Subgroups — C₂³×#88144 #16769586

Order 3328.

8414 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88144	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
3	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#6101	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	28	yes	—
13	C₂²×C₄⋊Dic₁₃	C₂²	1	56	yes	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	—
15	C₂²×#6101	C₂²	1	56	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	28	yes	—
17	C₂×#88144	C₂²	1	448	yes	direct product factor
18	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	—	2	16	no	—
20	C₂³×C₄×C₁₃	Q₈	1	16	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	28	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
23	C₂³×Dic₁₃	—	2	8	no	—
24	C₂³×Dic₁₃	—	2	8	no	—
25	C₂³×Dic₁₃	—	2	16	no	—
26	C₂³×Dic₁₃	C₂³	1	14	yes	—
27	C₂³×Dic₁₃	C₂³	1	14	yes	—
28	C₂³×Dic₁₃	C₂³	1	28	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	—
30	C₂×C₄⋊Dic₁₃	C₂³	1	224	yes	—
31	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	—
32	C₂×#6101	C₂³	1	224	yes	—
33	C₂×C₄×Dic₁₃	C₂³	1	112	yes	—
34	#88144	C₂³	1	512	yes	direct product factor
35	C₂³×#125	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
37	C₂⁴×C₁₃	#41	1	8	yes	—
38	C₂⁴×C₁₃	#41	1	8	yes	—
39	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
40	C₂²×C₄×C₁₃	—	2	112	no	—
41	C₂²×C₄×C₁₃	C₂×Q₈	1	112	yes	—
42	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	—
43	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
44	C₂²×Dic₁₃	—	2	56	no	—
45	C₂²×Dic₁₃	—	2	56	no	—
46	C₂²×Dic₁₃	—	2	112	no	—
47	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
48	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
49	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
50	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
51	C₄⋊Dic₁₃	C₂⁴	1	128	yes	—
52	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
53	#6101	C₂⁴	1	128	yes	—
54	C₄×Dic₁₃	C₂⁴	1	64	yes	—
55	C₂²×#125	—	13	56	no	—
56	C₂³×C₄⋊C₄	—	13	2	no	—
57	C₂³×C₄⋊C₄	—	13	2	no	—
58	C₂³×C₄⋊C₄	—	13	1	no	—
59	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
60	C₂³×C₄²	—	13	1	no	—
61	C₂³×C₁₃	#125	1	64	yes	complemented
62	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
63	C₂³×C₁₃	C₂×#41	1	28	yes	—
64	C₂³×C₁₃	C₂×#41	1	28	yes	—
65	C₂³×C₁₃	C₂⁵	1	7	yes	—
66	C₂×C₄×C₁₃	—	2	112	no	—
67	C₂×C₄×C₁₃	C₂²×Q₈	1	112	yes	—
68	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
69	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
70	C₂×Dic₁₃	—	2	56	no	—
71	C₂×Dic₁₃	—	2	56	no	—
72	C₂×Dic₁₃	—	2	112	no	—
73	C₂×Dic₁₃	C₂⁵	1	8	yes	—
74	C₂×Dic₁₃	C₂⁵	1	8	yes	—
75	C₂×Dic₁₃	C₂⁵	1	16	yes	—
76	C₂⁴×C₄	—	13	1	no	—
77	C₂⁴×C₄	—	13	1	no	—
78	C₂⁴×C₄	—	13	2	no	—
79	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
80	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
81	C₂×#125	—	13	448	no	—
82	C₂²×C₄⋊C₄	—	13	56	no	—
83	C₂²×C₄⋊C₄	—	13	28	no	—
84	C₂²×C₄⋊C₄	—	13	56	no	—
85	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	—
86	C₂²×C₄²	—	13	28	no	—
87	C₂²×C₁₃	C₂×#125	1	112	yes	complemented
88	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
89	C₂²×C₁₃	C₂²×#41	1	14	yes	—
