Subgroups — #13123918

Order 1664.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718496	C₂	1	2	yes	maximal
5	#11718494	C₂	1	1	yes	maximal, characteristic
6	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718489	C₂	1	2	yes	maximal, complemented
8	#11718366	C₂	1	2	yes	maximal
9	#11718362	C₂	1	2	yes	maximal, complemented
10	#11718346	C₂	1	1	yes	maximal, characteristic
11	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₄⋊C₄×D₁₃	C₄	1	2	yes	—
16	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
17	#88141	C₄	1	1	yes	characteristic
18	#88141	C₄	1	2	yes	—
19	C₄²×D₁₃	C₂²	1	1	yes	characteristic
20	C₄²×D₁₃	C₂²	1	2	yes	—
21	C₄×Dic₂₆	C₂²	1	2	yes	—
22	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₃×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	#88060	C₂²	1	2	yes	—
28	#88060	C₂²	1	1	yes	characteristic
29	#88060	C₂²	1	1	yes	characteristic
30	#88055	C₂²	1	2	yes	—
31	#88055	C₂²	1	2	yes	—
32	#88054	C₂²	1	1	yes	characteristic
33	#88054	C₂²	1	2	yes	—
34	#88054	C₂²	1	2	yes	—
35	#88054	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	2	yes	—
37	C₄⋊#1055	C₂²	1	1	yes	characteristic
38	C₄⋊#1055	C₂²	1	2	yes	—
39	#88043	C₂²	1	2	yes	—
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₁₃	—	2	4	no	—
43	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
49	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	2	yes	—
57	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	#6101	C₂×C₄	1	2	yes	—
60	#6101	C₂×C₄	1	2	yes	—
61	#6101	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₃	C₂³	1	2	yes	—
68	C₂×#1055	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	2	yes	—
71	#2983	—	13	1	no	maximal, Sylow
72	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₃	—	2	4	no	—
85	C₄×D₁₃	—	2	1	no	—
86	C₄×D₁₃	—	2	1	no	—
87	C₄×D₁₃	—	4	1	no	—
88	C₄×D₁₃	—	4	2	no	—
89	C₄×D₁₃	M₁₆	1	4	yes	—
90	Dic₂₆	—	2	2	no	—
91	Dic₂₆	—	2	4	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	—	2	2	no	—
95	#1055	—	2	1	no	—
96	#1055	—	2	2	no	—
97	#1055	—	2	1	no	—
98	C₂×C₄×Q₈	—	13	1	no	—
99	#446	—	13	1	no	—
100	#446	—	13	2	no	—
101	#446	—	13	1	no	—
102	#446	—	13	2	no	—
103	#446	—	13	1	no	—
104	#446	D₁₃	1	1	yes	characteristic, complemented
105	#432	—	13	2	no	—
106	#432	—	13	2	no	—
107	#431	—	13	1	no	—
108	#431	—	13	1	no	—
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	1	no	—
111	C₂×D₁₃	—	2	1	no	—
112	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
117	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
118	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	Dic₁₃	—	2	1	no	—
120	Dic₁₃	—	2	2	no	—
121	Dic₁₃	—	2	1	no	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	C₂×M₁₆	1	2	yes	—
124	C₂²×Q₈	—	13	1	no	—
125	C₄×Q₈	—	13	1	no	—
126	C₄×Q₈	—	13	2	no	—
127	C₄×Q₈	—	13	1	no	—
128	C₄×Q₈	—	13	2	no	—
129	C₄×Q₈	—	13	1	no	—
130	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	13	1	no	—
132	C₂×C₄⋊C₄	—	13	2	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	13	2	no	—
135	C₄⋊C₈	—	13	1	no	—
136	C₄⋊C₈	—	13	1	no	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	—	13	2	no	—
139	C₄⋊C₈	—	13	1	no	—
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
143	C₂²⋊C₈	—	13	2	no	—
144	C₂²⋊C₈	—	13	1	no	—
145	C₂²⋊C₈	—	13	1	no	—
146	#97	—	13	2	no	—
147	#97	—	13	2	no	—
148	#97	—	13	2	no	—
149	#97	C₂×D₁₃	1	2	yes	—
150	C₄×C₈	—	13	2	no	—
151	C₄×C₈	—	13	1	no	—
152	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₁₃	#519	1	1	yes	characteristic
154	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₂×C₁₃	#568	1	1	yes	characteristic
156	D₁₃	—	2	2	no	—
157	C₂×Q₈	—	13	2	no	—
158	C₂×Q₈	—	13	1	no	—
159	C₂×Q₈	—	26	4	no	—
160	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	C₂²×D₁₃	1	2	yes	—
170	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₄⋊C₄	—	13	2	no	—
173	C₄⋊C₄	—	13	2	no	—
174	C₄⋊C₄	—	13	2	no	—
175	C₄⋊C₄	—	13	2	no	—
176	C₄⋊C₄	—	13	1	no	—
177	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
178	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
179	C₄²	—	13	1	no	—
180	C₄²	—	13	2	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	2	no	—
183	C₄²	—	13	1	no	—
184	C₄²	—	13	2	no	—
185	C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	C₄²	C₂²×D₁₃	1	2	yes	—
187	C₁₃	#2983	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	13	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	26	4	no	—
191	Q₈	—	26	2	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	26	4	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	52	1	no	—
202	C₂×C₄	—	52	2	no	—
203	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	26	1	no	—
212	C₈	—	26	2	no	—
213	C₈	—	26	1	no	—
214	C₂²	—	13	2	no	—
215	C₂²	—	26	1	no	—
216	C₂²	—	26	1	no	—
217	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	13	2	no	—
222	C₄	—	26	2	no	—
223	C₄	—	26	1	no	—
224	C₄	—	26	2	no	—
225	C₄	—	26	1	no	—
226	C₄	M₁₆×D₁₃	1	2	yes	—
227	C₄	#88254	1	1	yes	characteristic
228	C₄	#88257	1	1	yes	characteristic
229	C₂	—	26	2	no	—
230	C₂	#11719277	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
232	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#446	C₂
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#446	D₁₃
hypocenter	C₁₃	#2983
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. #11718685	C₂	semidirect
6. #11718492	C₂	semidirect
7. #11718489	C₂	semidirect
9. #11718362	C₂	semidirect
11. #11718340	C₂	semidirect
104. #446	D₁₃	semidirect
187. C₁₃	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₃	2
C₁₃×#446	2
#11718685	2
#11718496	2
#11718494	2
#11718492	2
#11718489	2
#11718366	2
#11718362	2
#11718346	2
#11718340	2
#2983	13