Subgroups — C₂³×#88059 #16726608

Order 3328.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6099	C₂	1	2	yes	maximal
2	C₂²×#88059	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₄	1	14	yes	—
6	C₂³×C₈×C₁₃	C₄	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	14	yes	—
8	C₂²×#6099	—	2	14	no	—
9	C₂²×#6099	C₂²	1	28	yes	—
10	C₂³×#1055	—	2	2	no	—
11	C₂×#88059	C₂²	1	112	yes	direct product factor
12	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	D₄	1	7	yes	—
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	Q₈	1	7	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	84	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	28	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	28	yes	—
22	C₂×#6099	—	2	84	no	—
23	C₂×#6099	C₂³	1	56	yes	—
24	C₂²×#1055	—	2	28	no	—
25	#88059	C₂³	1	64	yes	direct product factor
26	C₂³×#108	—	13	1	no	maximal, Sylow
27	C₂⁴×C₁₃	—	2	8	no	—
28	C₂⁴×C₁₃	C₄⋊C₄	1	7	yes	—
29	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂⁴×C₁₃	C₂×Q₈	1	7	yes	—
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	42	yes	—
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	7	yes	—
33	C₂²×C₄×C₁₃	C₄⋊C₄	1	7	yes	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
35	C₂²×C₄×C₁₃	C₂²×C₄	1	7	yes	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
38	C₂²×C₄×C₁₃	C₂×Q₈	1	21	yes	—
39	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	56	yes	—
41	C₂×C₈×C₁₃	C₂²×C₄	1	56	yes	—
42	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
43	#6099	—	2	56	no	—
44	#6099	C₂⁴	1	16	yes	—
45	C₂×#1055	—	2	56	no	—
46	C₂³×M₁₆	—	13	2	no	—
47	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
48	C₂²×#108	—	13	28	no	—
49	C₂³×C₁₃	—	2	28	no	—
50	C₂³×C₁₃	—	2	28	no	—
51	C₂³×C₁₃	#108	1	8	yes	complemented
52	C₂³×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
53	C₂³×C₁₃	C₂×C₄⋊C₄	1	7	yes	—
54	C₂³×C₁₃	C₂²×Q₈	1	7	yes	—
55	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	42	yes	—
56	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
58	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
59	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
60	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
61	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
62	C₂×C₄×C₁₃	C₂²×Q₈	1	7	yes	—
63	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
64	C₈×C₁₃	C₂³×C₄	1	16	yes	—
65	#1055	—	2	16	no	—
66	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
67	C₂²×M₁₆	—	13	28	no	—
68	C₂²×M₁₆	—	26	14	no	—
69	C₂³×C₈	—	26	2	no	—
70	C₂³×C₈	Dic₁₃	1	2	yes	—
71	C₂³×C₈	Dic₁₃	1	14	yes	—
72	C₂³×C₈	C₂×D₁₃	1	14	yes	—
73	C₂×#108	—	13	112	no	—
74	C₂²×C₁₃	—	2	42	no	—
75	C₂²×C₁₃	—	2	14	no	—
76	C₂²×C₁₃	C₂×#108	1	28	yes	complemented
77	C₂²×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
78	C₂²×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
79	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
80	C₄×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
81	C₄×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
82	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
