Subgroups — #15780944

Order 3328.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116601	C₂	1	2	yes	maximal
2	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
3	#13114715	C₂	1	1	yes	maximal, characteristic
4	#13112923	C₂	1	2	yes	maximal
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₄	1	1	yes	characteristic
10	#11718767	C₄	1	1	yes	characteristic
11	#11718762	C₂²	1	2	yes	—
12	C₂×#88052	C₂²	1	2	yes	—
13	C₁₃×#346	—	2	1	no	—
14	C₄×C₁₃×C₁₆	—	2	1	no	—
15	#11718115	C₄	1	2	yes	—
16	#11718103	—	2	4	no	—
17	#11718055	—	2	1	no	—
18	#11718055	—	2	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	#88180	C₂×C₄	1	2	yes	—
23	C₂×#6099	C₂×C₄	1	2	yes	—
24	C₁₃×M₃₂	—	2	1	no	—
25	C₁₃×M₃₂	—	2	1	no	—
26	C₁₃×M₃₂	D₄	1	2	yes	—
27	C₁₃×M₃₂	Q₈	1	2	yes	—
28	C₂×C₁₃×C₁₆	—	2	1	no	—
29	C₂×C₁₃×C₁₆	—	2	1	no	—
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	#88059	C₈	1	2	yes	—
35	#88059	C₈	1	2	yes	—
36	#88058	—	2	1	no	—
37	#88057	—	2	1	no	—
38	#88055	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	#88052	—	2	2	no	—
41	#88052	—	2	2	no	—
42	C₂×#6090	—	2	2	no	—
43	#13164	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	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	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
58	C₄²×C₁₃	—	2	1	no	—
59	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	#6099	—	2	2	no	—
65	#6099	C₂×C₈	1	4	yes	—
66	C₂×#1055	—	2	2	no	—
67	C₁₃×C₁₆	—	2	2	no	—
68	C₁₃×C₁₆	—	2	2	no	—
69	#6090	—	4	2	no	—
70	#2153	—	13	2	no	—
71	C₂×#364	—	13	1	no	—
72	C₄×M₃₂	D₁₃	1	1	yes	characteristic
73	#1772	—	13	1	no	—
74	#1378	—	13	2	no	—
75	C₂³×C₁₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
85	C₂×Dic₁₃	—	4	2	no	—
86	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
87	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
88	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
90	C₈×C₁₃	#121	1	1	yes	characteristic
91	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₈×C₁₃	#125	1	1	yes	characteristic
93	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
94	#1055	—	2	4	no	—
95	C₂×M₃₂	—	13	2	no	—
96	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
97	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
98	C₂×#108	—	13	1	no	—
99	#427	—	13	1	no	—
100	#405	—	13	2	no	—
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	#364	—	26	4	no	—
103	#364	—	26	1	no	—
104	#364	—	26	1	no	—
105	#346	—	2	1	no	—
106	C₄×C₁₆	—	2	1	no	—
107	#339	—	13	2	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#424	1	1	yes	characteristic
110	C₂²×C₁₃	#436	1	1	yes	characteristic
111	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	4	1	no	—
114	C₄×C₁₃	#389	1	1	yes	characteristic
115	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
117	C₄×C₁₃	#446	1	1	yes	characteristic
118	Dic₁₃	—	8	2	no	—
119	C₂×M₁₆	—	13	2	no	—
120	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	#117	—	13	2	no	—
123	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
124	M₃₂	—	2	1	no	—
125	M₃₂	—	2	1	no	—
126	M₃₂	—	26	2	no	—
127	M₃₂	—	26	2	no	—
128	M₃₂	Dic₂₆	1	2	yes	—
129	M₃₂	D₅₂	1	2	yes	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	26	2	no	—
133	#108	—	13	2	no	—
134	#108	—	13	2	no	—
135	#107	—	26	1	no	—
136	#106	—	26	1	no	—
137	#97	—	26	2	no	—
138	C₄×C₈	—	26	2	no	—
139	C₄×C₈	Dic₂₆	1	1	yes	characteristic
140	C₄×C₈	Dic₂₆	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1939	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
149	M₁₆	—	13	4	no	—
150	M₁₆	—	26	2	no	—
151	C₂×C₈	—	26	2	no	—
152	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
153	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
154	C₂×C₈	#6094	1	1	yes	characteristic
155	C₂×C₈	#6094	1	1	yes	characteristic
156	C₂×C₈	#6101	1	1	yes	characteristic
157	C₂×C₈	#6101	1	1	yes	characteristic
158	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
160	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
161	C₂×C₈	#6127	1	1	yes	characteristic
162	C₂×C₈	#6127	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
164	C₄⋊C₄	—	26	2	no	—
165	C₂²⋊C₄	—	26	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	26	2	no	—
168	C₄²	#6101	1	1	yes	characteristic
169	C₄²	#6101	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	52	2	no	—
173	C₁₃	#13164	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88155	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
180	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
181	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#6094	1	1	yes	characteristic
183	C₂×C₄	#88155	1	1	yes	characteristic
184	C₂×C₄	C₂×#6101	1	1	yes	characteristic
185	C₈	—	26	4	no	—
186	C₈	#88054	1	1	yes	characteristic
187	C₈	#88054	1	1	yes	characteristic
188	C₈	#88054	1	1	yes	characteristic
189	C₈	#88054	1	1	yes	characteristic
190	C₈	#88123	1	1	yes	characteristic
191	C₈	#88124	1	1	yes	characteristic
192	C₈	#88178	1	1	yes	characteristic
193	C₈	#88184	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	C₈×Dic₂₆	1	1	yes	characteristic
196	C₂²	#11718349	1	1	yes	characteristic
197	C₂²	#11718763	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	104	2	no	—
201	C₄	#11718340	1	1	yes	characteristic
202	C₄	C₈×D₅₂	1	1	yes	characteristic
203	C₄	#11718589	1	1	yes	characteristic
204	C₄	C₂×#88054	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#13115554	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88184
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂×C₁₃	#1939
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#13164
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
173. C₁₃	#13164	semidirect

Maximal subgroups

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

subgroup	index
#13116601	2
C₂×#11718055	2
#13114715	2
#13112923	2
C₄×C₁₃×M₃₂	2
#13164	13