Subgroups — C₂³×#88178 #16769562

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	C₂²×#88178	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#6101	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂²⋊Dic₁₃	C₂	1	2	yes	maximal
6	C₂⁵×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	—	2	16	no	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
13	C₂⁴×Dic₁₃	C₂²	1	2	yes	—
14	C₂³×Dic₂₆	—	2	16	no	—
15	C₂³×Dic₂₆	C₂²	1	28	yes	complemented
16	C₂²×C₂²⋊Dic₁₃	C₂²	1	56	yes	—
17	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	complemented
18	C₂×#88178	C₂²	1	448	yes	direct product factor
19	C₂²×#6101	C₂²	1	56	yes	complemented
20	C₂⁵×C₁₃	—	2	8	no	—
21	C₂⁵×C₁₃	—	2	8	no	—
22	C₂⁵×C₁₃	Q₈	1	16	yes	complemented
23	C₂⁵×C₁₃	C₂³	1	14	yes	—
24	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	—	2	8	no	—
26	C₂³×C₄×C₁₃	—	2	224	no	—
27	C₂³×C₄×C₁₃	D₄	1	16	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	112	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
31	C₂³×Dic₁₃	—	2	16	no	—
32	C₂³×Dic₁₃	—	2	16	no	—
33	C₂³×Dic₁₃	C₂³	1	28	yes	—
34	C₂³×Dic₁₃	C₂³	1	28	yes	—
35	C₂²×Dic₂₆	—	2	224	no	—
36	C₂²×Dic₂₆	C₂³	1	112	yes	complemented
37	C₂×C₂²⋊Dic₁₃	C₂³	1	224	yes	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	complemented
39	#88178	C₂³	1	512	yes	direct product factor
40	C₂×#6101	C₂³	1	224	yes	complemented
41	C₂³×C₂²⋊Q₈	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	—	2	128	no	—
43	C₂⁴×C₁₃	—	2	56	no	—
44	C₂⁴×C₁₃	—	2	56	no	—
45	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
46	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
47	C₂⁴×C₁₃	C₂×Q₈	1	112	yes	complemented
48	C₂⁴×C₁₃	#41	1	8	yes	—
49	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
50	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
51	C₂²×C₄×C₁₃	—	2	56	no	—
52	C₂²×C₄×C₁₃	—	2	448	no	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	112	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	64	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
57	C₂²×Dic₁₃	—	2	112	no	—
58	C₂²×Dic₁₃	—	2	112	no	—
59	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
60	C₂²×Dic₁₃	C₂⁴	1	56	yes	—
61	C₂×Dic₂₆	—	2	448	no	—
62	C₂×Dic₂₆	C₂⁴	1	64	yes	complemented
63	C₂²⋊Dic₁₃	C₂⁴	1	128	yes	—
64	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
65	#6101	C₂⁴	1	128	yes	complemented
66	C₂⁴×Q₈	—	13	1	no	—
67	C₂⁵×C₄	D₁₃	1	1	yes	characteristic
68	C₂²×C₂²⋊Q₈	—	13	56	no	—
69	C₂³×C₄⋊C₄	—	13	2	no	—
70	C₂³×C₄⋊C₄	—	13	1	no	—
71	C₂³×C₂²⋊C₄	—	13	2	no	—
72	C₂³×C₁₃	—	2	448	no	—
73	C₂³×C₁₃	—	2	56	no	—
74	C₂³×C₁₃	—	2	56	no	—
75	C₂³×C₁₃	C₂²⋊Q₈	1	64	yes	complemented
76	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
77	C₂³×C₁₃	C₂²×Q₈	1	28	yes	—
