Subgroups — #20252419

Order 6016.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689198	C₂	1	2	yes	maximal, complemented
3	#15689197	C₂	1	2	yes	maximal, complemented
4	#15689194	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₄₇	C₂	1	2	yes	maximal, complemented
6	#15689177	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689173	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689087	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689085	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
14	C₂×#10625329	C₂²	1	2	yes	complemented
15	C₂×#10625324	C₂²	1	2	yes	complemented
16	C₂×C₈×D₄₇	C₂²	1	2	yes	complemented
17	C₄×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₄₇⋊C₄	C₄	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₄₇	C₄	1	1	yes	characteristic
27	C₂×(C₂×D₄₇)⋊C₄	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
29	Dic₄₇⋊C₈	C₂²	1	2	yes	complemented
30	Dic₄₇⋊C₈	C₂²	1	2	yes	complemented
31	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
33	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
34	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	complemented
35	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	complemented
36	#13093934	C₂²	1	2	yes	complemented
37	C₈×Dic₄₇	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
42	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	complemented
43	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₄₇	—	2	4	no	—
53	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₄₇	—	4	2	no	—
61	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	C₂³	1	2	yes	—
67	C₂×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
72	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
73	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
74	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
75	(C₂×D₄₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₄₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
84	#10625329	—	4	2	no	—
85	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
86	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
87	#10625324	—	4	2	no	—
88	C₈×D₄₇	—	4	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	4	1	no	—
92	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₄₇	—	2	2	no	—
96	C₂²×D₄₇	—	2	2	no	—
97	C₂²×D₄₇	—	4	1	no	—
98	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	4	1	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₄₇	—	2	2	no	—
111	C₂²⋊D₄₇	—	2	2	no	—
112	C₂²⋊D₄₇	—	2	2	no	—
113	C₂²⋊D₄₇	M₁₆	1	4	yes	complemented
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	4	1	no	—
117	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	C₄×D₄₇	—	2	2	no	—
124	C₄×D₄₇	—	2	2	no	—
125	C₄×D₄₇	—	4	1	no	—
126	C₄×D₄₇	—	4	1	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	—	2	2	no	—
129	C₄₇⋊C₈	—	2	2	no	—
130	C₄₇⋊C₈	—	2	2	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	4	4	no	—
139	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
140	C₂²×C₄₇	#131	1	2	yes	—
141	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×D₄₇	—	2	1	no	—
143	C₂×D₄₇	—	2	1	no	—
144	C₂×D₄₇	—	2	1	no	—
145	C₂×D₄₇	—	2	1	no	—
146	C₂×D₄₇	—	2	1	no	—
147	C₂×D₄₇	—	4	4	no	—
148	C₂×D₄₇	C₂×M₁₆	1	2	yes	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
153	Dic₄₇	—	2	1	no	—
154	Dic₄₇	—	2	1	no	—
155	Dic₄₇	—	2	1	no	—
156	Dic₄₇	—	2	1	no	—
157	Dic₄₇	—	2	1	no	—
158	Dic₄₇	C₂×M₁₆	1	2	yes	—
159	#2974	—	47	1	no	maximal, Sylow
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	4	1	no	—
163	C₂×C₄₇	#518	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
166	D₄₇	—	2	2	no	—
167	D₄₇	—	4	1	no	—
168	C₂×C₄×D₄	—	47	1	no	—
169	#436	—	47	2	no	—
170	#435	—	47	2	no	—
171	#435	—	47	2	no	—
172	C₈×D₄	—	47	2	no	—
173	#432	—	47	1	no	—
174	#431	—	47	1	no	—
175	#407	—	47	1	no	—
176	#407	—	47	1	no	—
177	C₂×C₂²⋊C₈	—	47	1	no	—
178	C₂×C₂²⋊C₈	D₄₇	1	1	yes	characteristic, complemented
179	C₄₇	#2974	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₄	—	47	1	no	—
181	C₂³×C₄	—	47	1	no	—
182	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
183	C₂×M₁₆	—	47	2	no	—
