Subgroups — #26199493

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254534	C₂	1	2	yes	maximal
2	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
3	#20252322	C₂	1	1	yes	maximal, characteristic
4	#20242883	C₂	1	4	yes	maximal, complemented
5	#20242759	C₂	1	4	yes	maximal, complemented
6	#20242059	C₂	1	2	yes	maximal
7	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₈₈	—	2	1	no	—
9	C₂²×Dic₁₈₈	—	2	1	no	—
10	#15689680	C₂²	1	1	yes	characteristic
11	C₂×#13094024	C₂²	1	2	yes	—
12	#15689521	—	2	2	no	—
13	#15689503	—	2	1	no	—
14	#15689503	—	2	1	no	—
15	#15689503	C₂²	1	2	yes	—
16	#15689503	C₂²	1	2	yes	—
17	#15689474	—	2	2	no	—
18	#15689454	—	2	4	no	—
19	#15689409	C₂²	1	8	yes	—
20	#15689253	—	2	1	no	—
21	#15689253	C₂²	1	2	yes	—
22	C₈×Dic₉₄	—	2	1	no	—
23	C₈×Dic₉₄	C₂²	1	2	yes	—
24	#15689149	—	2	1	no	—
25	#15689147	C₂²	1	1	yes	characteristic
26	#15689078	—	2	1	no	—
27	#15688868	C₂²	1	8	yes	—
28	C₄₇×#431	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#335	C₂²	1	2	yes	—
31	C₄₇×#334	C₂²	1	2	yes	—
32	C₂²×Dic₉₄	—	2	2	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	—	2	2	no	—
35	C₂×Dic₁₈₈	—	2	2	no	—
36	C₂×Dic₁₈₈	—	2	2	no	—
37	C₂×Dic₁₈₈	—	4	4	no	—
38	C₂×Dic₁₈₈	—	4	4	no	—
39	C₂×#10625329	—	2	1	no	—
40	#13094024	—	2	2	no	—
41	#13094024	—	2	2	no	—
42	#13094024	—	2	2	no	—
43	#13094024	D₄	1	1	yes	characteristic
44	#13094024	D₄	1	1	yes	characteristic
45	#13094024	C₂³	1	4	yes	—
46	#13094016	—	2	1	no	—
47	#13094003	—	2	1	no	—
48	#13094003	D₄	1	2	yes	—
49	C₄×Dic₉₄	—	2	1	no	—
50	C₄×Dic₉₄	C₂³	1	2	yes	—
51	#13093983	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	2	no	—
53	C₂×C₄⋊Dic₄₇	—	2	2	no	—
54	#13093956	—	2	8	no	—
55	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
56	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
57	Dic₄₇⋊C₈	—	2	2	no	—
58	Dic₄₇⋊C₈	—	2	2	no	—
59	#13093949	—	2	8	no	—
60	#13093934	—	2	2	no	—
61	C₈×Dic₄₇	—	2	2	no	—
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
66	C₂²⋊C₈×C₄₇	—	2	2	no	—
67	C₄×C₈×C₄₇	—	2	2	no	—
68	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂×Dic₉₄	—	2	1	no	—
77	C₂×Dic₉₄	—	2	4	no	—
78	C₂×Dic₉₄	—	4	4	no	—
79	C₂×Dic₉₄	—	4	4	no	—
80	C₂×Dic₉₄	—	4	2	no	—
81	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	1	no	—
95	C₄⋊Dic₄₇	—	2	1	no	—
96	C₄⋊Dic₄₇	—	2	1	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	4	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
102	Dic₄₇⋊C₄	—	2	2	no	—
103	Dic₄₇⋊C₄	—	2	2	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	#10625329	—	4	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	Dic₁₈₈	—	4	4	no	—
108	Dic₁₈₈	—	4	4	no	—
109	Dic₁₈₈	—	4	4	no	—
110	Dic₁₈₈	—	4	4	no	—
111	C₂³×C₄₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	8	4	no	—
126	Dic₉₄	—	2	4	no	—
127	Dic₉₄	—	4	2	no	—
128	Dic₉₄	—	4	4	no	—
129	Dic₉₄	—	4	4	no	—
130	Dic₉₄	—	8	4	no	—
131	C₈×C₄₇	—	2	1	no	—
132	C₈×C₄₇	—	2	1	no	—
133	C₈×C₄₇	—	4	4	no	—
134	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
135	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
136	C₄₇⋊C₈	—	4	2	no	—
137	#19204	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	#460	1	2	yes	—
141	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	#460	1	2	yes	—
145	C₄×C₄₇	#473	1	2	yes	—
146	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
147	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₄₇	#578	1	2	yes	—
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	—	8	4	no	—
152	#3229	—	47	2	no	—
153	C₂×#494	—	47	1	no	—
154	#2960	—	47	1	no	—
155	#1690	—	47	4	no	—
156	#1670	—	47	4	no	—
157	#1564	D₄₇	1	1	yes	characteristic
158	#1546	—	47	2	no	—
159	C₂×C₄₇	—	2	2	no	—
160	C₂×C₄₇	C₂×#460	1	1	yes	characteristic
161	C₂×C₄₇	#3059	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
163	C₂²×Q₁₆	—	94	1	no	—
164	C₂²×Q₁₆	—	94	1	no	—
165	#533	—	47	1	no	—
166	C₂×C₄⋊Q₈	—	47	2	no	—
167	#498	—	94	2	no	—
168	#494	—	47	2	no	—
169	#494	—	47	2	no	—
170	#494	—	94	1	no	—
171	#494	—	94	1	no	—
172	#487	—	94	2	no	—
173	#483	—	94	4	no	—
174	#475	—	47	8	no	—
