	—	2	2	no	—
67	C₂²⋊C₄×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₂²⋊Dic₄₇	—	2	4	no	—
70	C₂²⋊Dic₄₇	—	2	4	no	—
71	C₂²⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	—	2	2	no	—
74	C₄⋊Dic₄₇	—	2	4	no	—
75	C₄⋊Dic₄₇	D₄	1	4	yes	complemented
76	Dic₄₇⋊C₄	—	2	4	no	—
77	Dic₄₇⋊C₄	—	2	4	no	—
78	Dic₄₇⋊C₄	—	4	2	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	2	1	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	C₂×D₄	1	2	yes	—
90	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
92	C₂³×C₄₇	#41	1	2	yes	—
93	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
94	D₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	8	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	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#41	1	2	yes	—
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	—	2	2	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	4	8	no	—
124	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
125	Dic₉₄	—	4	4	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	4	1	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	—	4	1	no	—
138	C₂²×C₄₇	—	4	1	no	—
139	C₂²×C₄₇	C₂²⋊Q₈	1	4	yes	complemented
140	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	4	1	no	—
150	C₄×C₄₇	—	4	1	no	—
151	C₄×C₄₇	#121	1	4	yes	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	4	2	no	—
154	Dic₄₇	—	4	2	no	—
155	Dic₄₇	—	4	2	no	—
156	#2417	—	47	1	no	maximal, Sylow
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	—	4	2	no	—
159	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
160	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₄₇	#537	1	1	yes	characteristic
162	C₂×C₄₇	#546	1	1	yes	characteristic
163	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
164	C₂×C₄₇	#552	1	1	yes	characteristic
165	C₂×C₄₇	#554	1	1	yes	characteristic
166	C₂×C₂²⋊Q₈	—	47	1	no	—
167	C₂×C₂²⋊Q₈	—	47	1	no	—
168	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
169	C₂²×C₄⋊C₄	—	47	1	no	—
170	#398	—	47	2	no	—
171	#396	—	47	2	no	—
172	#391	—	47	1	no	—
173	#389	—	47	1	no	—
174	#386	—	47	2	no	—
175	#380	—	47	1	no	—
176	#380	—	47	2	no	—
177	C₄₇	#2417	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	47	1	no	—
179	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
180	C₂³×C₄	—	47	2	no	—
181	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
182	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
183	C₂²⋊Q₈	—	94	4	no	—
184	C₂²⋊Q₈	—	94	4	no	—
185	C₄×D₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	47	1	no	—
187	C₂×C₄⋊C₄	—	47	1	no	—
188	C₂×C₄⋊C₄	—	47	2	no	—
189	C₂×C₄⋊C₄	—	47	2	no	—
190	C₂×C₄⋊C₄	—	47	2	no	—
191	C₂×C₄⋊C₄	—	47	1	no	—
192	C₂×C₄⋊C₄	—	47	2	no	—
193	C₂×C₄⋊C₄	—	94	4	no	—
194	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	47	2	no	—
196	C₂×C₂²⋊C₄	—	47	2	no	—
197	C₂×C₂²⋊C₄	—	47	2	no	—
198	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
201	#95	—	47	2	no	—
202	#95	—	47	2	no	—
203	#95	—	47	2	no	—
204	#95	—	47	2	no	—
205	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
206	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×Q₈	—	94	2	no	—
208	C₂×Q₈	—	94	2	no	—
209	C₂×Q₈	—	94	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	Dic₉₄	1	4	yes	complemented
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	47	2	no	—
219	C₂²×C₄	—	47	2	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	188	2	no	—
225	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	47	4	no	—
233	C₄⋊C₄	—	94	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	94	2	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₄⋊C₄	—	94	2	no	—
238	C₄⋊C₄	—	188	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	94	4	no	—
242	C₂²⋊C₄	—	94	4	no	—
243	C₂²⋊C₄	—	188	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	C₂×Dic₉₄	1	2	yes	—
256	C₂³	C₂×Dic₉₄	1	2	yes	—
257	C₂³	#10625350	1	2	yes	—
258	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
259	C₂³	C₂³×D₄₇	1	1	yes	characteristic
260	Q₈	—	188	4	no	—
261	D₄	—	2	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	47	4	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	—	94	2	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	94	2	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	188	8	no	—
287	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
288	C₂×C₄	D₄×D₄₇	1	2	yes	—
289	C₂×C₄	#10625352	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	#13094003	1	4	yes	complemented
306	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
307	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
308	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
309	C₂²	C₂×#10625352	1	1	yes	characteristic
310	C₂²	C₂×#10625350	1	1	yes	characteristic
311	C₂²	#13094070	1	1	yes	characteristic
312	C₂²	#13094071	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	94	4	no	—
318	C₄	—	188	2	no	—
319	C₄	—	188	2	no	—
320	C₄	—	188	2	no	—
321	C₄	#13093996	1	4	yes	—
322	C₂	—	2	4	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#13093996	1	1	yes	characteristic
325	C₂	C₂×#13094003	1	1	yes	characteristic
326	C₂	#15689701	1	1	yes	characteristic
327	C₂	#15689769	1	1	yes	characteristic
328	C₂	D₄×Dic₉₄	1	1	yes	characteristic
329	C₂	#15689813	1	1	yes	characteristic
330	C₂	#15689830	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×D₄×C₄₇	C₂
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#2417
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13094003	C₂	semidirect
2. C₂×#13094003	C₂	semidirect
3. C₂²×C₄⋊Dic₄₇	C₂	semidirect
4. #15689056	C₂	semidirect
5. #15689046	C₂	semidirect
6. #15689029	C₂	semidirect
7. #15689021	C₂	semidirect
9. #15688983	C₂	semidirect
10. #15688981	C₂	semidirect
43. C₂×D₄×C₄₇	Q₈	semidirect
75. C₄⋊Dic₄₇	D₄	semidirect
139. C₂²×C₄₇	C₂²⋊Q₈	semidirect
177. C₄₇	#2417	semidirect
212. C₂×D₄	Dic₉₄	semidirect
305. C₂²	#13094003	semidirect