Subgroups — #20257204

Order 6016.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689853	C₂	1	2	yes	maximal, complemented
2	#15689848	C₂	1	2	yes	maximal, complemented
3	#15689834	C₂	1	2	yes	maximal, complemented
4	#15689828	C₂	1	4	yes	maximal, complemented
5	#15689788	C₂	1	2	yes	maximal, complemented
6	Q₈×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
7	#15689754	C₂	1	2	yes	maximal, complemented
8	#15689750	C₂	1	4	yes	maximal, complemented
9	#15689680	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689678	C₂	1	2	yes	maximal, complemented
11	C₄⋊Q₈×D₄₇	C₂	1	2	yes	maximal, complemented
12	#15689604	C₂	1	2	yes	maximal
13	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#10625350	C₂²	1	1	yes	characteristic
16	C₂×Q₈×D₄₇	C₂²	1	4	yes	—
17	#13094024	C₂²	1	1	yes	characteristic
18	#13094024	C₂²	1	1	yes	characteristic
19	#13094023	C₂²	1	4	yes	—
20	#13094023	C₂²	1	4	yes	—
21	#13094023	C₂²	1	4	yes	—
22	#13094022	C₂²	1	8	yes	—
23	#13094022	C₂²	1	4	yes	—
24	#13094016	C₂²	1	2	yes	—
25	#13094015	C₂²	1	4	yes	—
26	#13094015	C₂²	1	4	yes	—
27	#13094014	C₂²	1	8	yes	—
28	#13094004	C₂²	1	4	yes	—
29	#13094003	C₂²	1	8	yes	—
30	#13094003	C₂²	1	2	yes	—
31	#13094003	C₂²	1	2	yes	—
32	#13094002	C₂²	1	8	yes	—
33	#13094000	C₂²	1	8	yes	—
34	#13094000	C₂²	1	4	yes	—
35	Q₈×Dic₄₇	C₂²	1	4	yes	—
36	C₄×Dic₉₄	C₂²	1	2	yes	—
37	C₄×Dic₉₄	C₂²	1	2	yes	—
38	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
40	#13093989	C₂²	1	4	yes	—
41	#13093989	C₂²	1	4	yes	—
42	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
43	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
44	#13093986	C₂²	1	4	yes	—
45	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
46	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	—
47	C₄×C₂²⋊D₄₇	C₂²	1	2	yes	—
48	#13093983	C₂²	1	1	yes	characteristic
49	#13093983	C₂²	1	2	yes	—
50	#13093982	C₂²	1	4	yes	—
51	#13093981	C₂²	1	2	yes	—
52	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
53	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
54	C₄²×D₄₇	C₂²	1	2	yes	—
55	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
56	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
57	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
58	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
59	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
63	C₂×Q₈×C₄₇	—	4	4	no	—
64	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
65	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
71	Q₈×D₄₇	—	4	8	no	—
72	#10625350	Q₈	1	8	yes	complemented
73	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₄₇	C₂³	1	2	yes	—
75	C₂×C₄×D₄₇	C₂³	1	2	yes	—
76	C₂×C₄×D₄₇	C₂³	1	8	yes	—
77	C₂×C₄×D₄₇	C₂³	1	2	yes	—
78	C₂×Dic₉₄	C₂³	1	2	yes	—
79	C₂×Dic₉₄	C₂³	1	2	yes	—
80	C₂×Dic₉₄	C₂³	1	8	yes	—
81	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
83	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
84	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
85	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
86	C₄²×C₄₇	C₂³	1	1	yes	characteristic
87	C₄²×C₄₇	C₂³	1	1	yes	characteristic
88	C₄²×C₄₇	C₂³	1	1	yes	characteristic
89	C₄²×C₄₇	C₂³	1	1	yes	characteristic
90	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
91	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
92	(C₂×D₄₇)⋊C₄	C₂³	1	2	yes	—
93	(C₂×D₄₇)⋊C₄	C₂³	1	2	yes	—
94	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
95	C₄⋊Dic₄₇	C₂³	1	2	yes	—
96	C₄⋊Dic₄₇	C₂³	1	2	yes	—