78	C₂³×C₁₃	C₂²×Q₈	1	112	yes	complemented
79	C₂³×C₁₃	C₂×#41	1	28	yes	—
80	C₂³×C₁₃	C₂⁵	1	8	yes	—
81	C₂³×C₁₃	C₂⁵	1	7	yes	—
82	C₂×C₄×C₁₃	—	2	56	no	—
83	C₂×C₄×C₁₃	—	2	128	no	—
84	C₂×C₄×C₁₃	C₂²×D₄	1	112	yes	—
85	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
86	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
87	C₂×Dic₁₃	—	2	112	no	—
88	C₂×Dic₁₃	—	2	112	no	—
89	C₂×Dic₁₃	C₂⁵	1	16	yes	—
90	C₂×Dic₁₃	C₂⁵	1	16	yes	—
91	Dic₂₆	—	2	128	no	—
92	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
93	C₂³×Q₈	—	13	28	no	—
94	C₂³×Q₈	—	26	16	no	—
95	C₂⁴×C₄	—	2	16	no	—
96	C₂⁴×C₄	—	13	2	no	—
97	C₂⁴×C₄	—	13	2	no	—
98	C₂⁴×C₄	C₂×D₁₃	1	28	yes	—
99	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
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	—	13	28	no	—
104	C₂²×C₂²⋊C₄	—	13	56	no	—
105	C₂²×C₁₃	—	2	8	no	—
106	C₂²×C₁₃	—	2	8	no	—
107	C₂²×C₁₃	—	2	224	no	—
108	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	112	yes	complemented
109	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
110	C₂²×C₁₃	C₂³×Q₈	1	14	yes	—
111	C₂²×C₁₃	C₂³×Q₈	1	16	yes	complemented
112	C₂²×C₁₃	C₂²×#41	1	14	yes	—
113	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
114	C₄×C₁₃	—	2	8	no	—
115	C₄×C₁₃	C₂³×D₄	1	16	yes	—
116	Dic₁₃	—	2	16	no	—
117	Dic₁₃	—	2	16	no	—
118	C₂⁵	—	2	8	no	—
119	C₂⁵	—	2	8	no	—
120	C₂⁵	Dic₂₆	1	16	yes	complemented
121	C₂⁵	C₂²×D₁₃	1	14	yes	—
122	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
123	C₂²×Q₈	—	13	112	no	—
124	C₂²×Q₈	—	26	224	no	—
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	—	2	224	no	—
127	C₂³×C₄	—	13	28	no	—
128	C₂³×C₄	—	13	28	no	—
129	C₂³×C₄	—	26	16	no	—
130	C₂³×C₄	—	26	16	no	—
131	C₂³×C₄	C₂²⋊D₁₃	1	16	yes	—
132	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
133	C₂³×C₄	C₂²×D₁₃	1	112	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	—	13	112	no	—
138	C₂×C₂²⋊C₄	—	13	224	no	—
139	C₂×C₁₃	—	2	16	no	—
140	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	28	yes	complemented
141	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
142	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
143	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
144	C₂⁴	—	2	128	no	—
145	C₂⁴	—	2	56	no	—
146	C₂⁴	—	2	56	no	—
147	C₂⁴	C₂×Dic₂₆	1	8	yes	—
148	C₂⁴	C₂×Dic₂₆	1	112	yes	complemented
149	C₂⁴	#6127	1	8	yes	—
150	C₂⁴	C₂×C₂²⋊D₁₃	1	8	yes	—
151	C₂⁴	C₂³×D₁₃	1	28	yes	—
152	C₂⁴	C₂³×D₁₃	1	7	yes	—
153	C₂×Q₈	—	13	64	no	—
154	C₂×Q₈	—	26	448	no	—
155	C₂²×C₄	—	2	56	no	—
156	C₂²×C₄	—	2	448	no	—
157	C₂²×C₄	—	13	56	no	—
158	C₂²×C₄	—	13	56	no	—
159	C₂²×C₄	—	26	112	no	—
160	C₂²×C₄	—	26	112	no	—
161	C₂²×C₄	C₂×C₂²⋊D₁₃	1	112	yes	—
162	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
163	C₂²×C₄	C₂³×D₁₃	1	64	yes	—
