Subgroups — #15850854

Order 3328.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125957	C₂	1	2	yes	maximal, complemented
2	#13125415	C₂	1	1	yes	maximal, characteristic
3	#13125293	C₂	1	2	yes	maximal, complemented
4	#13114506	C₂	1	4	yes	maximal, complemented
5	#13114453	C₂	1	2	yes	maximal, complemented
6	#13114429	C₂	1	2	yes	maximal, complemented
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×#88196	—	2	4	no	—
11	#11719182	C₂²	1	2	yes	—
12	C₁₃×#500	—	2	2	no	—
13	C₁₃×#500	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	2	yes	—
15	C₂×C₁₃×#107	—	2	2	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718776	—	2	2	no	—
18	#11718753	C₂²	1	4	yes	—
19	#11718749	C₂²	1	4	yes	—
20	#11718748	—	2	4	no	—
21	#11718747	—	2	4	no	—
22	#11718746	C₂²	1	4	yes	—
23	#11718739	C₂²	1	4	yes	—
24	#11718628	—	2	4	no	—
25	#11718586	C₂²	1	1	yes	characteristic
26	C₄×#6099	C₂²	1	1	yes	characteristic
27	#11718356	C₂²	1	1	yes	characteristic
28	#11718355	C₂²	1	2	yes	—
29	C₄×D₁₀₄	—	2	2	no	—
30	#11718343	C₂²	1	1	yes	characteristic
31	#11718046	C₂²	1	2	yes	—
32	C₈×#1055	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃	—	2	2	no	—
34	C₂×#6127	—	2	2	no	—
35	C₂²×C₈×C₁₃	D₄	1	2	yes	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	2	2	no	—
38	C₁₃×C₄⋊Q₈	—	2	2	no	—
39	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6107	—	2	8	no	—
44	#88196	—	4	16	no	—
45	C₂×#6106	—	2	8	no	—
46	#88184	—	2	2	no	—
47	#88184	—	2	2	no	—
48	C₂×#6099	D₄	1	2	yes	—
49	C₂×D₁₀₄	—	2	2	no	—
50	#88129	—	2	2	no	—
51	#88128	C₂³	1	2	yes	—
52	C₄×D₅₂	—	2	2	no	—
53	C₄×D₅₂	—	2	2	no	—
54	#88123	C₂³	1	2	yes	—
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×#107	—	2	4	no	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²⋊#1055	D₄	1	2	yes	—
61	#88061	—	2	4	no	—
62	#88058	—	2	2	no	—
63	#88049	—	2	8	no	—
64	#88047	—	2	8	no	—
65	C₄⋊#1055	D₄	1	2	yes	—
66	#88043	D₄	1	2	yes	—
67	C₄×#1055	C₂³	1	2	yes	—
68	C₄×#1055	C₂³	1	2	yes	—
69	#19630	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	—	2	4	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	4	8	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₃	—	4	2	no	—
78	#6127	—	4	8	no	—
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	—	2	2	no	—
81	C₂×D₅₂	—	2	2	no	—
82	C₂×C₄×D₁₃	—	4	2	no	—
83	C₂×Dic₂₆	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#6107	—	4	16	no	—
96	#6106	—	4	16	no	—
97	#6103	—	4	2	no	—
98	#6103	—	4	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	#6099	D₈	1	8	yes	complemented
102	C₂×#1055	C₂×D₄	1	4	yes	—
103	C₂×#1055	C₂×D₄	1	4	yes	—
104	D₁₀₄	—	4	4	no	—
105	#3226	—	13	2	no	—
106	#3169	—	13	1	no	—
107	C₂×#500	D₁₃	1	1	yes	characteristic, complemented
108	#3152	—	13	2	no	—
109	#1722	—	13	4	no	—
110	#1713	—	13	2	no	—
111	#1706	—	13	2	no	—
112	C₈×M₁₆	—	13	1	no	—
113	C₂³×C₁₃	#127	1	1	yes	characteristic
114	C₂²×D₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	8	no	—
116	Q₈×C₁₃	—	8	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₁₃	—	8	4	no	—
129	C₂×Dic₁₃	—	4	2	no	—
130	D₅₂	—	4	4	no	—
131	D₅₂	—	4	4	no	—
132	D₅₂	—	4	4	no	—
133	C₄×D₁₃	—	8	4	no	—
134	Dic₂₆	—	4	4	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	#137	1	4	yes	complemented
137	#1055	—	2	4	no	—
138	#1055	C₂×D₈	1	8	yes	—
139	C₂×#137	—	26	4	no	—
140	#532	—	13	2	no	—
141	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
142	#500	—	2	2	no	—
143	#500	—	13	4	no	—
144	#500	—	13	1	no	—
145	#500	C₂×D₁₃	1	2	yes	—
146	#500	C₂×D₁₃	1	2	yes	—
147	#497	—	26	4	no	—
148	#494	—	13	4	no	—
149	#493	—	13	1	no	—
150	#492	—	13	4	no	—
151	#486	—	13	4	no	—
152	#486	—	13	2	no	—
153	#480	—	26	4	no	—
154	#474	—	26	4	no	—
155	#466	—	26	2	no	—
156	C₄×D₈	—	26	2	no	—
157	#431	—	13	1	no	—
158	C₂×#107	—	2	2	no	—
159	C₄×M₁₆	—	13	1	no	—