83	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
84	C₂⁵	Dic₂₆	1	1	yes	characteristic
85	C₂³×C₄	Dic₂₆	1	7	yes	—
86	C₂³×C₄	D₅₂	1	7	yes	—
87	C₂³×C₄	D₅₂	1	1	yes	characteristic
88	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
89	C₂³×C₄	C₂×Dic₁₃	1	7	yes	—
90	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
91	C₂×M₁₆	—	13	56	no	—
92	C₂×M₁₆	—	26	84	no	—
93	C₂²×C₈	—	26	28	no	—
94	C₂²×C₈	C₂×Dic₁₃	1	84	yes	—
95	C₂²×C₈	C₂×Dic₁₃	1	28	yes	—
96	C₂²×C₈	C₂²×D₁₃	1	28	yes	—
97	#108	—	13	64	no	—
98	C₂×C₁₃	—	2	7	no	—
99	C₂×C₁₃	—	2	1	no	—
100	C₂×C₁₃	C₂²×#108	1	14	yes	complemented
101	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
102	C₂⁴	—	2	8	no	—
103	C₂⁴	C₄⋊Dic₁₃	1	7	yes	—
104	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
105	C₂⁴	C₂×Dic₂₆	1	7	yes	—
106	C₂²×C₄	C₄⋊Dic₁₃	1	42	yes	—
107	C₂²×C₄	C₄⋊Dic₁₃	1	7	yes	—
108	C₂²×C₄	C₄⋊Dic₁₃	1	7	yes	—
109	C₂²×C₄	C₂×Dic₂₆	1	21	yes	—
110	C₂²×C₄	C₂×D₅₂	1	7	yes	—
111	C₂²×C₄	C₂×D₅₂	1	21	yes	—
112	C₂²×C₄	C₂²×Dic₁₃	1	21	yes	—
113	C₂²×C₄	C₂²×Dic₁₃	1	7	yes	—
114	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
115	M₁₆	—	13	16	no	—
116	M₁₆	—	26	56	no	—
117	C₂×C₈	—	26	56	no	—
118	C₂×C₈	C₂²×Dic₁₃	1	56	yes	—
119	C₂×C₈	C₂²×Dic₁₃	1	56	yes	—
120	C₂×C₈	C₂³×D₁₃	1	8	yes	—
121	C₁₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
122	C₂³	—	2	28	no	—
123	C₂³	—	2	28	no	—
124	C₂³	#88059	1	8	yes	direct product factor
125	C₂³	C₂×C₄⋊Dic₁₃	1	21	yes	—
126	C₂³	C₂×C₄⋊Dic₁₃	1	7	yes	—
127	C₂³	C₂²×Dic₂₆	1	7	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₁₃	1	42	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₁₃	1	21	yes	—
130	C₂×C₄	C₂×C₄⋊Dic₁₃	1	21	yes	—
131	C₂×C₄	C₂²×Dic₂₆	1	7	yes	—
132	C₂×C₄	C₂²×D₅₂	1	7	yes	—
133	C₂×C₄	C₂²×D₅₂	1	7	yes	—
134	C₂×C₄	C₂³×Dic₁₃	1	7	yes	—
135	C₂×C₄	C₂³×Dic₁₃	1	7	yes	—
136	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
137	C₈	—	26	16	no	—
138	C₈	C₂³×Dic₁₃	1	16	yes	—
139	C₂²	—	2	42	no	—
140	C₂²	—	2	14	no	—
141	C₂²	C₂×#88059	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊Dic₁₃	1	7	yes	—
143	C₂²	C₂²×C₄⋊Dic₁₃	1	7	yes	—
144	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊Dic₁₃	1	7	yes	—
146	C₄	C₂²×C₄⋊Dic₁₃	1	7	yes	—
147	C₄	C₂³×D₅₂	1	1	yes	characteristic
148	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
149	C₂	—	2	7	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂²×#88059	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₄⋊C₄
prime radical	C₂⁵×C₁₃	Q₈
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂³×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#88059	C₂	direct (N × H)
11. C₂×#88059	C₂²	direct (N × H)
25. #88059	C₂³	direct (N × H)
51. C₂³×C₁₃	#108	semidirect
76. C₂²×C₁₃	C₂×#108	semidirect
100. C₂×C₁₃	C₂²×#108	semidirect
121. C₁₃	C₂³×#108	semidirect
124. C₂³	#88059	direct (N × H)
141. C₂²	C₂×#88059	direct (N × H)
151. C₂	C₂²×#88059	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
C₂²×#88059	2
C₂⁴×C₈×C₁₃	2
C₂³×#108	13

Subgroups — C23×#88059 #16726608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88059 #16726608