164	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
165	C₄⋊C₄	—	13	128	no	—
166	C₄⋊C₄	—	13	64	no	—
167	C₂²⋊C₄	—	13	128	no	—
168	C₁₃	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	2	448	no	—
170	C₂³	—	2	56	no	—
171	C₂³	—	2	56	no	—
172	C₂³	#88178	1	64	yes	direct product factor
173	C₂³	C₂²×Dic₂₆	1	28	yes	—
174	C₂³	C₂²×Dic₂₆	1	112	yes	complemented
175	C₂³	C₂×#6127	1	28	yes	—
176	C₂³	C₂²×C₂²⋊D₁₃	1	28	yes	—
177	C₂³	C₂⁴×D₁₃	1	8	yes	—
178	C₂³	C₂⁴×D₁₃	1	7	yes	—
179	Q₈	—	26	128	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	2	128	no	—
182	C₂×C₄	—	13	16	no	—
183	C₂×C₄	—	13	16	no	—
184	C₂×C₄	—	26	112	no	—
185	C₂×C₄	—	26	112	no	—
186	C₂×C₄	C₂²×C₂²⋊D₁₃	1	112	yes	—
187	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
188	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	8	no	—
192	C₂²	C₂×#88178	1	112	yes	direct product factor
193	C₂²	C₂³×Dic₂₆	1	14	yes	—
194	C₂²	C₂³×Dic₂₆	1	16	yes	complemented
195	C₂²	C₂²×#6127	1	14	yes	—
196	C₂²	C₂³×C₂²⋊D₁₃	1	14	yes	—
197	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	26	16	no	—
200	C₄	—	26	16	no	—
201	C₄	C₂³×C₂²⋊D₁₃	1	16	yes	—
202	C₂	—	2	16	no	—
203	C₂	C₂²×#88178	1	28	yes	direct product factor
204	C₂	C₂⁴×C₂²⋊D₁₃	1	1	yes	characteristic
205	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
206	C₂	C₂³×#6127	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂⁵×C₄×C₁₃	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁶×C₁₃	C₂²
hypercenter	C₂⁵×C₄	D₁₃
hypocenter	C₁₃	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₂₆	C₂	semidirect
2. C₂²×#88178	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₁₃	C₂	semidirect
4. C₂³×#6101	C₂	semidirect
15. C₂³×Dic₂₆	C₂²	semidirect
17. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
18. C₂×#88178	C₂²	direct (N × H)
19. C₂²×#6101	C₂²	semidirect
22. C₂⁵×C₁₃	Q₈	semidirect
36. C₂²×Dic₂₆	C₂³	semidirect
38. C₂×C₄⋊Dic₁₃	C₂³	semidirect
39. #88178	C₂³	direct (N × H)
40. C₂×#6101	C₂³	semidirect
47. C₂⁴×C₁₃	C₂×Q₈	semidirect
62. C₂×Dic₂₆	C₂⁴	semidirect
64. C₄⋊Dic₁₃	C₂⁴	semidirect
65. #6101	C₂⁴	semidirect
75. C₂³×C₁₃	C₂²⋊Q₈	semidirect
78. C₂³×C₁₃	C₂²×Q₈	semidirect
108. C₂²×C₁₃	C₂×C₂²⋊Q₈	semidirect
111. C₂²×C₁₃	C₂³×Q₈	semidirect
120. C₂⁵	Dic₂₆	semidirect
140. C₂×C₁₃	C₂²×C₂²⋊Q₈	semidirect
148. C₂⁴	C₂×Dic₂₆	semidirect
168. C₁₃	C₂³×C₂²⋊Q₈	semidirect
172. C₂³	#88178	direct (N × H)
174. C₂³	C₂²×Dic₂₆	semidirect
192. C₂²	C₂×#88178	direct (N × H)
194. C₂²	C₂³×Dic₂₆	semidirect
203. C₂	C₂²×#88178	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
C₂²×#88178	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×C₂²⋊Dic₁₃	2
C₂⁵×C₄×C₁₃	2
C₂³×C₂²⋊Q₈	13

Subgroups — C23×#88178 #16769562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88178 #16769562