175	#446	—	47	2	no	—
176	#446	—	94	1	no	—
177	C₈×Q₈	—	47	2	no	—
178	C₈×Q₈	—	94	1	no	—
179	#431	C₂×D₄₇	1	2	yes	—
180	#424	—	94	1	no	—
181	#423	—	47	1	no	—
182	#405	—	94	1	no	—
183	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
184	#335	C₂×D₄₇	1	2	yes	—
185	#334	C₂×D₄₇	1	2	yes	—
186	#332	—	47	8	no	—
187	C₄₇	#19204	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	94	2	no	—
189	C₂×Q₁₆	—	94	2	no	—
190	C₂×Q₁₆	—	94	2	no	—
191	C₂×Q₁₆	—	94	2	no	—
192	C₂×Q₁₆	—	94	2	no	—
193	C₂×Q₁₆	—	188	4	no	—
194	C₂×Q₁₆	—	188	4	no	—
195	C₂×M₁₆	—	94	1	no	—
196	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
197	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
198	C₄⋊Q₈	—	47	1	no	—
199	C₄⋊Q₈	—	47	1	no	—
200	C₄⋊Q₈	—	47	4	no	—
201	C₄⋊Q₈	—	94	2	no	—
202	C₄⋊Q₈	—	94	2	no	—
203	C₄⋊Q₈	—	94	2	no	—
204	#125	—	94	1	no	—
205	C₂²⋊Q₈	—	47	2	no	—
206	C₂²⋊Q₈	—	94	1	no	—
207	C₄×Q₈	—	47	2	no	—
208	C₄×Q₈	—	94	1	no	—
209	#117	—	94	1	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	#106	—	94	8	no	—
214	C₄⋊C₈	—	47	2	no	—
215	C₄⋊C₈	—	94	2	no	—
216	C₄⋊C₈	—	94	2	no	—
217	C₄⋊C₈	—	94	1	no	—
218	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
219	#103	—	94	8	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	#97	—	94	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	—	94	2	no	—
224	C₄×C₈	D₁₈₈	1	1	yes	characteristic
225	C₄×C₈	D₁₈₈	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₂×Q₈	—	47	2	no	—
229	C₂×Q₈	—	94	1	no	—
230	C₂×Q₈	—	94	4	no	—
231	C₂×Q₈	—	188	4	no	—
232	C₂×Q₈	—	188	4	no	—
233	C₂×Q₈	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	#10625354	1	1	yes	characteristic
238	Q₁₆	—	188	4	no	—
239	Q₁₆	—	188	4	no	—
240	Q₁₆	—	188	4	no	—
241	Q₁₆	—	188	4	no	—
242	M₁₆	—	188	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	#10625325	1	4	yes	—
247	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
250	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
251	C₄⋊C₄	—	47	2	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	1	no	—
254	C₄⋊C₄	—	94	1	no	—
255	C₄⋊C₄	—	94	1	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	94	4	no	—
259	C₄⋊C₄	—	188	2	no	—
260	C₄⋊C₄	—	188	2	no	—
261	C₂²⋊C₄	—	94	2	no	—
262	C₄²	—	94	2	no	—
263	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
264	C₄²	D₄×D₄₇	1	1	yes	characteristic
265	C₄²	#10625354	1	1	yes	characteristic
266	C₄²	C₂³×D₄₇	1	1	yes	characteristic
267	C₂³	#13093999	1	1	yes	characteristic
268	Q₈	—	94	4	no	—
269	Q₈	—	376	4	no	—
270	Q₈	—	188	4	no	—
271	Q₈	—	188	4	no	—
272	Q₈	—	188	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	376	4	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	#13093999	1	1	yes	characteristic
280	C₂×C₄	#13093999	1	1	yes	characteristic
281	C₂×C₄	#13093999	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625325	1	2	yes	—
283	C₂×C₄	#13094052	1	2	yes	—
284	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
286	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	4	4	no	—
290	C₈	—	188	2	no	—
291	C₈	C₂×#10625325	1	2	yes	—
292	C₈	C₂×#10625325	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#15689324	1	2	yes	—
296	C₂²	C₂×#13093999	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	376	4	no	—
300	C₄	—	188	2	no	—
301	C₄	—	188	2	no	—
302	C₄	#15689324	1	2	yes	—
303	C₄	#15689399	1	2	yes	—
304	C₄	C₂²×#10625325	1	1	yes	characteristic
305	C₄	C₂×#13094052	1	1	yes	characteristic
306	C₄	#15689937	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×#15689324	1	1	yes	characteristic
309	C₂	#20253044	1	1	yes	characteristic
310	C₂	C₂×#15689399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093999
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂³×C₄₇	#121
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#19204
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #20242883	C₂	semidirect
5. #20242759	C₂	semidirect
187. C₄₇	#19204	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20254534	2
C₂×#15689503	2
#20252322	2
#20242883	2
#20242759	2
#20242059	2
C₄₇×#1564	2
#19204	47