Subgroups — C₂³×#88151 #16769563

Order 3328.

14078 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88151	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
5	C₂³×#6103	C₂	1	2	yes	maximal
6	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₁₃	—	2	16	no	—
13	C₂³×C₄×D₁₃	C₂²	1	28	yes	—
14	C₂³×Dic₂₆	—	2	16	no	—
15	C₂³×Dic₂₆	C₂²	1	28	yes	complemented
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	28	yes	complemented
17	C₂²×#6103	C₂²	1	56	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	56	yes	complemented
19	C₂×#88151	C₂²	1	448	yes	direct product factor
20	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
21	C₂⁴×D₁₃	—	2	8	no	—
22	C₂⁴×D₁₃	—	2	8	no	—
23	C₂⁴×D₁₃	Q₈	1	16	yes	complemented
24	C₂⁴×D₁₃	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₃	—	2	16	no	—
26	C₂³×C₄×C₁₃	D₄	1	16	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	28	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
29	C₂³×Dic₁₃	—	2	16	no	—
30	C₂³×Dic₁₃	—	2	8	no	—
31	C₂³×Dic₁₃	C₂³	1	14	yes	—
32	C₂³×Dic₁₃	C₂³	1	28	yes	—
33	C₂²×C₄×D₁₃	—	2	224	no	—
34	C₂²×C₄×D₁₃	C₂³	1	112	yes	—
35	C₂²×Dic₂₆	—	2	224	no	—
36	C₂²×Dic₂₆	C₂³	1	112	yes	complemented
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	complemented
38	C₂×#6103	C₂³	1	224	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	224	yes	complemented
40	#88151	C₂³	1	512	yes	direct product factor
41	C₂³×C₂²⋊Q₈	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
43	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
44	C₂⁴×C₁₃	#41	1	8	yes	—
45	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
46	C₂³×D₁₃	—	2	128	no	—
47	C₂³×D₁₃	—	2	56	no	—
48	C₂³×D₁₃	—	2	56	no	—
49	C₂³×D₁₃	C₂×Q₈	1	112	yes	complemented
50	C₂³×D₁₃	C₂⁴	1	28	yes	—
51	C₂²×C₄×C₁₃	—	2	112	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	112	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
55	C₂²×Dic₁₃	—	2	112	no	—
56	C₂²×Dic₁₃	—	2	56	no	—
57	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
58	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
59	C₂×C₄×D₁₃	—	2	448	no	—
60	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
61	C₂×Dic₂₆	—	2	448	no	—
62	C₂×Dic₂₆	C₂⁴	1	64	yes	complemented
63	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	complemented
64	#6103	C₂⁴	1	128	yes	—
65	C₄⋊Dic₁₃	C₂⁴	1	128	yes	complemented
66	C₂⁴×Q₈	—	13	1	no	—
67	C₂⁵×C₄	—	13	1	no	—
68	C₂²×C₂²⋊Q₈	—	13	56	no	—
69	C₂³×C₄⋊C₄	—	13	2	no	—
70	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
71	C₂³×C₂²⋊C₄	—	13	2	no	—
72	C₂³×C₁₃	C₂²⋊Q₈	1	64	yes	complemented
73	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
74	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
75	C₂³×C₁₃	C₂×#41	1	28	yes	—
76	C₂³×C₁₃	C₂⁵	1	7	yes	—
77	C₂²×D₁₃	—	2	448	no	—
78	C₂²×D₁₃	—	2	56	no	—
79	C₂²×D₁₃	—	2	56	no	—
80	C₂²×D₁₃	C₂²×Q₈	1	112	yes	complemented
81	C₂²×D₁₃	C₂⁵	1	8	yes	—
82	C₂×C₄×C₁₃	—	2	112	no	—
83	C₂×C₄×C₁₃	C₂²×D₄	1	112	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
85	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
86	C₂×Dic₁₃	—	2	112	no	—
87	C₂×Dic₁₃	—	2	56	no	—
88	C₂×Dic₁₃	C₂⁵	1	8	yes	—
89	C₂×Dic₁₃	C₂⁵	1	16	yes	—
90	C₄×D₁₃	—	2	128	no	—
91	Dic₂₆	—	2	128	no	—
92	C₂⁶	—	13	1	no	—
93	C₂³×Q₈	—	13	28	no	—
94	C₂³×Q₈	—	26	16	no	—
95	C₂⁴×C₄	—	13	1	no	—
96	C₂⁴×C₄	—	13	2	no	—
97	C₂⁴×C₄	—	13	28	no	—
98	C₂⁴×C₄	—	26	16	no	—
99	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
100	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
101	C₂×C₂²⋊Q₈	—	13	448	no	—
102	C₂²×C₄⋊C₄	—	13	56	no	—
103	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	complemented
104	C₂²×C₂²⋊C₄	—	13	56	no	—
105	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
106	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
107	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
108	C₂²×C₁₃	C₂²×#41	1	14	yes	—
109	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
110	C₂×D₁₃	—	2	8	no	—
111	C₂×D₁₃	—	2	8	no	—
112	C₂×D₁₃	—	2	224	no	—
113	C₂×D₁₃	C₂³×Q₈	1	16	yes	complemented
114	C₄×C₁₃	—	2	16	no	—
115	C₄×C₁₃	C₂³×D₄	1	16	yes	—
116	Dic₁₃	—	2	16	no	—
117	Dic₁₃	—	2	8	no	—
118	C₂⁵	—	13	16	no	—
119	C₂⁵	—	13	14	no	—
120	C₂⁵	—	26	8	no	—
