Subgroups — #15774491

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116590	C₂	1	1	yes	maximal, characteristic
2	D₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13113118	C₂	1	1	yes	maximal, characteristic
4	#13112906	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112905	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112905	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1370	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88038	—	2	1	no	—
9	C₂×C₁₆×D₁₃	—	2	1	no	—
10	#11718346	C₄	1	1	yes	characteristic
11	#11718346	C₄	1	1	yes	characteristic
12	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₈²×C₁₃	C₂²	1	1	yes	characteristic
16	#11718093	—	2	1	no	—
17	#11718093	—	2	1	no	—
18	C₄⋊#6090	C₂²	1	1	yes	characteristic
19	C₄⋊#6090	C₂²	1	1	yes	characteristic
20	#11718046	C₄	1	1	yes	characteristic
21	#11718046	C₄	1	1	yes	characteristic
22	C₈×#1055	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
24	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
25	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×C₁₆	—	2	1	no	—
27	C₂×C₁₃×C₁₆	—	2	1	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	#88060	—	2	1	no	—
32	#88060	C₈	1	1	yes	characteristic
33	#88060	C₈	1	1	yes	characteristic
34	#88054	—	2	1	no	—
35	#88054	C₈	1	1	yes	characteristic
36	#88054	C₈	1	1	yes	characteristic
37	C₈×Dic₁₃	D₄	1	1	yes	characteristic
38	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
39	C₂×#6090	—	2	1	no	—
40	C₂×#6090	—	2	1	no	—
41	C₄×#1055	C₂×C₄	1	1	yes	characteristic
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	C₄×#1055	C₂×C₄	1	1	yes	characteristic
44	#88038	—	4	2	no	—
45	C₁₆×D₁₃	—	4	2	no	—
46	#12005	—	13	1	no	maximal, Sylow
47	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	—	2	1	no	—
51	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
58	C₄×Dic₁₃	M₁₆	1	1	yes	characteristic
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#1055	C₂×C₈	1	1	yes	characteristic
63	C₂×#1055	C₂×C₈	1	1	yes	characteristic
64	C₈×D₁₃	—	2	2	no	—
65	C₈×D₁₃	D₈	1	2	yes	—
66	C₈×D₁₃	Q₁₆	1	2	yes	—
67	C₁₃×C₁₆	—	4	1	no	—
68	C₁₃×C₁₆	—	4	1	no	—
69	#6090	—	4	1	no	—
70	#6090	—	4	1	no	—
71	#2150	—	13	1	no	—
72	C₂×C₄⋊C₁₆	—	13	1	no	—
73	#1450	—	13	1	no	—
74	#1370	—	13	1	no	—
75	#1370	—	13	1	no	—
76	#1370	—	13	1	no	—
77	#1370	D₁₃	1	1	yes	characteristic, complemented
78	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
83	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
85	C₄×D₁₃	—	2	1	no	—
86	C₄×D₁₃	—	2	1	no	—
87	C₄×D₁₃	—	4	1	no	—
88	C₄×D₁₃	#107	1	2	yes	—
89	C₄×D₁₃	#108	1	2	yes	—
90	C₈×C₁₃	—	2	1	no	—
91	C₈×C₁₃	—	2	1	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	4	1	no	—
94	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
96	#1055	—	2	1	no	—
97	#1055	—	2	1	no	—
98	#1055	—	2	1	no	—
99	#1055	—	4	1	no	—
100	#1055	#107	1	2	yes	—
101	C₂×M₃₂	—	26	1	no	—
102	C₂²×C₁₆	—	26	1	no	—
103	#431	—	13	1	no	—
104	#431	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₄⋊C₁₆	—	13	1	no	—
107	C₄⋊C₁₆	—	13	1	no	—
108	C₄⋊C₁₆	—	26	1	no	—
109	C₄⋊C₁₆	—	26	1	no	—
110	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
111	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
112	#322	—	13	1	no	—
113	#322	—	13	1	no	—
114	C₈²	—	13	1	no	—
115	C₈²	C₂×D₁₃	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	—	2	1	no	—
119	C₂×D₁₃	#335	1	2	yes	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
122	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
123	C₄×C₁₃	#504	1	1	yes	characteristic
124	C₄×C₁₃	#504	1	1	yes	characteristic
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	#335	1	2	yes	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂×C₄²	—	13	1	no	—
131	M₃₂	—	52	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	26	1	no	—
135	C₂×C₁₆	—	26	1	no	—
136	C₂×C₁₆	—	52	2	no	—
137	C₄⋊C₈	—	13	1	no	—
138	C₄⋊C₈	—	13	1	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₂²⋊C₈	—	13	1	no	—
141	C₂²⋊C₈	—	13	1	no	—
142	C₂²⋊C₈	—	26	1	no	—
143	C₄×C₈	—	13	1	no	—
144	C₄×C₈	—	13	1	no	—
145	C₄×C₈	—	13	1	no	—
146	C₄×C₈	—	13	1	no	—
147	C₄×C₈	—	13	1	no	—
148	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
152	C₂×C₁₃	#2151	1	1	yes	characteristic
153	C₂×C₁₃	#2153	1	1	yes	characteristic
154	D₁₃	—	2	2	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	13	1	no	—
160	C₂×C₈	—	13	2	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	1	no	—
167	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
168	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
169	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
171	C₄²	—	13	1	no	—
172	C₄²	—	13	1	no	—
173	C₄²	—	13	1	no	—
174	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
175	C₁₆	—	4	1	no	—
176	C₁₆	—	4	1	no	—
177	C₁₆	—	52	1	no	—
178	C₁₆	—	52	1	no	—
179	C₁₃	#12005	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	13	1	no	—
181	C₂×C₄	—	13	1	no	—
182	C₂×C₄	—	13	2	no	—
183	C₂×C₄	—	13	2	no	—
184	C₂×C₄	—	13	1	no	—
185	C₂×C₄	—	13	1	no	—
186	C₂×C₄	—	26	1	no	—
187	C₂×C₄	—	26	1	no	—
188	C₂×C₄	—	52	1	no	—
189	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	1	no	—
196	C₈	—	13	2	no	—
197	C₈	—	26	1	no	—
198	C₈	—	26	1	no	—
199	C₈	—	26	1	no	—
200	C₈	—	52	1	no	—
201	C₈	D₈×D₁₃	1	1	yes	characteristic
202	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
203	C₂²	—	13	2	no	—
204	C₂²	—	26	1	no	—
205	C₂²	—	26	1	no	—
206	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	13	2	no	—
209	C₄	—	26	1	no	—
210	C₄	—	26	1	no	—
211	C₄	D₁₃×#107	1	1	yes	characteristic
212	C₄	D₁₃×#108	1	1	yes	characteristic
213	C₄	#11718562	1	1	yes	characteristic
214	C₄	#11718568	1	1	yes	characteristic
215	C₂	—	26	2	no	—
216	C₂	D₁₃×#335	1	1	yes	characteristic
217	C₂	#13116593	1	1	yes	characteristic
218	C₂	#13116600	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1370	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂²×D₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1370	D₁₃
hypocenter	C₁₃	#12005
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

Complements

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

N	H ≅ G/N	type
4. #13112906	C₂	semidirect
5. #13112905	C₂	semidirect
6. #13112905	C₂	semidirect
7. C₁₃×#1370	C₂	semidirect
77. #1370	D₁₃	semidirect
179. C₁₃	#12005	semidirect

Maximal subgroups

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

subgroup	index
#13116590	2
D₁₃×C₄⋊C₁₆	2
#13113118	2
#13112906	2
#13112905	2
C₁₃×#1370	2
#12005	13