Subgroups — C₂³×#88062 #16726607

Order 3328.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 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	1	yes	maximal, characteristic
2	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88062	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×M₁₆	—	2	7	no	—
7	C₂²×C₁₃×M₁₆	C₂²	1	14	yes	—
8	C₂³×C₈×C₁₃	—	2	1	no	—
9	C₂²×#6099	—	2	7	no	—
10	C₂²×#6099	C₂²	1	14	yes	—
11	C₂³×#1055	—	2	1	no	—
12	C₂³×#1055	C₄	1	14	yes	—
13	C₂³×#1055	C₄	1	2	yes	—
14	C₂³×#1055	C₂²	1	14	yes	—
15	C₂×#88062	C₂²	1	112	yes	direct product factor
16	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	D₄	1	7	yes	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	Q₈	1	7	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
23	C₂×C₁₃×M₁₆	—	2	42	no	—
24	C₂×C₁₃×M₁₆	C₂³	1	28	yes	—
25	C₂²×C₈×C₁₃	—	2	14	no	—
26	C₂×#6099	—	2	42	no	—
27	C₂×#6099	C₂³	1	28	yes	—
28	C₂²×#1055	—	2	14	no	—
29	C₂²×#1055	C₂×C₄	1	84	yes	—
30	C₂²×#1055	C₂×C₄	1	28	yes	—
31	C₂²×#1055	C₂³	1	28	yes	—
32	#88062	C₂³	1	64	yes	direct product factor
33	C₂³×#108	—	13	1	no	maximal, Sylow
34	C₂⁴×C₁₃	—	2	8	no	—
35	C₂⁴×C₁₃	C₄⋊C₄	1	7	yes	—
36	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂⁴×C₁₃	C₂×Q₈	1	7	yes	—
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	42	yes	—
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	7	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	7	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	7	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	21	yes	—
46	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
47	C₁₃×M₁₆	—	2	28	no	—
48	C₁₃×M₁₆	C₂⁴	1	8	yes	—
49	C₂×C₈×C₁₃	—	2	28	no	—
50	#6099	—	2	28	no	—
51	#6099	C₂⁴	1	8	yes	—
52	C₂×#1055	—	2	28	no	—
53	C₂×#1055	C₂²×C₄	1	56	yes	—
54	C₂×#1055	C₂²×C₄	1	56	yes	—
55	C₂×#1055	C₂⁴	1	8	yes	—
56	C₂³×M₁₆	—	13	1	no	—
57	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
58	C₂⁴×C₈	—	13	1	no	—
59	C₂²×#108	—	13	28	no	—
60	C₂³×C₁₃	—	2	28	no	—
61	C₂³×C₁₃	—	2	28	no	—
62	C₂³×C₁₃	#108	1	8	yes	complemented
63	C₂³×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
64	C₂³×C₁₃	C₂×C₄⋊C₄	1	7	yes	—
65	C₂³×C₁₃	C₂²×Q₈	1	7	yes	—
66	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	42	yes	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	21	yes	—
69	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
70	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
71	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
72	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
73	C₂×C₄×C₁₃	C₂²×Q₈	1	7	yes	—
74	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
75	C₈×C₁₃	—	2	8	no	—
76	#1055	—	2	8	no	—
77	#1055	C₂³×C₄	1	16	yes	—
78	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
79	C₂²×M₁₆	—	2	7	no	—
80	C₂²×M₁₆	—	13	14	no	—
81	C₂²×M₁₆	—	26	7	no	—
82	C₂²×M₁₆	C₂×D₁₃	1	14	yes	—
83	C₂³×C₈	—	2	1	no	—
84	C₂³×C₈	—	13	2	no	—
85	C₂³×C₈	—	13	14	no	—
86	C₂³×C₈	—	13	14	no	—
87	C₂³×C₈	—	26	1	no	—
88	C₂×#108	—	13	112	no	—
89	C₂²×C₁₃	—	2	42	no	—
90	C₂²×C₁₃	—	2	14	no	—
91	C₂²×C₁₃	C₂×#108	1	28	yes	complemented
92	C₂²×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
93	C₂²×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
94	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