121	C₂⁵	—	26	8	no	—
122	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
123	C₂²×Q₈	—	13	112	no	—
124	C₂²×Q₈	—	26	224	no	—
125	C₂³×C₄	—	2	16	no	—
126	C₂³×C₄	—	13	28	no	—
127	C₂³×C₄	—	13	14	no	—
128	C₂³×C₄	—	13	112	no	—
129	C₂³×C₄	—	26	8	no	—
130	C₂³×C₄	—	26	16	no	—
131	C₂³×C₄	—	26	224	no	—
132	C₂³×C₄	D₅₂	1	16	yes	—
133	C₂³×C₄	C₂²×D₁₃	1	28	yes	—
134	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
135	C₂²⋊Q₈	—	13	512	no	—
136	C₂×C₄⋊C₄	—	13	224	no	—
137	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	complemented
138	C₂×C₂²⋊C₄	—	13	224	no	—
139	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
140	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
141	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
143	D₁₃	—	2	16	no	—
144	C₂⁴	—	13	112	no	—
145	C₂⁴	—	13	28	no	—
146	C₂⁴	—	26	128	no	—
147	C₂⁴	—	26	56	no	—
148	C₂⁴	—	26	56	no	—
149	C₂⁴	C₂×D₅₂	1	8	yes	—
150	C₂⁴	#6129	1	8	yes	—
151	C₂⁴	Q₈×D₁₃	1	8	yes	—
152	C₂⁴	C₂³×D₁₃	1	7	yes	—
153	C₂×Q₈	—	13	64	no	—
154	C₂×Q₈	—	26	448	no	—
155	C₂²×C₄	—	2	112	no	—
156	C₂²×C₄	—	13	56	no	—
157	C₂²×C₄	—	13	28	no	—
158	C₂²×C₄	—	13	64	no	—
159	C₂²×C₄	—	26	56	no	—
160	C₂²×C₄	—	26	112	no	—
161	C₂²×C₄	—	26	448	no	—
162	C₂²×C₄	C₂×D₅₂	1	112	yes	—
163	C₂²×C₄	C₂³×D₁₃	1	56	yes	—
164	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
165	C₄⋊C₄	—	13	128	no	—
166	C₄⋊C₄	C₂³×D₁₃	1	64	yes	complemented
167	C₂²⋊C₄	—	13	128	no	—
168	C₁₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	13	112	no	—
170	C₂³	—	13	8	no	—
171	C₂³	—	26	448	no	—
172	C₂³	—	26	56	no	—
173	C₂³	—	26	56	no	—
174	C₂³	#88151	1	64	yes	direct product factor
175	C₂³	C₂²×D₅₂	1	28	yes	—
176	C₂³	C₂×#6129	1	28	yes	—
177	C₂³	C₂×Q₈×D₁₃	1	28	yes	—
178	C₂³	C₂⁴×D₁₃	1	7	yes	—
179	Q₈	—	26	128	no	—
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	13	16	no	—
182	C₂×C₄	—	13	8	no	—
183	C₂×C₄	—	26	56	no	—
184	C₂×C₄	—	26	112	no	—
185	C₂×C₄	—	26	128	no	—
186	C₂×C₄	C₂²×D₅₂	1	112	yes	—
187	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
188	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
189	C₂²	—	13	16	no	—
190	C₂²	—	26	224	no	—
191	C₂²	—	26	8	no	—
192	C₂²	—	26	8	no	—
193	C₂²	C₂×#88151	1	112	yes	direct product factor
194	C₂²	C₂³×D₅₂	1	14	yes	—
195	C₂²	C₂²×#6129	1	14	yes	—
196	C₂²	C₂²×Q₈×D₁₃	1	14	yes	—
197	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	26	8	no	—
200	C₄	—	26	16	no	—
201	C₄	C₂³×D₅₂	1	16	yes	—
202	C₂	—	26	16	no	—
203	C₂	C₂²×#88151	1	28	yes	direct product factor
204	C₂	C₂⁴×D₅₂	1	1	yes	characteristic
205	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
206	C₂	C₂³×#6129	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	C₂⁵×C₁₃	C₂³
prime radical	C₂⁵×D₁₃	C₂²
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×C₂²⋊Q₈
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. C₂⁴×Dic₂₆	C₂	semidirect
3. C₂²×#88151	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₃	C₂	semidirect
6. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
15. C₂³×Dic₂₆	C₂²	semidirect
16. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
19. C₂×#88151	C₂²	direct (N × H)
23. C₂⁴×D₁₃	Q₈	semidirect
36. C₂²×Dic₂₆	C₂³	semidirect
37. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
39. C₂×C₄⋊Dic₁₃	C₂³	semidirect
40. #88151	C₂³	direct (N × H)
49. C₂³×D₁₃	C₂×Q₈	semidirect
62. C₂×Dic₂₆	C₂⁴	semidirect
63. C₁₃×C₄⋊C₄	C₂⁴	semidirect
65. C₄⋊Dic₁₃	C₂⁴	semidirect
70. C₂³×C₄⋊C₄	D₁₃	semidirect
72. C₂³×C₁₃	C₂²⋊Q₈	semidirect
80. C₂²×D₁₃	C₂²×Q₈	semidirect
103. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
105. C₂²×C₁₃	C₂×C₂²⋊Q₈	semidirect
113. C₂×D₁₃	C₂³×Q₈	semidirect
137. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
139. C₂×C₁₃	C₂²×C₂²⋊Q₈	semidirect
166. C₄⋊C₄	C₂³×D₁₃	semidirect
168. C₁₃	C₂³×C₂²⋊Q₈	semidirect
174. C₂³	#88151	direct (N × H)
193. C₂²	C₂×#88151	direct (N × H)
203. C₂	C₂²×#88151	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×Dic₂₆	2
C₂⁴×C₄×D₁₃	2
C₂²×#88151	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6103	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×C₂²⋊Q₈	13

Subgroups — C23×#88151 #16769563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88151 #16769563