Subgroups — C₂³×#88075 #16726431

Order 3328.

5502 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88075	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	—	2	8	no	—
6	C₂³×Q₈×C₁₃	—	2	7	no	—
7	C₂³×Q₈×C₁₃	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₁₃	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₁₃	—	2	1	no	—
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₁₃	—	2	1	no	—
12	C₂²×C₄⋊Dic₁₃	—	2	7	no	—
13	C₂²×C₄⋊Dic₁₃	C₂²	1	14	yes	—
14	C₂³×#1055	—	2	8	no	—
15	C₂³×#1055	C₂²	1	14	yes	—
16	C₂×#88075	C₂²	1	112	yes	direct product factor
17	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×Q₈×C₁₃	—	2	112	no	—
19	C₂²×Q₈×C₁₃	—	2	56	no	—
20	C₂²×Q₈×C₁₃	—	2	42	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	112	yes	complemented
22	C₂²×Q₈×C₁₃	C₂³	1	28	yes	—
23	C₂³×C₄×C₁₃	—	2	16	no	—
24	C₂³×C₄×C₁₃	—	2	14	no	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₁₃	D₄	1	7	yes	—
27	C₂³×C₄×C₁₃	D₄	1	8	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
29	C₂³×Dic₁₃	—	2	14	no	—
30	C₂³×Dic₁₃	—	4	8	no	—
31	C₂×C₄⋊Dic₁₃	—	2	42	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	28	yes	—
33	C₂²×#1055	—	2	56	no	—
34	C₂²×#1055	C₂³	1	28	yes	—
35	#88075	C₂³	1	64	yes	direct product factor
36	C₂³×#103	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂²⋊C₄	1	8	yes	—
38	C₂⁴×C₁₃	SD₁₆	1	8	yes	—
39	C₂⁴×C₁₃	Q₁₆	1	8	yes	—
40	C₂⁴×C₁₃	C₂×D₄	1	7	yes	—
41	C₂×Q₈×C₁₃	—	2	336	no	—
42	C₂×Q₈×C₁₃	—	2	56	no	—
43	C₂×Q₈×C₁₃	—	2	28	no	—
44	C₂×Q₈×C₁₃	C₂²×C₄	1	112	yes	complemented
45	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₁₃	—	2	112	no	—
47	C₂²×C₄×C₁₃	—	2	28	no	—
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	56	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	28	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	28	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
53	C₂²×Dic₁₃	—	2	28	no	—
54	C₂²×Dic₁₃	—	4	56	no	—
55	C₄⋊Dic₁₃	—	2	28	no	—
56	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
57	C₂×#1055	—	2	56	no	—
58	C₂×#1055	C₂⁴	1	8	yes	—
59	C₂⁴×Q₈	D₁₃	1	1	yes	characteristic
60	C₂⁴×C₈	—	13	1	no	—
61	C₂³×C₄⋊C₄	—	13	1	no	—
62	C₂²×#103	—	13	28	no	—
63	C₂³×C₁₃	#103	1	64	yes	complemented
64	C₂³×C₁₃	C₂×C₂²⋊C₄	1	28	yes	—
65	C₂³×C₁₃	C₂×SD₁₆	1	28	yes	—
66	C₂³×C₁₃	C₂×Q₁₆	1	28	yes	—
67	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
68	Q₈×C₁₃	—	2	112	no	—
69	Q₈×C₁₃	—	2	8	no	—
70	Q₈×C₁₃	C₂³×C₄	1	16	yes	complemented
71	C₂×C₄×C₁₃	—	2	112	no	—
72	C₂×C₄×C₁₃	—	2	8	no	—
73	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	84	yes	—
74	C₂×C₄×C₁₃	C₂³×C₄	1	14	yes	—
75	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
76	C₂×C₄×C₁₃	C₂²×D₄	1	14	yes	—
77	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₁₃	—	2	8	no	—
79	C₂×Dic₁₃	—	4	56	no	—
80	#1055	—	2	8	no	—
81	C₂³×Q₈	—	2	8	no	—
82	C₂³×Q₈	—	2	7	no	—
83	C₂³×Q₈	Dic₁₃	1	16	yes	complemented
84	C₂³×Q₈	C₂×D₁₃	1	14	yes	—
85	C₂⁴×C₄	—	2	1	no	—
86	C₂⁴×C₄	—	26	1	no	—
87	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
88	C₂³×C₈	—	13	14	no	—
89	C₂³×C₈	—	26	8	no	—
90	C₂²×C₄⋊C₄	—	13	14	no	—
91	C₂²×C₄⋊C₄	—	26	7	no	—
92	C₂×#103	—	13	112	no	—
93	C₂²×C₁₃	C₂×#103	1	112	yes	complemented
94	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	14	yes	—
95	C₂²×C₁₃	C₂²×SD₁₆	1	14	yes	—
96	C₂²×C₁₃	C₂²×Q₁₆	1	14	yes	—
97	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₁₃	—	2	16	no	—
99	C₄×C₁₃	C₂²×C₂²⋊C₄	1	14	yes	—
100	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
102	Dic₁₃	—	4	8	no	—
103	C₂⁵	C₂²⋊D₁₃	1	1	yes	characteristic
104	C₂²×Q₈	—	2	112	no	—