90	C₂²×C₁₃	C₂²×#41	1	14	yes	—
91	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
92	C₄×C₁₃	—	2	16	no	—
93	C₄×C₁₃	C₂³×Q₈	1	16	yes	—
94	Dic₁₃	—	2	8	no	—
95	Dic₁₃	—	2	8	no	—
96	Dic₁₃	—	2	16	no	—
97	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
98	C₂³×C₄	—	2	16	no	—
99	C₂³×C₄	—	13	28	no	—
100	C₂³×C₄	—	13	14	no	—
101	C₂³×C₄	—	13	14	no	—
102	C₂³×C₄	—	26	8	no	—
103	C₂³×C₄	—	26	8	no	—
104	C₂³×C₄	—	26	16	no	—
105	C₂³×C₄	Dic₂₆	1	16	yes	—
106	C₂³×C₄	C₂²×D₁₃	1	28	yes	—
107	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
108	#125	—	13	512	no	—
109	C₂×C₄⋊C₄	—	13	224	no	—
110	C₂×C₄⋊C₄	—	13	224	no	—
111	C₂×C₄⋊C₄	—	13	112	no	—
112	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	—
113	C₂×C₄²	—	13	112	no	—
114	C₂×C₁₃	C₂²×#125	1	28	yes	complemented
115	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
116	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
117	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
118	C₂⁴	C₂×Dic₂₆	1	8	yes	—
119	C₂⁴	#6129	1	8	yes	—
120	C₂⁴	#6131	1	8	yes	—
121	C₂⁴	C₂³×D₁₃	1	7	yes	—
122	C₂²×C₄	—	2	112	no	—
123	C₂²×C₄	—	13	28	no	—
124	C₂²×C₄	—	13	28	no	—
125	C₂²×C₄	—	13	56	no	—
126	C₂²×C₄	—	26	56	no	—
127	C₂²×C₄	—	26	112	no	—
128	C₂²×C₄	—	26	56	no	—
129	C₂²×C₄	C₂×Dic₂₆	1	112	yes	—
130	C₂²×C₄	C₂³×D₁₃	1	56	yes	—
131	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
132	C₄⋊C₄	—	13	128	no	—
133	C₄⋊C₄	—	13	128	no	—
134	C₄⋊C₄	—	13	64	no	—
135	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
136	C₄²	—	13	64	no	—
137	C₁₃	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#88144	1	64	yes	direct product factor
139	C₂³	C₂²×Dic₂₆	1	28	yes	—
140	C₂³	C₂×#6129	1	28	yes	—
141	C₂³	C₂×#6131	1	28	yes	—
142	C₂³	C₂⁴×D₁₃	1	7	yes	—
143	C₂×C₄	—	2	112	no	—
144	C₂×C₄	—	13	8	no	—
145	C₂×C₄	—	13	8	no	—
146	C₂×C₄	—	13	16	no	—
147	C₂×C₄	—	26	56	no	—
148	C₂×C₄	—	26	56	no	—
149	C₂×C₄	—	26	112	no	—
150	C₂×C₄	C₂²×Dic₂₆	1	112	yes	—
151	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
152	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
153	C₂²	C₂×#88144	1	112	yes	direct product factor
154	C₂²	C₂³×Dic₂₆	1	14	yes	—
155	C₂²	C₂²×#6129	1	14	yes	—
156	C₂²	C₂²×#6131	1	14	yes	—
157	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
158	C₄	—	2	16	no	—
159	C₄	—	26	8	no	—
160	C₄	—	26	8	no	—
161	C₄	—	26	16	no	—
162	C₄	C₂³×Dic₂₆	1	16	yes	—
163	C₂	C₂²×#88144	1	28	yes	direct product factor
164	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
165	C₂	C₂³×#6129	1	1	yes	characteristic
166	C₂	C₂³×#6131	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#125
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88144	C₂	direct (N × H)
17. C₂×#88144	C₂²	direct (N × H)
34. #88144	C₂³	direct (N × H)
61. C₂³×C₁₃	#125	semidirect
87. C₂²×C₁₃	C₂×#125	semidirect
114. C₂×C₁₃	C₂²×#125	semidirect
137. C₁₃	C₂³×#125	semidirect
138. C₂³	#88144	direct (N × H)
153. C₂²	C₂×#88144	direct (N × H)
163. C₂	C₂²×#88144	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88144	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×#125	13

Subgroups — C23×#88144 #16769586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88144 #16769586