Subgroups — #16015489

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718539	C₂	1	2	yes	maximal
2	C₂×#11718519	C₂	1	2	yes	maximal
3	#13123198	C₂	1	2	yes	maximal
4	#13115000	C₂	1	1	yes	maximal, characteristic
5	#13114998	C₂	1	2	yes	maximal
6	#13114994	C₂	1	2	yes	maximal
7	#13114991	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
10	C₂×#88144	C₂²	1	4	yes	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#382	C₂²	1	2	yes	—
15	C₂×#88058	C₂²	1	2	yes	—
16	C₂×#88057	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	#11718637	C₂²	1	4	yes	—
22	C₂×#88048	C₂²	1	4	yes	—
23	C₂×#88047	C₂²	1	4	yes	—
24	#11718597	C₂²	1	2	yes	—
25	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	#11718539	—	2	8	no	—
28	#11718519	—	2	8	no	—
29	#11718308	—	2	8	no	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	—	2	4	no	—
35	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
36	C₂×#6101	—	2	4	no	—
37	C₂×#6101	—	2	8	no	—
38	C₂×C₄×Dic₁₃	D₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂²×#1055	C₂³	1	1	yes	characteristic
42	C₂²×#1055	C₂³	1	1	yes	characteristic
43	#88144	—	2	8	no	—
44	#88144	—	2	8	no	—
45	C₁₃×#107	—	2	4	no	—
46	C₁₃×#106	—	2	4	no	—
47	C₄×C₈×C₁₃	Q₈	1	4	yes	—
48	C₁₃×#95	—	2	4	no	—
49	#88071	—	2	4	no	—
50	#88058	—	2	4	no	—
51	#88057	—	2	4	no	—
52	C₈×Dic₁₃	D₄	1	4	yes	—
53	C₈×Dic₁₃	D₄	1	4	yes	—
54	C₈×Dic₁₃	Q₈	1	4	yes	—
55	C₈×Dic₁₃	Q₈	1	4	yes	—
56	#88048	—	2	8	no	—
57	#88047	—	2	8	no	—
58	C₄×#1055	Q₈	1	4	yes	—
59	#33311	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
65	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
76	C₄⋊Dic₁₃	—	2	4	no	—
77	C₄⋊Dic₁₃	—	2	4	no	—
78	C₄⋊Dic₁₃	—	4	8	no	—
79	#6101	—	4	8	no	—
80	#6101	—	4	16	no	—
81	C₄×Dic₁₃	—	2	8	no	—
82	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
83	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
84	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
85	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
86	C₂×#1055	C₂×D₄	1	2	yes	—
87	C₂×#1055	C₂×D₄	1	2	yes	—
88	C₂×#1055	C₂×Q₈	1	2	yes	—
89	C₂×#1055	C₂×Q₈	1	2	yes	—
90	C₂×#1055	C₂×Q₈	1	2	yes	—
91	C₂×#1055	C₂×Q₈	1	2	yes	—
92	C₂×#499	—	13	2	no	—
93	C₂×#499	—	13	2	no	—
94	#2859	—	13	2	no	—
95	#1846	—	13	1	no	—
96	#1846	—	13	2	no	—
97	#1846	D₁₃	1	1	yes	characteristic
98	#1845	—	13	2	no	—
99	#1844	—	13	2	no	—
100	C₄²×C₈	—	13	1	no	—
101	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	8	no	—
105	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	4	4	no	—
114	C₂×Dic₁₃	—	4	8	no	—
115	C₂×Dic₁₃	#127	1	4	yes	—
116	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
117	C₂×Dic₁₃	#135	1	8	yes	—
118	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
119	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
120	#1055	C₄⋊Q₈	1	4	yes	—
121	#1055	C₄⋊Q₈	1	4	yes	—
122	C₂×#125	—	13	4	no	—
123	#499	—	26	8	no	—
124	#499	—	26	8	no	—
125	C₂×#107	—	13	2	no	—
126	C₂×#107	—	13	4	no	—
127	C₂×#107	C₂×D₁₃	1	2	yes	—
128	C₂×#106	—	13	2	no	—
129	C₂×#106	—	13	4	no	—
130	C₂×#106	C₂×D₁₃	1	2	yes	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	C₂×C₄×C₈	—	13	1	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	#389	—	26	8	no	—
