Subgroups — #13125362

Order 1664.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#544	C₂	1	1	yes	maximal, characteristic
2	#11719224	C₂	1	1	yes	maximal, characteristic
3	#11718741	C₂	1	2	yes	maximal, complemented
4	#11718740	C₂	1	2	yes	maximal, complemented
5	#11718726	C₂	1	2	yes	maximal, complemented
6	#11718725	C₂	1	2	yes	maximal, complemented
7	#11718659	C₂	1	1	yes	maximal, characteristic
8	#11718658	C₂	1	1	yes	maximal, characteristic
9	#11718656	C₂	1	1	yes	maximal, characteristic
10	#11718625	C₂	1	1	yes	maximal, characteristic
11	C₂×#88048	C₂	1	1	yes	maximal, characteristic
12	C₁₃×C₄⋊Q₈	—	2	1	no	—
13	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₃×#125	—	2	1	no	—
15	C₁₃×#125	C₂²	1	2	yes	—
16	C₁₃×C₂²⋊Q₈	—	2	2	no	—
17	C₁₃×#117	C₂²	1	1	yes	characteristic
18	C₁₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#88180	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂²×#1055	C₂²	1	1	yes	characteristic
24	#88144	—	2	1	no	—
25	#88142	—	2	1	no	—
26	#88132	—	2	2	no	—
27	#88124	C₂²	1	2	yes	—
28	#88123	C₂²	1	2	yes	—
29	#88048	C₂²	1	2	yes	—
30	#88048	C₂²	1	1	yes	characteristic
31	#88048	C₂²	1	1	yes	characteristic
32	#88048	C₂²	1	2	yes	—
33	#88048	C₂²	1	2	yes	—
34	#88048	C₂²	1	2	yes	—
35	#88047	C₂²	1	2	yes	—
36	#88047	C₂²	1	1	yes	characteristic
37	#88047	C₂²	1	1	yes	characteristic
38	#88047	C₂²	1	2	yes	—
39	#88043	C₂²	1	2	yes	—
40	C₄×#1055	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	1	no	—
45	C₂×Dic₂₆	—	2	2	no	—
46	C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
55	C₁₃×C₂²⋊C₄	—	2	1	no	—
56	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	C₂³	1	2	yes	—
59	C₂²⋊Dic₁₃	—	2	1	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	1	no	—
62	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₁₃	C₂³	1	2	yes	—
64	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
65	#6101	—	2	2	no	—
66	#6101	—	2	2	no	—
67	C₄×Dic₁₃	—	2	1	no	—
68	#6099	Q₈	1	4	yes	complemented
69	C₂×#1055	Q₈	1	4	yes	complemented
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	C₂×#1055	C₂³	1	1	yes	characteristic
72	C₂×#1055	C₂³	1	2	yes	—
73	#3163	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	8	1	no	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	8	1	no	—
90	Dic₂₆	—	8	2	no	—
91	#1055	C₂×Q₈	1	2	yes	—
92	#1055	C₂×Q₈	1	2	yes	—
93	#1055	C₂×Q₈	1	4	yes	—
94	#544	—	13	1	no	—
95	#544	D₁₃	1	1	yes	characteristic
96	#501	—	13	2	no	—
97	#501	—	13	2	no	—
98	#499	—	13	2	no	—
99	#498	—	13	2	no	—
100	#428	—	13	1	no	—
101	#428	—	13	1	no	—
102	#427	—	13	1	no	—
103	C₂×#106	—	13	1	no	—
104	#405	—	13	1	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	C₄⋊Q₈	1	2	yes	—
108	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	4	1	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	—	4	1	no	—
114	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
115	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	Dic₁₃	—	4	1	no	—
118	Dic₁₃	—	4	2	no	—
119	Dic₁₃	—	4	1	no	—
120	C₂×M₁₆	—	13	1	no	—
121	C₂²×C₈	—	13	1	no	—
122	C₄⋊Q₈	—	2	1	no	—
123	C₄⋊Q₈	—	13	2	no	—
124	C₄⋊Q₈	—	26	1	no	—
125	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
126	#125	—	2	1	no	—
127	#125	—	13	2	no	—
128	#125	—	26	1	no	—
129	#125	C₂×D₁₃	1	2	yes	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₂²⋊Q₈	—	26	2	no	—
132	#117	—	13	1	no	—
133	#117	C₂×D₁₃	1	1	yes	characteristic
134	#117	C₂×D₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	13	1	no	—
136	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
137	#107	—	13	1	no	—
138	#107	—	13	1	no	—
139	#107	—	13	2	no	—
140	#107	—	13	2	no	—
141	#106	—	13	2	no	—
142	#106	—	13	2	no	—
143	#106	—	13	1	no	—
144	#106	—	13	1	no	—
145	#106	—	13	2	no	—
146	#106	—	13	2	no	—
147	#97	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₂×C₁₃	—	2	2	no	—
150	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₃	#577	1	1	yes	characteristic
152	C₂×C₁₃	#577	1	1	yes	characteristic
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	26	2	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	26	1	no	—
157	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	M₁₆	—	13	4	no	—
159	C₂×C₈	—	13	4	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	13	2	no	—
168	C₄⋊C₄	—	13	1	no	—
169	C₄⋊C₄	—	13	1	no	—
170	C₄⋊C₄	—	26	1	no	—
171	C₄⋊C₄	—	26	2	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
176	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
177	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²⋊C₄	—	2	1	no	—
180	C₂²⋊C₄	—	26	1	no	—
181	C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
182	C₄²	—	2	1	no	—
183	C₄²	—	26	1	no	—
184	C₄²	C₂²×D₁₃	1	2	yes	—
185	C₁₃	#3163	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	104	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	26	1	no	—
194	C₂×C₄	—	26	1	no	—
195	C₂×C₄	—	26	2	no	—
196	C₂×C₄	—	104	1	no	—
197	C₂×C₄	Q₈×D₁₃	1	2	yes	—
198	C₂×C₄	Q₈×D₁₃	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	C₈	—	13	2	no	—
204	C₈	—	13	2	no	—
205	C₈	—	13	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#88198	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	52	1	no	—
216	C₄	—	52	2	no	—
217	C₄	—	52	1	no	—
218	C₄	#88198	1	2	yes	—
219	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#88198	1	1	yes	characteristic
223	C₂	#11719541	1	1	yes	characteristic
224	C₂	#11719542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#544	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#544	D₁₃
hypocenter	C₁₃	#3163
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
3. #11718741	C₂	semidirect
4. #11718740	C₂	semidirect
5. #11718726	C₂	semidirect
6. #11718725	C₂	semidirect
68. #6099	Q₈	semidirect
69. C₂×#1055	Q₈	semidirect
185. C₁₃	#3163	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#544	2
#11719224	2
#11718741	2
#11718740	2
#11718726	2
#11718725	2
#11718659	2
#11718658	2
#11718656	2
#11718625	2
C₂×#88048	2
#3163	13