95	C₄×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
96	C₄×C₁₃	C₂²×C₄⋊C₄	1	7	yes	—
97	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
98	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
99	C₂⁵	Dic₂₆	1	1	yes	characteristic
100	C₂³×C₄	Dic₂₆	1	7	yes	—
101	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
102	C₂³×C₄	C₄×D₁₃	1	7	yes	—
103	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
104	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
106	C₂×M₁₆	—	2	42	no	—
107	C₂×M₁₆	—	13	28	no	—
108	C₂×M₁₆	—	26	42	no	—
109	C₂×M₁₆	C₂²×D₁₃	1	28	yes	—
110	C₂²×C₈	—	2	14	no	—
111	C₂²×C₈	—	13	84	no	—
112	C₂²×C₈	—	13	28	no	—
113	C₂²×C₈	—	13	28	no	—
114	C₂²×C₈	—	26	14	no	—
115	#108	—	13	64	no	—
116	C₂×C₁₃	—	2	7	no	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	C₂²×#108	1	14	yes	complemented
119	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
120	C₂⁴	—	2	8	no	—
121	C₂⁴	#6101	1	7	yes	—
122	C₂⁴	#6101	1	1	yes	characteristic
123	C₂⁴	C₂×Dic₂₆	1	7	yes	—
124	C₂²×C₄	#6101	1	42	yes	—
125	C₂²×C₄	#6101	1	7	yes	—
126	C₂²×C₄	#6101	1	7	yes	—
127	C₂²×C₄	C₂×Dic₂₆	1	21	yes	—
128	C₂²×C₄	C₂×C₄×D₁₃	1	21	yes	—
129	C₂²×C₄	C₂×C₄×D₁₃	1	7	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₁₃	1	7	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
132	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
133	M₁₆	—	2	28	no	—
134	M₁₆	—	13	8	no	—
135	M₁₆	—	26	28	no	—
136	M₁₆	C₂³×D₁₃	1	8	yes	—
137	C₂×C₈	—	2	28	no	—
138	C₂×C₈	—	13	56	no	—
139	C₂×C₈	—	13	56	no	—
140	C₂×C₈	—	13	8	no	—
141	C₂×C₈	—	26	28	no	—
142	C₁₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	2	28	no	—
144	C₂³	—	2	28	no	—
145	C₂³	#88062	1	8	yes	direct product factor
146	C₂³	C₂×#6101	1	21	yes	—
147	C₂³	C₂×#6101	1	7	yes	—
148	C₂³	C₂²×Dic₂₆	1	7	yes	—
149	C₂×C₄	C₂×#6101	1	42	yes	—
150	C₂×C₄	C₂×#6101	1	21	yes	—
151	C₂×C₄	C₂×#6101	1	21	yes	—
152	C₂×C₄	C₂²×Dic₂₆	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
154	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
156	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
157	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
158	C₈	—	2	8	no	—
159	C₈	—	13	16	no	—
160	C₈	—	26	8	no	—
161	C₂²	—	2	42	no	—
162	C₂²	—	2	14	no	—
163	C₂²	C₂×#88062	1	28	yes	direct product factor
164	C₂²	C₂²×#6101	1	7	yes	—
165	C₂²	C₂²×#6101	1	7	yes	—
166	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
167	C₄	C₂²×#6101	1	7	yes	—
168	C₄	C₂²×#6101	1	7	yes	—
169	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#88062	1	14	yes	direct product factor
174	C₂	C₂³×#6101	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂⁴×C₁₃	C₄⋊C₄
prime radical	C₂⁵×C₁₃	Q₈
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂³×#108
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#88062	C₂	direct (N × H)
15. C₂×#88062	C₂²	direct (N × H)
32. #88062	C₂³	direct (N × H)
62. C₂³×C₁₃	#108	semidirect
91. C₂²×C₁₃	C₂×#108	semidirect
118. C₂×C₁₃	C₂²×#108	semidirect
142. C₁₃	C₂³×#108	semidirect
145. C₂³	#88062	direct (N × H)
163. C₂²	C₂×#88062	direct (N × H)
173. C₂	C₂²×#88062	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
C₂⁴×#1055	2
C₂²×#88062	2
C₂³×C₁₃×M₁₆	2
C₂³×#108	13

Subgroups — C23×#88062 #16726607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88062 #16726607