Subgroups — #16015532

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718741	C₂	1	2	yes	maximal
2	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
3	#13120956	C₂	1	2	yes	maximal
4	#13115003	C₂	1	4	yes	maximal
5	#13114999	C₂	1	2	yes	maximal
6	#13114994	C₂	1	2	yes	maximal
7	#13114522	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×#88124	C₂²	1	2	yes	—
11	C₁₃×#500	—	2	4	no	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718890	—	2	4	no	—
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88057	C₂²	1	2	yes	—
19	C₂×#88055	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	#11718741	—	2	8	no	—
22	#11718641	—	2	4	no	—
23	#11718638	C₂²	1	4	yes	—
24	#11718637	C₂²	1	4	yes	—
25	C₂×#88048	C₂²	1	4	yes	—
26	C₂×#88047	C₂²	1	4	yes	—
27	#11718593	C₂²	1	1	yes	characteristic
28	C₂×#88043	C₂²	1	1	yes	characteristic
29	#11718344	—	2	4	no	—
30	C₂²×Q₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	—	2	4	no	—
34	C₁₃×C₄⋊Q₈	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6101	—	2	2	no	—
41	C₂×#6101	—	2	2	no	—
42	C₂×#6101	—	2	8	no	—
43	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×#1055	C₂³	1	2	yes	—
45	#88124	—	2	4	no	—
46	#88124	—	2	4	no	—
47	C₁₃×#107	—	2	4	no	—
48	C₁₃×#107	—	2	4	no	—
49	C₄×C₈×C₁₃	D₄	1	4	yes	—
50	#88071	—	2	8	no	—
51	#88071	D₄	1	2	yes	—
52	#88071	D₄	1	2	yes	—
53	#88058	—	2	2	no	—
54	#88058	—	2	2	no	—
55	#88057	—	2	4	no	—
56	#88055	Q₈	1	8	yes	complemented
57	C₈×Dic₁₃	Q₈	1	8	yes	complemented
58	#88048	—	2	8	no	—
59	#88047	—	2	8	no	—
60	#88043	D₄	1	4	yes	—
61	#33319	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	4	4	no	—
63	C₂×Q₈×C₁₃	—	8	4	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₄⋊Dic₁₃	—	2	2	no	—
81	C₄⋊Dic₁₃	—	2	2	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	#6101	—	4	4	no	—
85	#6101	—	4	4	no	—
86	#6101	—	8	8	no	—
87	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
88	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
89	C₂×#1055	C₂×D₄	1	4	yes	—
90	C₂×#1055	C₂×Q₈	1	4	yes	—
91	C₂×#1055	C₂×Q₈	1	4	yes	—
92	C₂×#501	—	13	2	no	—
93	C₂×#500	D₁₃	1	1	yes	characteristic
94	C₂×#499	—	13	1	no	—
95	#2660	—	13	2	no	—
96	#1847	—	13	4	no	—
97	#1846	—	13	2	no	—
98	#1845	—	13	2	no	—
99	#1726	—	13	1	no	—
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₁₃	—	8	8	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	#127	1	2	yes	—
107	C₂×C₄×C₁₃	#127	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	—	4	4	no	—
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	C₄⋊Q₈	1	8	yes	—
119	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
120	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
121	#1055	C₄⋊Q₈	1	8	yes	—
122	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
123	C₂×#125	—	13	2	no	—
124	#501	—	26	8	no	—
125	#500	—	2	4	no	—
126	#499	—	26	4	no	—
127	C₂×#107	—	13	1	no	—
128	C₂×#107	—	13	1	no	—
129	C₂×#107	—	13	4	no	—
130	C₂×#107	C₂×D₁₃	1	2	yes	—
131	C₂×#107	C₂×D₁₃	1	2	yes	—
132	C₂×#106	—	13	2	no	—
133	C₂×#106	—	13	4	no	—
134	C₂×#97	—	13	2	no	—
135	C₂×#97	—	13	1	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	#398	—	26	4	no	—
139	#395	—	26	4	no	—
140	#384	—	13	4	no	—
141	#382	—	13	4	no	—
142	#376	—	13	1	no	—
143	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
145	C₂²×C₁₃	#576	1	2	yes	—
