Subgroups — #15756722

Order 3328.

354 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115007	C₂	1	2	yes	maximal
2	#13113595	C₂	1	1	yes	maximal, characteristic
3	#13113594	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	C₂×#88062	—	2	4	no	—
9	#11718585	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#335	—	2	2	no	—
13	#11718071	—	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	#88062	—	4	8	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
37	C₄×#1055	D₄	1	2	yes	—
38	C₄×#1055	Q₈	1	2	yes	—
39	#8823	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₁₃×M₁₆	—	4	4	no	—
44	C₂×C₈×C₁₃	—	2	4	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	#6099	—	4	4	no	—
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	C₄⋊C₄	1	2	yes	—
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	D₈	1	4	yes	—
60	C₂×#1055	Q₁₆	1	4	yes	—
61	#1848	—	13	2	no	—
62	#1567	—	13	2	no	—
63	#1567	—	13	1	no	—
64	#1567	D₁₃	1	1	yes	characteristic
65	C₂×C₈²	—	13	1	no	—
66	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
77	C₈×C₁₃	—	2	2	no	—
78	C₈×C₁₃	—	8	2	no	—
79	C₈×C₁₃	#108	1	4	yes	complemented
80	#1055	—	2	2	no	—
81	#1055	—	8	2	no	—
82	#1055	#107	1	4	yes	—
83	#1055	#107	1	4	yes	—
84	#1055	#108	1	4	yes	complemented
85	C₂×#108	—	26	4	no	—
86	#423	—	13	2	no	—
87	#423	C₂×D₁₃	1	2	yes	—
88	C₂×C₄×C₈	—	13	1	no	—
89	C₂×C₄×C₈	—	13	1	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	#335	—	2	2	no	—
92	#335	—	26	4	no	—
93	#335	—	26	2	no	—
94	C₈²	—	13	1	no	—
95	C₈²	—	13	1	no	—
96	C₈²	—	13	1	no	—
97	C₈²	—	13	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	#384	1	1	yes	characteristic
101	C₂²×C₁₃	#500	1	2	yes	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	C₂×#107	1	2	yes	—
105	C₄×C₁₃	C₂×#108	1	2	yes	—
106	C₄×C₁₃	C₂×#108	1	2	yes	—
107	C₄×C₁₃	#493	1	1	yes	characteristic
108	C₄×C₁₃	#494	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	26	2	no	—
111	C₂²×C₈	—	13	2	no	—
112	C₂²×C₈	—	13	2	no	—
113	C₂²×C₈	Dic₂₆	1	2	yes	—
114	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
115	#108	—	52	8	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	26	2	no	—
119	C₄⋊C₈	—	26	2	no	—
120	C₄×C₈	—	13	1	no	—
121	C₄×C₈	—	13	1	no	—
122	C₄×C₈	—	13	1	no	—
123	C₄×C₈	—	13	1	no	—
124	C₄×C₈	—	13	2	no	—
125	C₄×C₈	—	13	2	no	—
126	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
127	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	#1845	1	1	yes	characteristic
131	C₂×C₁₃	#1848	1	1	yes	characteristic
132	C₂×C₁₃	#1848	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
136	M₁₆	—	4	4	no	—
137	M₁₆	—	52	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	13	2	no	—
141	C₂×C₈	—	13	2	no	—
142	C₂×C₈	—	13	2	no	—
143	C₂×C₈	—	13	4	no	—
144	C₂×C₈	—	13	4	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	26	4	no	—
147	C₂×C₈	—	52	2	no	—
148	C₂×C₈	#6101	1	2	yes	—
149	C₂×C₈	#6101	1	2	yes	—
150	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₄²	D₄×D₁₃	1	1	yes	characteristic
152	C₄²	Q₈×D₁₃	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	C₁₃	#8823	1	1	yes	characteristic, complemented, Sylow
155	C₂³	#88142	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	#88142	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
161	C₂×C₄	D₈×D₁₃	1	2	yes	—
162	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
163	C₂×C₄	C₂×#6101	1	1	yes	characteristic
164	C₂×C₄	#88194	1	1	yes	characteristic
165	C₂×C₄	#88198	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	—	13	4	no	—
169	C₈	—	13	4	no	—
170	C₈	—	13	4	no	—
171	C₈	—	26	2	no	—
172	C₈	—	104	2	no	—
173	C₈	#88062	1	4	yes	complemented
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#11718535	1	2	yes	—
177	C₂²	#11718637	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	D₁₃×#107	1	2	yes	—
181	C₄	D₁₃×#108	1	2	yes	—
182	C₄	C₂×#88062	1	2	yes	—
183	C₄	#11718812	1	1	yes	characteristic
184	C₄	#11718842	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13114994	1	1	yes	characteristic
187	C₂	#13115006	1	1	yes	characteristic
188	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	#8823
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
79. C₈×C₁₃	#108	semidirect
84. #1055	#108	semidirect
154. C₁₃	#8823	semidirect
173. C₈	#88062	semidirect

Maximal subgroups

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

subgroup	index
#13115007	2
#13113595	2
#13113594	2
C₂×C₈×#1055	2
C₁₃×#1567	2
#8823	13