97	C₄⋊Dic₄₇	C₂³	1	2	yes	—
98	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₄₇	C₂³	1	2	yes	—
101	C₄⋊Dic₄₇	C₂³	1	8	yes	—
102	Dic₄₇⋊C₄	C₂³	1	8	yes	—
103	Dic₄₇⋊C₄	C₂³	1	8	yes	—
104	Dic₄₇⋊C₄	C₂³	1	2	yes	—
105	Dic₄₇⋊C₄	C₂³	1	2	yes	—
106	Dic₄₇⋊C₄	C₂³	1	8	yes	—
107	Dic₄₇⋊C₄	C₂³	1	4	yes	—
108	Dic₄₇⋊C₄	C₂³	1	4	yes	—
109	C₄×Dic₄₇	C₂³	1	2	yes	—
110	C₄×Dic₄₇	C₂³	1	2	yes	—
111	C₄×Dic₄₇	C₂³	1	2	yes	—
112	C₄×Dic₄₇	C₂³	1	8	yes	—
113	C₄×Dic₄₇	C₂³	1	4	yes	—
114	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂²×D₄₇	C₂⁴	1	2	yes	—
116	Q₈×C₄₇	—	2	8	no	—
117	Q₈×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	complemented
122	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
123	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₄₇	C₂×Q₈	1	8	yes	complemented
130	C₂×Dic₄₇	C₂⁴	1	4	yes	—
131	C₂×Dic₄₇	C₂⁴	1	2	yes	—
132	C₂×Dic₄₇	C₂⁴	1	2	yes	—
133	C₂×Dic₄₇	C₂⁴	1	4	yes	—
134	C₂×Dic₄₇	C₂⁴	1	2	yes	—
135	D₁₈₈	C₂×Q₈	1	4	yes	complemented
136	C₄×D₄₇	—	2	4	no	—
137	C₄×D₄₇	—	2	4	no	—
138	C₄×D₄₇	—	4	8	no	—
139	C₄×D₄₇	C₂×Q₈	1	8	yes	complemented
140	Dic₉₄	—	2	4	no	—
141	Dic₉₄	—	2	4	no	—
142	Dic₉₄	—	4	8	no	—
143	Dic₉₄	C₂×Q₈	1	4	yes	complemented
144	C₂²×C₄₇	—	2	1	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
147	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
148	C₂×D₄₇	—	2	2	no	—
149	C₂×D₄₇	—	2	2	no	—
150	C₂×D₄₇	C₂²×Q₈	1	4	yes	—
151	C₄×C₄₇	—	2	8	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
155	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
156	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
157	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
158	Dic₄₇	—	2	4	no	—
159	Dic₄₇	—	2	2	no	—
160	Dic₄₇	—	2	4	no	—
161	Dic₄₇	—	2	2	no	—
162	Dic₄₇	C₂²×Q₈	1	4	yes	—
163	#3503	—	47	1	no	maximal, Sylow
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
166	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
168	D₄₇	—	2	4	no	—
169	Q₈²	—	47	2	no	—
170	#557	—	47	2	no	—
171	#554	—	47	2	no	—
172	#554	—	47	4	no	—
173	D₄×Q₈	—	47	4	no	—
174	D₄×Q₈	—	47	2	no	—
175	#544	—	47	4	no	—
176	#544	—	47	2	no	—
177	#533	—	47	2	no	—
178	#533	—	47	1	no	—
179	C₂×C₄⋊Q₈	—	47	2	no	—
180	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
181	#520	—	47	2	no	—
182	C₄×#41	—	47	1	no	—
183	C₄₇	#3503	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	47	1	no	—
185	C₂²×Q₈	—	47	4	no	—
186	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
187	C₄⋊Q₈	—	47	4	no	—
188	C₄⋊Q₈	—	47	8	no	—
189	C₄⋊Q₈	—	47	4	no	—
190	C₄⋊Q₈	—	47	4	no	—
191	C₄⋊Q₈	—	47	4	no	—
192	C₄⋊Q₈	—	47	1	no	—
193	C₄⋊Q₈	—	47	1	no	—
194	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
195	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
196	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
197	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
198	#125	—	47	4	no	—
199	#125	—	47	8	no	—
200	#125	—	47	4	no	—
201	#125	—	47	2	no	—
202	C₂²⋊Q₈	—	47	4	no	—
203	C₂²⋊Q₈	—	47	8	no	—
204	C₂²⋊Q₈	—	47	2	no	—
205	C₂²⋊Q₈	—	47	2	no	—
206	C₂²⋊Q₈	—	47	8	no	—
207	C₂²⋊Q₈	—	47	4	no	—
208	C₂²⋊Q₈	—	47	8	no	—
209	C₄×Q₈	—	47	4	no	—
210	C₄×Q₈	—	47	4	no	—
211	C₄×Q₈	—	47	4	no	—
212	C₄×Q₈	—	47	1	no	—
213	C₄×Q₈	—	47	1	no	—
214	C₄×Q₈	—	47	2	no	—
215	C₄×Q₈	—	47	2	no	—
216	C₄×D₄	—	47	2	no	—
217	C₄×D₄	—	47	2	no	—
218	C₄×D₄	—	47	8	no	—
219	C₄×D₄	—	47	4	no	—
220	C₄×D₄	—	47	1	no	—
221	C₄×D₄	—	47	1	no	—
222	#117	—	47	1	no	—
223	#117	—	47	2	no	—
224	#117	—	47	4	no	—
225	#117	—	47	2	no	—