160	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
161	#322	—	13	2	no	—
162	C₈²	—	13	2	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	#493	1	2	yes	—
166	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
167	C₂×D₁₃	—	8	2	no	—
168	C₂×D₁₃	—	8	2	no	—
169	C₂×D₁₃	—	8	2	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	8	4	no	—
173	C₄×C₁₃	#493	1	2	yes	—
174	C₄×C₁₃	C₂²×D₈	1	2	yes	—
175	C₄×C₁₃	C₂×#137	1	2	yes	—
176	C₄×C₁₃	#578	1	2	yes	—
177	Dic₁₃	—	8	2	no	—
178	C₂×#41	—	26	2	no	—
179	C₂²×Q₈	—	2	2	no	—
180	#137	—	52	16	no	—
181	C₂×Q₁₆	—	26	8	no	—
182	C₂×SD₁₆	—	26	8	no	—
183	C₂×D₈	—	26	2	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	13	2	no	—
190	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
191	#127	—	13	2	no	—
192	#124	—	26	2	no	—
193	#121	—	26	2	no	—
194	#121	—	26	2	no	—
195	C₄×D₄	—	26	2	no	—
196	C₄×D₄	—	26	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
200	#107	—	2	4	no	—
201	#107	—	2	4	no	—
202	#107	—	26	8	no	—
203	#107	—	26	2	no	—
204	C₄⋊C₈	—	13	2	no	—
205	#102	—	26	8	no	—
206	#102	—	26	4	no	—
207	C₂²⋊C₈	—	13	2	no	—
208	#97	—	13	2	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	2	no	—
211	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
216	C₂×C₁₃	#3152	1	1	yes	characteristic
217	C₂×C₁₃	#3156	1	1	yes	characteristic
218	D₁₃	—	16	2	no	—
219	#41	—	52	8	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	8	no	—
223	C₂×Q₈	—	26	2	no	—
224	C₂×D₄	—	26	2	no	—
225	C₂×D₄	—	26	2	no	—
226	C₂×D₄	—	26	2	no	—
227	C₂×D₄	—	52	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	52	2	no	—
230	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	Q₁₆	—	52	16	no	—
234	SD₁₆	—	52	16	no	—
235	D₈	—	52	4	no	—
236	M₁₆	—	13	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	26	2	no	—
247	C₄⋊C₄	—	26	2	no	—
248	C₂²⋊C₄	—	52	2	no	—
249	C₂²⋊C₄	—	52	2	no	—
250	C₄²	D₄×D₁₃	1	1	yes	characteristic
251	C₄²	D₄×D₁₃	1	1	yes	characteristic
252	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
253	C₄²	C₂³×D₁₃	1	1	yes	characteristic
254	C₁₃	#19630	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	52	2	no	—
256	C₂³	#88194	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	Q₈	—	8	4	no	—
259	Q₈	—	52	4	no	—
260	D₄	—	52	4	no	—
261	D₄	—	52	4	no	—
262	D₄	—	52	4	no	—
263	D₄	—	104	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	104	4	no	—
270	C₂×C₄	D₈×D₁₃	1	4	yes	—
271	C₂×C₄	#88194	1	1	yes	characteristic
272	C₂×C₄	#88194	1	1	yes	characteristic
273	C₂×C₄	#88194	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	13	8	no	—
279	C₈	—	26	4	no	—
280	C₈	#88196	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	104	2	no	—
284	C₂²	—	104	2	no	—
285	C₂²	—	104	2	no	—
286	C₂²	#11718812	1	2	yes	—
287	C₂²	C₂×#88194	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	—	104	2	no	—
292	C₄	#11718812	1	2	yes	—
293	C₄	C₂×D₈×D₁₃	1	2	yes	—
294	C₄	#11719471	1	2	yes	—
295	C₄	C₂×#88196	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	208	2	no	—
298	C₂	C₂×#11718812	1	1	yes	characteristic
299	C₂	#13125293	1	1	yes	characteristic
300	C₂	#13125325	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	#13125415	C₂
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#19630
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. #13125957	C₂	semidirect
3. #13125293	C₂	semidirect
4. #13114506	C₂	semidirect
5. #13114453	C₂	semidirect
6. #13114429	C₂	semidirect
7. C₈×#6099	C₂	semidirect
8. C₂×C₁₃×#500	C₂	semidirect
101. #6099	D₈	semidirect
107. C₂×#500	D₁₃	semidirect
136. C₈×C₁₃	#137	semidirect
254. C₁₃	#19630	semidirect
280. C₈	#88196	semidirect

Maximal subgroups

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

subgroup	index
#13125957	2
#13125415	2
#13125293	2
#13114506	2
#13114453	2
#13114429	2
C₈×#6099	2
C₂×C₁₃×#500	2
#19630	13