Subgroups — #13127205

Order 1664.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719246	C₂	1	2	yes	maximal
2	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718777	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718777	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88058	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718490	C₂	1	2	yes	maximal, complemented
8	#11718384	C₂	1	2	yes	maximal, complemented
9	#11718382	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718344	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	#88189	—	2	2	no	—
16	#88180	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	#88178	C₂²	1	2	yes	—
19	#88178	C₂²	1	2	yes	—
20	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
21	#88143	—	2	2	no	—
22	#88133	—	2	2	no	—
23	#88131	—	2	2	no	—
24	#88124	C₂²	1	2	yes	—
25	C₄×Dic₂₆	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	#88058	C₂²	1	2	yes	—
31	#88058	C₂²	1	1	yes	characteristic
32	#88058	C₂²	1	1	yes	characteristic
33	#88058	C₂²	1	1	yes	characteristic
34	#88057	C₂²	1	2	yes	—
35	#88057	C₂²	1	2	yes	—
36	#88056	C₂²	1	2	yes	—
37	#88056	C₂²	1	2	yes	—
38	#88056	C₂²	1	2	yes	—
39	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	2	no	—
43	C₂×Dic₂₆	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
52	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₁₃	—	2	2	no	—
55	C₂²⋊Dic₁₃	—	2	2	no	—
56	C₂²⋊Dic₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	C₂³	1	2	yes	—
59	C₄⋊Dic₁₃	C₂³	1	2	yes	—
60	C₄⋊Dic₁₃	C₂³	1	2	yes	—
61	#6101	—	2	2	no	—
62	#6101	—	2	2	no	—
63	#6101	—	2	2	no	—
64	#6101	—	2	2	no	—
65	#6101	—	2	2	no	—
66	C₄×Dic₁₃	—	2	2	no	—
67	C₄×Dic₁₃	—	2	2	no	—
68	Dic₅₂	—	2	2	no	—
69	#3336	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
72	D₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	4	4	no	—
87	Dic₂₆	—	2	2	no	—
88	Dic₂₆	—	2	2	no	—
89	C₈×C₁₃	—	2	1	no	—
90	C₈×C₁₃	—	2	1	no	—
91	C₈×C₁₃	—	2	1	no	—
92	C₈×C₁₃	#41	1	2	yes	—
93	#552	—	13	2	no	—
94	#499	—	13	1	no	—
95	#485	—	13	2	no	—
96	#483	—	13	2	no	—
97	#479	—	13	2	no	—
98	#467	—	13	1	no	—
99	#467	—	13	1	no	—
100	C₄×Q₁₆	—	13	1	no	—
101	C₈×D₄	D₁₃	1	1	yes	characteristic
102	#427	—	13	1	no	—
103	C₂×#107	—	13	1	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	#135	1	2	yes	—
110	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
117	Dic₁₃	—	4	2	no	—
118	Dic₁₃	—	4	2	no	—
119	Dic₁₃	—	4	2	no	—
120	Dic₁₃	—	4	2	no	—
121	C₂×Q₁₆	—	13	1	no	—
122	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
124	#126	—	26	2	no	—
125	#125	—	13	2	no	—
126	#125	—	26	2	no	—
127	#123	—	26	2	no	—
128	C₂²⋊Q₈	—	13	2	no	—
129	C₂²⋊Q₈	—	13	2	no	—
130	C₄×Q₈	—	13	2	no	—
131	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
132	#117	—	13	2	no	—
133	#117	—	26	2	no	—
134	C₂×C₄⋊C₄	—	13	2	no	—
135	#107	—	13	1	no	—
136	#107	—	13	1	no	—
137	#107	—	13	2	no	—
138	#107	—	13	1	no	—
139	#106	—	13	2	no	—
140	#106	—	13	2	no	—
141	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
142	#103	—	13	2	no	—
143	#103	—	13	2	no	—
144	#103	—	13	2	no	—
145	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
146	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂×C₁₃	—	2	2	no	—
149	C₂×C₁₃	—	4	1	no	—
150	C₂×C₁₃	#547	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
152	C₂×C₁₃	#578	1	1	yes	characteristic
153	C₂×Q₈	—	13	2	no	—
154	C₂×D₄	D₅₂	1	1	yes	characteristic
155	C₂²×C₄	—	26	2	no	—
156	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	Q₁₆	—	26	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₄⋊C₄	—	13	2	no	—
166	C₄⋊C₄	—	13	2	no	—
167	C₄⋊C₄	—	13	2	no	—
168	C₄⋊C₄	—	26	2	no	—
169	C₄⋊C₄	—	26	2	no	—
170	C₄⋊C₄	—	26	2	no	—
171	C₄⋊C₄	—	26	2	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	D₅₂	1	1	yes	characteristic
175	C₂²⋊C₄	—	26	2	no	—
176	C₂²⋊C₄	—	26	2	no	—
177	C₂²⋊C₄	—	26	2	no	—
178	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
179	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
180	C₄²	—	26	2	no	—
181	C₄²	—	26	2	no	—
182	C₄²	C₂²×D₁₃	1	1	yes	characteristic
183	C₁₃	#3336	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×D₅₂	1	1	yes	characteristic
185	C₂³	C₂×D₅₂	1	1	yes	characteristic
186	Q₈	—	26	2	no	—
187	Q₈	—	26	2	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	2	no	—
195	C₂×C₄	—	26	2	no	—
196	C₂×C₄	—	26	2	no	—
197	C₂×C₄	—	52	4	no	—
198	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
199	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
200	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
201	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#6129	1	2	yes	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#88158	1	2	yes	—
213	C₂²	C₂²×D₅₂	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	52	2	no	—
219	C₄	—	52	2	no	—
220	C₄	—	52	2	no	—
221	C₄	—	52	2	no	—
222	C₄	C₂×#6129	1	1	yes	characteristic
223	C₄	#88257	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#11719260	1	1	yes	characteristic
227	C₂	C₂×#88158	1	1	yes	characteristic
228	C₂	#11719458	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₄×C₁₃	D₄
hypercenter	C₈×D₄	D₁₃
hypocenter	C₁₃	#3336
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
3. #11718777	C₂	semidirect
4. #11718777	C₂	semidirect
5. #11718767	C₂	semidirect
6. C₂×#88058	C₂	semidirect
7. #11718490	C₂	semidirect
8. #11718384	C₂	semidirect
9. #11718382	C₂	semidirect
10. C₄×Dic₅₂	C₂	semidirect
11. #11718344	C₂	semidirect
183. C₁₃	#3336	semidirect

Maximal subgroups

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

subgroup	index
#11719246	2
C₈×D₄×C₁₃	2
#11718777	2
#11718767	2
C₂×#88058	2
#11718490	2
#11718384	2
#11718382	2
C₄×Dic₅₂	2
#11718344	2
#3336	13