226	C₂×C₄⋊C₄	—	47	4	no	—
227	C₂×C₄⋊C₄	—	47	4	no	—
228	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
232	#41	—	47	8	no	—
233	C₂×Q₈	—	4	4	no	—
234	C₂×Q₈	—	47	1	no	—
235	C₂×Q₈	—	47	2	no	—
236	C₂×Q₈	—	47	2	no	—
237	C₂×Q₈	—	47	8	no	—
238	C₂×Q₈	—	188	8	no	—
239	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
240	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
241	C₂×D₄	—	47	2	no	—
242	C₂×D₄	—	47	1	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	47	2	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	8	no	—
247	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
250	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
251	C₄⋊C₄	—	47	8	no	—
252	C₄⋊C₄	—	47	8	no	—
253	C₄⋊C₄	—	47	2	no	—
254	C₄⋊C₄	—	47	2	no	—
255	C₄⋊C₄	—	47	4	no	—
256	C₄⋊C₄	—	47	8	no	—
257	C₄⋊C₄	—	47	4	no	—
258	C₄⋊C₄	—	47	2	no	—
259	C₄⋊C₄	—	47	2	no	—
260	C₄⋊C₄	—	47	2	no	—
261	C₄⋊C₄	—	47	1	no	—
262	C₄⋊C₄	—	47	1	no	—
263	C₄⋊C₄	—	47	2	no	—
264	C₄⋊C₄	—	47	8	no	—
265	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
266	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
267	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
268	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
269	C₂²⋊C₄	—	47	4	no	—
270	C₂²⋊C₄	—	47	2	no	—
271	C₂²⋊C₄	—	47	8	no	—
272	C₂²⋊C₄	—	47	2	no	—
273	C₂²⋊C₄	—	47	2	no	—
274	C₄²	—	47	2	no	—
275	C₄²	—	47	2	no	—
276	C₄²	—	47	2	no	—
277	C₄²	—	47	8	no	—
278	C₄²	—	47	4	no	—
279	C₄²	C₂²×D₄₇	1	1	yes	characteristic
280	C₄²	C₂²×D₄₇	1	1	yes	characteristic
281	C₄²	C₂²×D₄₇	1	1	yes	characteristic
282	C₄²	C₂²×D₄₇	1	1	yes	characteristic
283	C₂³	—	47	2	no	—
284	C₂³	C₂³×D₄₇	1	1	yes	characteristic
285	Q₈	—	2	8	no	—
286	Q₈	—	4	4	no	—
287	Q₈	—	47	4	no	—
288	Q₈	—	94	4	no	—
289	Q₈	—	94	4	no	—
290	Q₈	—	188	8	no	—
291	D₄	—	47	8	no	—
292	D₄	—	47	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	47	2	no	—
297	C₂×C₄	—	47	8	no	—
298	C₂×C₄	—	47	4	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	4	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	188	8	no	—
305	C₂×C₄	Q₈×D₄₇	1	4	yes	complemented
306	C₂×C₄	C₂³×D₄₇	1	8	yes	—
307	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	47	4	no	—
316	C₂²	—	94	2	no	—
317	C₂²	—	94	2	no	—
318	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
319	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	8	no	—
322	C₄	—	2	1	no	—
323	C₄	—	47	4	no	—
324	C₄	—	94	2	no	—
325	C₄	—	94	4	no	—
326	C₄	—	94	2	no	—
327	C₄	—	94	4	no	—
328	C₄	C₂×Q₈×D₄₇	1	2	yes	—
329	C₄	C₂×Q₈×D₄₇	1	2	yes	—
330	C₄	#13094069	1	2	yes	—
331	C₄	#13094072	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	94	4	no	—
334	C₂	C₂²×Q₈×D₄₇	1	1	yes	characteristic
335	C₂	C₂×#13094069	1	1	yes	characteristic
336	C₂	C₂×#13094072	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#3503
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15689853	C₂	semidirect
2. #15689848	C₂	semidirect
3. #15689834	C₂	semidirect
4. #15689828	C₂	semidirect
5. #15689788	C₂	semidirect
6. Q₈×C₂²⋊D₄₇	C₂	semidirect
7. #15689754	C₂	semidirect
8. #15689750	C₂	semidirect
9. #15689680	C₂	semidirect
10. #15689678	C₂	semidirect
11. C₄⋊Q₈×D₄₇	C₂	semidirect
14. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
72. #10625350	Q₈	semidirect
121. C₂×C₄×C₄₇	C₂×Q₈	semidirect
129. C₂²⋊D₄₇	C₂×Q₈	semidirect
135. D₁₈₈	C₂×Q₈	semidirect
139. C₄×D₄₇	C₂×Q₈	semidirect
143. Dic₉₄	C₂×Q₈	semidirect
180. C₂×C₄⋊Q₈	D₄₇	semidirect
183. C₄₇	#3503	semidirect
305. C₂×C₄	Q₈×D₄₇	semidirect