Subgroups — #15756726

Order 3328.

322 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114973	C₂	1	2	yes	maximal
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13113591	C₂	1	2	yes	maximal
4	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1567	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718585	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₁₃×#335	—	2	2	no	—
12	#11718150	—	2	4	no	—
13	#11718072	—	2	4	no	—
14	#11718050	—	2	2	no	—
15	C₈×#1055	C₄	1	1	yes	characteristic
16	C₈×#1055	C₄	1	1	yes	characteristic
17	C₈×#1055	C₄	1	1	yes	characteristic
18	C₈×#1055	C₄	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6099	—	2	2	no	—
23	C₂²×#1055	D₄	1	2	yes	—
24	C₂²×#1055	Q₈	1	2	yes	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	C₄×#1055	C₂×C₄	1	1	yes	characteristic
34	C₄×#1055	C₂×C₄	1	1	yes	characteristic
35	C₄×#1055	C₂×C₄	1	1	yes	characteristic
36	C₄×#1055	D₄	1	2	yes	—
37	C₄×#1055	Q₈	1	2	yes	—
38	#8824	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₁₃×M₁₆	—	4	4	no	—
43	C₂×C₈×C₁₃	—	2	4	no	—
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	#6099	—	4	4	no	—
52	C₂×#1055	—	2	2	no	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	2	4	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	C₄⋊C₄	1	2	yes	—
57	C₂×#1055	C₄⋊C₄	1	2	yes	—
58	C₂×#1055	C₄⋊C₄	1	2	yes	—
59	C₂×#1055	C₄⋊C₄	1	2	yes	—
60	#1837	—	13	2	no	—
61	#1567	—	13	1	no	—
62	#1567	D₁₃	1	1	yes	characteristic
63	#1566	—	13	2	no	—
64	C₂×C₈²	—	13	1	no	—
65	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#135	1	2	yes	—
75	C₂×C₄×C₁₃	#135	1	2	yes	—
76	C₈×C₁₃	—	2	2	no	—
77	C₈×C₁₃	—	8	2	no	—
78	C₈×C₁₃	#108	1	4	yes	—
79	#1055	—	2	2	no	—
80	#1055	—	2	2	no	—
81	#1055	—	2	2	no	—
82	#1055	—	8	2	no	—
83	#1055	#108	1	4	yes	—
84	#423	—	13	2	no	—
85	#423	C₂×D₁₃	1	2	yes	—
86	C₂×C₄×C₈	—	13	1	no	—
87	C₂×C₄×C₈	—	13	1	no	—
88	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	#341	—	26	4	no	—
90	#335	—	2	2	no	—
91	#335	—	26	2	no	—
92	#334	—	26	4	no	—
93	C₈²	—	13	1	no	—
94	C₈²	—	13	1	no	—
95	C₈²	—	13	1	no	—
96	C₈²	—	13	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	#384	1	1	yes	characteristic
100	C₂²×C₁₃	#499	1	2	yes	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	#427	1	2	yes	—
104	C₄×C₁₃	C₂×#108	1	2	yes	—
105	C₄×C₁₃	C₂×#108	1	2	yes	—
106	C₄×C₁₃	#495	1	1	yes	characteristic
107	C₄×C₁₃	#495	1	1	yes	characteristic
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	26	2	no	—
110	C₂²×C₈	—	13	2	no	—
111	C₂²×C₈	—	13	2	no	—
112	C₂²×C₈	Dic₂₆	1	2	yes	—
113	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₄⋊C₈	—	26	2	no	—
118	C₄×C₈	—	13	1	no	—
119	C₄×C₈	—	13	1	no	—
120	C₄×C₈	—	13	1	no	—
121	C₄×C₈	—	13	1	no	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	—	13	2	no	—
124	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
125	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
127	C₂×C₁₃	—	2	2	no	—
128	C₂×C₁₃	#1846	1	1	yes	characteristic
129	C₂×C₁₃	#1848	1	1	yes	characteristic
130	C₂×C₁₃	#1848	1	1	yes	characteristic
131	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
134	M₁₆	—	4	4	no	—
135	M₁₆	—	52	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	13	2	no	—
139	C₂×C₈	—	13	2	no	—
140	C₂×C₈	—	13	2	no	—
141	C₂×C₈	—	13	2	no	—
142	C₂×C₈	—	26	4	no	—
143	C₂×C₈	—	26	2	no	—
144	C₂×C₈	—	26	2	no	—
145	C₂×C₈	—	52	2	no	—
146	C₂×C₈	#6101	1	2	yes	—
147	C₂×C₈	#6101	1	2	yes	—
148	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
149	C₄²	D₄×D₁₃	1	1	yes	characteristic
150	C₄²	Q₈×D₁₃	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	C₁₃	#8824	1	1	yes	characteristic, complemented, Sylow
153	C₂³	#88142	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	#88142	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
159	C₂×C₄	#88166	1	2	yes	—
160	C₂×C₄	#88173	1	2	yes	—
161	C₂×C₄	C₂×#6101	1	1	yes	characteristic
162	C₂×C₄	#88194	1	1	yes	characteristic
163	C₂×C₄	#88198	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	13	4	no	—
167	C₈	—	26	2	no	—
168	C₈	—	26	2	no	—
169	C₈	—	26	2	no	—
170	C₈	—	104	2	no	—
171	C₈	#88062	1	4	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#11718539	1	2	yes	—
175	C₂²	#11718637	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#11718541	1	2	yes	—
179	C₄	D₁₃×#108	1	2	yes	—
180	C₄	C₂×#88062	1	2	yes	—
181	C₄	#11718816	1	1	yes	characteristic
182	C₄	#11718852	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	#13114999	1	1	yes	characteristic
185	C₂	#13115006	1	1	yes	characteristic
186	C₂	#13115007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1567	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1567	D₁₃
hypocenter	C₁₃	#8824
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
152. C₁₃	#8824	semidirect

Maximal subgroups

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

subgroup	index
#13114973	2
#13113595	2
#13113591	2
C₂×C₈×#1055	2
C₁₃×#1567	2
#8824	13