105	C₂²×Q₈	—	2	56	no	—
106	C₂²×Q₈	—	2	42	no	—
107	C₂²×Q₈	C₂×Dic₁₃	1	112	yes	complemented
108	C₂²×Q₈	C₂²×D₁₃	1	28	yes	—
109	C₂³×C₄	—	2	16	no	—
110	C₂³×C₄	—	2	14	no	—
111	C₂³×C₄	—	26	14	no	—
112	C₂³×C₄	—	52	8	no	—
113	C₂³×C₄	C₂×Dic₁₃	1	8	yes	—
114	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
115	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
116	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
117	C₂²×C₈	—	13	28	no	—
118	C₂²×C₈	—	26	56	no	—
119	C₂×C₄⋊C₄	—	13	28	no	—
120	C₂×C₄⋊C₄	—	26	42	no	—
121	#103	—	13	64	no	—
122	C₂×C₁₃	C₂²×#103	1	28	yes	complemented
123	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
125	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
126	C₂⁴	#6106	1	8	yes	—
127	C₂⁴	#6107	1	8	yes	—
128	C₂⁴	C₂²⋊Dic₁₃	1	8	yes	—
129	C₂⁴	C₂×C₂²⋊D₁₃	1	7	yes	—
130	C₂×Q₈	—	2	336	no	—
131	C₂×Q₈	—	2	56	no	—
132	C₂×Q₈	—	2	28	no	—
133	C₂×Q₈	C₂²×Dic₁₃	1	112	yes	complemented
134	C₂×Q₈	C₂³×D₁₃	1	8	yes	—
135	C₂²×C₄	—	2	112	no	—
136	C₂²×C₄	—	2	28	no	—
137	C₂²×C₄	—	26	28	no	—
138	C₂²×C₄	—	52	56	no	—
139	C₂²×C₄	C₂²⋊Dic₁₃	1	56	yes	—
140	C₂²×C₄	C₂²×Dic₁₃	1	28	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₁₃	1	28	yes	—
143	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
144	C₂×C₈	—	13	8	no	—
145	C₂×C₈	—	26	56	no	—
146	C₄⋊C₄	—	13	8	no	—
147	C₄⋊C₄	—	26	28	no	—
148	C₁₃	C₂³×#103	1	1	yes	characteristic, complemented, Sylow
149	C₂³	#88075	1	64	yes	direct product factor
150	C₂³	C₂×#6106	1	28	yes	—
151	C₂³	C₂×#6107	1	28	yes	—
152	C₂³	C₂×C₂²⋊Dic₁₃	1	28	yes	—
153	C₂³	C₂²×C₂²⋊D₁₃	1	7	yes	—
154	Q₈	—	2	112	no	—
155	Q₈	—	2	8	no	—
156	Q₈	C₂³×Dic₁₃	1	16	yes	complemented
157	C₂×C₄	—	2	112	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	26	8	no	—
160	C₂×C₄	—	52	56	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	84	yes	—
162	C₂×C₄	C₂³×Dic₁₃	1	14	yes	—
163	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₁₃	1	14	yes	—
165	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
166	C₈	—	26	8	no	—
167	C₂²	C₂×#88075	1	112	yes	direct product factor
168	C₂²	C₂²×#6106	1	14	yes	—
169	C₂²	C₂²×#6107	1	14	yes	—
170	C₂²	C₂²×C₂²⋊Dic₁₃	1	14	yes	—
171	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	52	8	no	—
174	C₄	C₂²×C₂²⋊Dic₁₃	1	14	yes	—
175	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
176	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₂	C₂²×#88075	1	28	yes	direct product factor
178	C₂	C₂³×C₂²⋊Dic₁₃	1	1	yes	characteristic
179	C₂	C₂³×#6106	1	1	yes	characteristic
180	C₂	C₂³×#6107	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂⁴×Q₈×C₁₃	C₂
socle, prime radical	C₂⁵×C₁₃	D₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂⁴×Q₈	D₁₃
hypocenter	C₁₃	C₂³×#103
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#88075	C₂	direct (N × H)
7. C₂³×Q₈×C₁₃	C₄	semidirect
16. C₂×#88075	C₂²	direct (N × H)
21. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
35. #88075	C₂³	direct (N × H)
44. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
63. C₂³×C₁₃	#103	semidirect
70. Q₈×C₁₃	C₂³×C₄	semidirect
83. C₂³×Q₈	Dic₁₃	semidirect
93. C₂²×C₁₃	C₂×#103	semidirect
107. C₂²×Q₈	C₂×Dic₁₃	semidirect
122. C₂×C₁₃	C₂²×#103	semidirect
133. C₂×Q₈	C₂²×Dic₁₃	semidirect
148. C₁₃	C₂³×#103	semidirect
149. C₂³	#88075	direct (N × H)
156. Q₈	C₂³×Dic₁₃	semidirect
167. C₂²	C₂×#88075	direct (N × H)
177. C₂	C₂²×#88075	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#1055	2
C₂³×C₄⋊Dic₁₃	2
C₂²×#88075	2
C₂⁴×Q₈×C₁₃	2
C₂³×#103	13

Subgroups — C23×#88075 #16726431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88075 #16726431