184	C₂×M₁₆	—	47	2	no	—
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	C₂×D₄₇	1	2	yes	complemented
187	C₄×D₄	—	47	1	no	—
188	C₄×D₄	—	47	1	no	—
189	C₄×D₄	—	47	1	no	—
190	C₄×D₄	—	47	1	no	—
191	C₄×D₄	—	47	1	no	—
192	C₄×D₄	—	47	1	no	—
193	C₄×D₄	—	47	1	no	—
194	C₄×D₄	—	47	1	no	—
195	C₂×C₄⋊C₄	—	47	1	no	—
196	C₂×C₂²⋊C₄	—	47	1	no	—
197	C₂×C₂²⋊C₄	—	47	1	no	—
198	C₂×C₄²	—	47	1	no	—
199	C₄⋊C₈	—	47	2	no	—
200	C₄⋊C₈	—	47	2	no	—
201	C₂²⋊C₈	—	47	1	no	—
202	C₂²⋊C₈	—	47	1	no	—
203	C₂²⋊C₈	—	47	2	no	—
204	C₂²⋊C₈	—	47	2	no	—
205	C₂²⋊C₈	—	47	2	no	—
206	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
207	C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
208	C₂²⋊C₈	C₂×D₄₇	1	2	yes	complemented
209	#97	—	47	2	no	—
210	C₄×C₈	—	47	2	no	—
211	C₂⁴	—	47	1	no	—
212	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
213	C₂×D₄	—	47	1	no	—
214	C₂×D₄	—	47	1	no	—
215	C₂×D₄	—	47	1	no	—
216	C₂×D₄	—	47	1	no	—
217	C₂×D₄	—	94	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	47	1	no	—
220	C₂²×C₄	—	47	1	no	—
221	C₂²×C₄	—	47	1	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	1	no	—
225	C₂²×C₄	—	47	1	no	—
226	C₂²×C₄	—	188	2	no	—
227	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
233	M₁₆	—	188	2	no	—
234	M₁₆	—	188	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	47	2	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	C₂²×D₄₇	1	2	yes	—
240	C₂×C₈	C₂²×D₄₇	1	2	yes	—
241	C₄⋊C₄	—	47	1	no	—
242	C₄⋊C₄	—	47	1	no	—
243	C₄⋊C₄	—	47	1	no	—
244	C₄⋊C₄	—	47	1	no	—
245	C₂²⋊C₄	—	47	1	no	—
246	C₂²⋊C₄	—	47	1	no	—
247	C₂²⋊C₄	—	47	1	no	—
248	C₂²⋊C₄	—	47	1	no	—
249	C₂²⋊C₄	—	47	1	no	—
250	C₂²⋊C₄	—	47	1	no	—
251	C₂²⋊C₄	—	47	1	no	—
252	C₂²⋊C₄	—	47	1	no	—
253	C₄²	—	47	1	no	—
254	C₄²	—	47	1	no	—
255	C₄²	—	47	1	no	—
256	C₄²	—	47	1	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	47	1	no	—
261	C₂³	—	47	1	no	—
262	C₂³	—	94	2	no	—
263	C₂³	—	94	2	no	—
264	C₂³	—	188	1	no	—
265	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
268	D₄	—	47	4	no	—
269	D₄	—	94	2	no	—
270	D₄	—	94	2	no	—
271	D₄	—	94	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	47	1	no	—
279	C₂×C₄	—	47	1	no	—
280	C₂×C₄	—	47	1	no	—
281	C₂×C₄	—	47	1	no	—
282	C₂×C₄	—	47	1	no	—
283	C₂×C₄	—	47	1	no	—
284	C₂×C₄	—	94	2	no	—
285	C₂×C₄	—	94	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	188	1	no	—
289	C₂×C₄	—	188	1	no	—
290	C₂×C₄	—	188	1	no	—
291	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	94	2	no	—
298	C₈	—	94	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	47	2	no	—
308	C₂²	—	94	1	no	—
309	C₂²	—	94	1	no	—
310	C₂²	—	94	1	no	—
311	C₂²	—	94	1	no	—
312	C₂²	—	94	1	no	—
313	C₂²	—	188	4	no	—
314	C₂²	M₁₆×D₄₇	1	2	yes	—
315	C₂²	#13094030	1	2	yes	—
316	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	47	2	no	—
321	C₄	—	94	1	no	—
322	C₄	—	94	1	no	—
323	C₄	—	94	1	no	—
324	C₄	—	94	1	no	—
325	C₄	—	94	1	no	—
326	C₄	#13094069	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	94	2	no	—
331	C₂	—	188	1	no	—
332	C₂	#15689596	1	1	yes	characteristic
333	C₂	C₂×M₁₆×D₄₇	1	1	yes	characteristic
334	C₂	C₂×#13094030	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₄₇	C₂
hypercenter	C₂×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#2974
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #15689198	C₂	semidirect
3. #15689197	C₂	semidirect
4. #15689194	C₂	semidirect
5. C₈×C₂²⋊D₄₇	C₂	semidirect
6. #15689177	C₂	semidirect
7. #15689173	C₂	semidirect
8. #15689087	C₂	semidirect
9. #15689085	C₂	semidirect
10. C₂²⋊C₈×D₄₇	C₂	semidirect
11. C₂×C₂²⋊C₈×C₄₇	C₂	semidirect
14. C₂×#10625329	C₂²	semidirect
15. C₂×#10625324	C₂²	semidirect
16. C₂×C₈×D₄₇	C₂²	semidirect
29. Dic₄₇⋊C₈	C₂²	semidirect
30. Dic₄₇⋊C₈	C₂²	semidirect
33. C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
34. (C₂×D₄₇)⋊C₈	C₂²	semidirect
35. (C₂×D₄₇)⋊C₈	C₂²	semidirect
36. #13093934	C₂²	semidirect
37. C₈×Dic₄₇	C₂²	semidirect
39. C₂²×C₈×C₄₇	C₂²	semidirect
42. C₂²⋊C₈×C₄₇	C₂²	semidirect
113. C₂²⋊D₄₇	M₁₆	semidirect
178. C₂×C₂²⋊C₈	D₄₇	semidirect
179. C₄₇	#2974	semidirect
186. C₂²×C₈	C₂×D₄₇	semidirect
208. C₂²⋊C₈	C₂×D₄₇	semidirect