Subgroups — #13123699

Order 1664.

456 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	C₂×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
2	#11719277	C₂	1	2	yes	maximal
3	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718691	C₂	1	2	yes	maximal, complemented
6	#11718541	C₂	1	1	yes	maximal, characteristic
7	#11718534	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718463	C₂	1	2	yes	maximal
9	#11718450	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718350	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×D₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	Q₈×Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6107	C₂²	1	2	yes	—
17	Q₁₆×D₁₃	C₄	1	8	yes	complemented
18	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
19	C₂×#6094	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
21	#88146	—	2	2	no	—
22	#88146	C₂²	1	2	yes	—
23	#88141	—	2	2	no	—
24	#88141	C₂²	1	2	yes	—
25	#88126	—	2	2	no	—
26	#88126	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆	—	2	2	no	—
28	C₄×Dic₂₆	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88075	C₂²	1	2	yes	—
33	#88058	C₂²	1	1	yes	characteristic
34	#88056	C₂²	1	2	yes	—
35	#88055	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
37	#88050	C₂²	1	2	yes	—
38	#88048	C₂²	1	2	yes	—
39	#88043	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
41	Q₈×D₁₃	—	2	4	no	—
42	Q₈×D₁₃	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₁₃	—	2	2	no	—
44	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₆	—	2	2	no	—
46	C₂×Dic₂₆	C₂³	1	2	yes	—
47	C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	#6107	C₂×C₄	1	8	yes	—
55	#6103	—	2	2	no	—
56	#6103	D₄	1	1	yes	characteristic
57	#6103	D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	C₂³	1	2	yes	—
60	#6101	—	2	2	no	—
61	#6101	—	2	2	no	—
62	#6101	D₄	1	1	yes	characteristic
63	#6101	D₄	1	1	yes	characteristic
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	2	2	no	—
66	C₄×Dic₁₃	—	2	2	no	—
67	C₄×Dic₁₃	—	2	2	no	—
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	Dic₅₂	C₂×C₄	1	4	yes	—
72	#6094	—	2	2	no	—
73	C₈×D₁₃	C₂×C₄	1	4	yes	—
74	#2950	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₃	—	2	8	no	—
89	C₄×D₁₃	C₂²×C₄	1	2	yes	—
90	C₄×D₁₃	#41	1	2	yes	—
91	Dic₂₆	—	2	2	no	—
92	Dic₂₆	—	2	8	no	—
93	Dic₂₆	C₂²×C₄	1	4	yes	—
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	#1055	—	2	1	no	—
97	#1055	C₂²×C₄	1	2	yes	—
98	C₂²×Q₁₆	—	13	1	no	—
99	#519	—	13	2	no	—
100	#441	—	13	2	no	—
101	#441	—	13	2	no	—
102	C₄×Q₁₆	—	13	1	no	—
103	C₄×Q₁₆	—	13	1	no	—
104	C₄×Q₁₆	—	13	1	no	—
105	C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
106	#427	—	13	1	no	—
107	#419	—	13	2	no	—
108	#405	—	13	1	no	—
109	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	1	no	—
111	C₂×D₁₃	—	2	1	no	—
112	C₂×D₁₃	C₄×D₄	1	2	yes	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	Dic₁₃	—	2	4	no	—
120	Dic₁₃	—	2	1	no	—
121	Dic₁₃	—	2	1	no	—
122	Dic₁₃	—	4	2	no	—
123	Dic₁₃	C₄×D₄	1	2	yes	—
124	C₂²×Q₈	—	13	2	no	—
125	C₂×Q₁₆	—	13	2	no	—
126	C₂×Q₁₆	—	13	8	no	—
127	C₂×Q₁₆	—	13	1	no	—
128	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
129	C₂×M₁₆	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₄×Q₈	—	13	2	no	—
132	C₄×Q₈	—	13	2	no	—
133	C₄×Q₈	—	13	2	no	—
134	C₄×Q₈	—	26	2	no	—
135	C₄×Q₈	—	26	2	no	—
136	C₄×Q₈	C₂×D₁₃	1	2	yes	—
137	#117	—	13	1	no	—
138	#117	—	13	2	no	—
139	#117	—	26	2	no	—
140	#117	—	26	2	no	—
141	#107	—	13	1	no	—
142	#107	C₂×D₁₃	1	1	yes	characteristic
143	#106	—	13	2	no	—
144	#103	—	13	2	no	—
145	#103	—	13	2	no	—
146	#103	—	13	2	no	—
147	#103	C₂×D₁₃	1	2	yes	—
148	#97	—	13	1	no	—
149	#97	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×C₁₃	#578	1	1	yes	characteristic
154	C₂×C₁₃	#578	1	1	yes	characteristic
155	D₁₃	—	2	2	no	—
156	C₂×Q₈	—	13	8	no	—
157	C₂×Q₈	—	13	2	no	—
158	C₂×Q₈	—	26	2	no	—
159	C₂×Q₈	—	26	4	no	—
160	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	26	2	no	—
163	Q₁₆	—	13	8	no	—
164	Q₁₆	—	13	4	no	—
165	Q₁₆	C₄×D₁₃	1	4	yes	—
166	M₁₆	—	26	2	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	4	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	13	1	no	—
174	C₄⋊C₄	—	13	1	no	—
175	C₄⋊C₄	—	13	2	no	—
176	C₄⋊C₄	—	26	2	no	—
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
180	C₂²⋊C₄	—	13	1	no	—
181	C₂²⋊C₄	—	13	1	no	—
182	C₂²⋊C₄	—	26	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	13	1	no	—
185	C₄²	—	26	2	no	—
186	C₄²	—	26	2	no	—
187	C₄²	—	26	2	no	—
188	C₄²	—	26	2	no	—
189	C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	C₁₃	#2950	1	1	yes	characteristic, complemented, Sylow
191	C₂³	—	13	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	13	4	no	—
194	Q₈	—	26	8	no	—
195	Q₈	—	26	2	no	—
196	Q₈	C₂×C₄×D₁₃	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	13	1	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	26	8	no	—
205	C₂×C₄	—	26	2	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
208	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	13	2	no	—
212	C₈	—	26	1	no	—
213	C₈	C₂×C₄×D₁₃	1	2	yes	—
214	C₂²	—	13	2	no	—
215	C₂²	—	26	1	no	—
216	C₂²	—	26	1	no	—
217	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	13	2	no	—
223	C₄	—	26	1	no	—
224	C₄	—	26	1	no	—
225	C₄	—	26	4	no	—
226	C₄	—	52	2	no	—
227	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
228	C₄	#41×D₁₃	1	1	yes	characteristic
229	C₂	—	26	2	no	—
230	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
231	C₂	#11719471	1	1	yes	characteristic
232	C₂	#11719476	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₃	C₂×D₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#2950
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₃×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₁₃	C₂	semidirect
5. #11718691	C₂	semidirect
7. #11718534	C₂	semidirect
9. #11718450	C₂	semidirect
10. C₄×Dic₅₂	C₂	semidirect
17. Q₁₆×D₁₃	C₄	semidirect
105. C₄×Q₁₆	D₁₃	semidirect
190. C₁₃	#2950	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₁₃	2
#11719277	2
C₄×C₁₃×Q₁₆	2
Q₁₆×Dic₁₃	2
#11718691	2
#11718541	2
#11718534	2
#11718463	2
#11718450	2
C₄×Dic₅₂	2
#11718350	2
#2950	13