138	#384	—	13	4	no	—
139	#382	—	13	2	no	—
140	#382	C₂×D₁₃	1	2	yes	—
141	C₄³	—	13	1	no	—
142	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×#135	1	4	yes	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	8	4	no	—
148	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	Dic₁₃	—	2	4	no	—
153	Dic₁₃	—	8	4	no	—
154	Dic₁₃	#499	1	8	yes	complemented
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
159	#125	—	26	8	no	—
160	#125	—	26	8	no	—
161	C₂×C₄⋊C₄	—	13	4	no	—
162	C₂×C₄⋊C₄	—	26	4	no	—
163	C₂×C₄⋊C₄	—	26	4	no	—
164	C₂×C₄⋊C₄	—	26	8	no	—
165	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	4	no	—
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	#107	—	2	4	no	—
171	#107	—	26	8	no	—
172	#107	—	26	4	no	—
173	#106	—	2	4	no	—
174	#106	—	26	8	no	—
175	#106	—	26	4	no	—
176	C₄×C₈	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	—	13	4	no	—
179	C₄×C₈	—	13	4	no	—
180	C₄×C₈	—	13	4	no	—
181	C₄×C₈	Dic₂₆	1	4	yes	—
182	#95	—	2	4	no	—
183	#95	—	26	4	no	—
184	C₂×C₁₃	#2868	1	1	yes	characteristic
185	C₂×C₁₃	#3147	1	2	yes	—
186	C₂×C₁₃	C₂×#499	1	2	yes	—
187	C₂×C₁₃	#3160	1	2	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	26	4	no	—
192	C₂²×C₄	D₄×D₁₃	1	2	yes	—
193	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
201	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
202	C₂×C₈	D₄×D₁₃	1	2	yes	—
203	C₂×C₈	D₄×D₁₃	1	2	yes	—
204	C₂×C₈	Q₈×D₁₃	1	2	yes	—
205	C₂×C₈	Q₈×D₁₃	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	26	4	no	—
209	C₄⋊C₄	—	26	4	no	—
210	C₄⋊C₄	—	52	8	no	—
211	C₄⋊C₄	—	52	8	no	—
212	C₄⋊C₄	—	52	16	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	13	2	no	—
215	C₄²	—	13	2	no	—
216	C₄²	—	13	2	no	—
217	C₄²	—	26	8	no	—
218	C₄²	C₂×Dic₂₆	1	2	yes	—
219	C₄²	Q₈×D₁₃	1	2	yes	—
220	C₁₃	#33311	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	8	no	—
227	C₂×C₄	—	13	4	no	—
228	C₂×C₄	—	26	4	no	—
229	C₂×C₄	—	52	8	no	—
230	C₂×C₄	—	52	4	no	—
231	C₂×C₄	#88142	1	4	yes	—
232	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
238	C₈	—	13	4	no	—
239	C₈	—	13	4	no	—
240	C₈	#88142	1	4	yes	—
241	C₈	#88142	1	4	yes	—
242	C₂²	C₂×#88142	1	1	yes	characteristic
243	C₂²	D₁₃×#127	1	1	yes	characteristic
244	C₂²	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
245	C₂²	D₁₃×#135	1	4	yes	—
246	C₄	—	2	4	no	—
247	C₄	—	8	4	no	—
248	C₄	—	13	8	no	—
249	C₄	—	26	4	no	—
250	C₄	—	104	4	no	—
251	C₄	C₂×#88142	1	1	yes	characteristic
252	C₄	C₂×#88142	1	1	yes	characteristic
253	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
254	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
255	C₂	#13123233	1	1	yes	characteristic
256	C₂	#13125261	1	2	yes	—
257	C₂	D₁₃×#499	1	2	yes	—
258	C₂	#13125348	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1846	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#1846	D₁₃
hypocenter	C₁₃	#33311
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
154. Dic₁₃	#499	semidirect
220. C₁₃	#33311	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718539	2
C₂×#11718519	2
#13123198	2
#13115000	2
#13114998	2
#13114994	2
#13114991	2
C₄×C₈×Dic₁₃	2
C₁₃×#1846	2
#33311	13