146	C₂²×C₁₃	#578	1	2	yes	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	8	4	no	—
150	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
151	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₁₃	—	2	8	no	—
154	Dic₁₃	—	8	2	no	—
155	Dic₁₃	—	8	2	no	—
156	C₂²×Q₈	—	2	2	no	—
157	C₂²×C₈	—	13	2	no	—
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	4	no	—
161	C₄⋊Q₈	—	2	4	no	—
162	#125	—	26	4	no	—
163	#125	—	26	4	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	13	2	no	—
166	C₂×C₄⋊C₄	—	13	2	no	—
167	C₂×C₄⋊C₄	—	26	2	no	—
168	C₂×C₄⋊C₄	—	26	2	no	—
169	C₂×C₄⋊C₄	—	26	8	no	—
170	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
171	C₂×C₄²	—	13	2	no	—
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	#107	—	2	4	no	—
174	#107	—	2	4	no	—
175	#107	—	26	8	no	—
176	#107	—	26	2	no	—
177	#107	—	26	2	no	—
178	#106	—	26	8	no	—
179	#106	—	26	4	no	—
180	#97	—	13	8	no	—
181	#97	—	13	4	no	—
182	C₄×C₈	—	13	8	no	—
183	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
184	#95	—	13	2	no	—
185	#95	—	13	2	no	—
186	#95	—	26	8	no	—
187	C₂×C₁₃	#2868	1	1	yes	characteristic
188	C₂×C₁₃	#3155	1	1	yes	characteristic
189	C₂×C₁₃	#3156	1	1	yes	characteristic
190	C₂×C₁₃	#3164	1	4	yes	—
191	C₂×Q₈	—	4	4	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	13	4	no	—
195	C₂²×C₄	—	26	2	no	—
196	C₂²×C₄	—	26	2	no	—
197	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
200	C₂×C₈	—	13	4	no	—
201	C₂×C₈	—	13	4	no	—
202	C₂×C₈	—	13	4	no	—
203	C₂×C₈	Q₈×D₁₃	1	4	yes	—
204	C₂×C₈	Q₈×D₁₃	1	4	yes	—
205	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
206	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	26	2	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	—	26	2	no	—
213	C₄⋊C₄	—	26	2	no	—
214	C₄⋊C₄	—	52	4	no	—
215	C₄⋊C₄	—	52	4	no	—
216	C₄⋊C₄	—	104	8	no	—
217	C₄²	—	13	4	no	—
218	C₄²	—	13	4	no	—
219	C₄²	D₄×D₁₃	1	2	yes	—
220	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
221	C₁₃	#33319	1	1	yes	characteristic, complemented, Sylow
222	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	13	8	no	—
229	C₂×C₄	—	26	4	no	—
230	C₂×C₄	—	26	4	no	—
231	C₂×C₄	—	52	4	no	—
232	C₂×C₄	—	52	4	no	—
233	C₂×C₄	—	52	2	no	—
234	C₂×C₄	—	52	2	no	—
235	C₂×C₄	#88194	1	2	yes	—
236	C₂×C₄	#88194	1	2	yes	—
237	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
239	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
240	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₈	—	13	8	no	—
242	C₈	#88198	1	4	yes	—
243	C₈	#88198	1	4	yes	—
244	C₂²	D₁₃×C₄⋊Q₈	1	2	yes	—
245	C₂²	#11719462	1	2	yes	—
246	C₂²	#11719471	1	2	yes	—
247	C₂²	C₂×#88194	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	—	26	8	no	—
252	C₄	—	104	2	no	—
253	C₄	—	104	2	no	—
254	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
255	C₄	C₂×#88198	1	1	yes	characteristic
256	C₄	C₂×#88198	1	1	yes	characteristic
257	C₂	#13123232	1	1	yes	characteristic
258	C₂	#13125317	1	1	yes	characteristic
259	C₂	#13125325	1	1	yes	characteristic
260	C₂	#13125369	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#33319
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
56. #88055	Q₈	semidirect
57. C₈×Dic₁₃	Q₈	semidirect
221. C₁₃	#33319	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718741	2
C₂×#11718344	2
#13120956	2
#13115003	2
#13114999	2
#13114994	2
#13114522	2
C₂×C₁₃×#500	2
#33319	13