Subgroups — #16069023

Order 3328.

4086 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719182	C₂	1	6	yes	maximal, complemented
2	#13123234	C₂	1	1	yes	maximal, characteristic, complemented
3	#13123231	C₂	1	1	yes	maximal, characteristic, complemented
4	#13123193	C₂	1	6	yes	maximal, complemented
5	#13123181	C₂	1	8	yes	maximal, complemented
6	C₄×#88128	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6127	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
11	C₂×#88184	C₂²	1	24	yes	—
12	#11719182	—	2	48	no	—
13	C₂×#88129	C₂²	1	12	yes	—
14	C₂×#88128	C₂²	1	8	yes	—
15	C₂×#88128	C₂²	1	6	yes	—
16	C₂×C₄×D₅₂	C₂²	1	24	yes	—
17	C₂×#88123	C₂²	1	6	yes	—
18	C₄³×C₁₃	C₂²	1	3	yes	—
19	C₄³×C₁₃	C₂²	1	1	yes	characteristic
20	C₄³×C₁₃	C₂²	1	4	yes	—
21	#11718600	C₂²	1	48	yes	—
22	#11718596	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂²⋊D₁₃	C₂³	1	8	yes	—
26	C₂×#6127	—	2	48	no	—
27	C₂²×D₅₂	C₂³	1	24	yes	—
28	C₂²×D₅₂	C₂³	1	4	yes	—
29	C₂²×C₄×D₁₃	C₂³	1	8	yes	—
30	C₂²×Dic₂₆	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₃	—	2	16	no	—
32	C₂×C₄²×C₁₃	D₄	1	24	yes	complemented
33	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
38	#88184	—	2	96	no	—
39	C₂×#6103	C₂³	1	48	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
41	#88129	—	2	48	no	—
42	#88128	—	2	24	no	—
43	#88128	—	2	32	no	—
44	C₄×D₅₂	—	2	96	no	—
45	#88123	—	2	24	no	—
46	#36644	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂³×D₁₃	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₁₃	—	2	24	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₃	—	2	48	no	—
57	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
58	#6127	—	4	64	no	—
59	C₂×D₅₂	—	2	48	no	—
60	C₂×D₅₂	—	2	24	no	—
61	C₂×D₅₂	—	2	96	no	—
62	C₂×C₄×D₁₃	—	2	48	no	—
63	C₂×Dic₂₆	—	2	24	no	—
64	C₄²×C₁₃	—	2	24	no	—
65	C₄²×C₁₃	C₂×D₄	1	12	yes	—
66	C₄²×C₁₃	C₂×D₄	1	24	yes	—
67	C₄²×C₁₃	C₂×D₄	1	12	yes	—
68	C₄²×C₁₃	#41	1	16	yes	complemented
69	#6103	—	2	96	no	—
70	C₄⋊Dic₁₃	—	2	48	no	—
71	C₂×#532	—	13	6	no	—
72	#2868	—	13	1	no	—
73	#2867	—	13	1	no	—
74	#2858	—	13	6	no	—
75	#2856	—	13	8	no	—
76	C₄×#127	—	13	8	no	—
77	C₂×C₄³	D₁₃	1	1	yes	characteristic, complemented
78	C₂³×C₁₃	—	2	4	no	—
79	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
80	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
81	C₂²×D₁₃	—	2	24	no	—
82	C₂²×D₁₃	—	2	32	no	—
83	C₂×C₄×C₁₃	—	2	12	no	—
84	C₂×C₄×C₁₃	—	2	48	no	—
85	C₂×C₄×C₁₃	#127	1	16	yes	complemented
86	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
90	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
91	C₂²⋊D₁₃	—	4	32	no	—
92	C₂×Dic₁₃	—	2	24	no	—
93	D₅₂	—	4	16	no	—
94	D₅₂	—	4	96	no	—
95	C₄×D₁₃	—	4	32	no	—
96	Dic₂₆	—	4	16	no	—
97	C₂²×#41	—	13	4	no	—
98	#532	—	26	48	no	—
99	C₂×C₄⋊Q₈	—	13	6	no	—
100	C₂×#127	—	13	8	no	—
101	C₂×#127	—	13	6	no	—
102	C₂×#124	—	13	12	no	—
103	C₂×#121	—	13	24	no	—
104	C₂×C₄×D₄	—	13	24	no	—
105	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
106	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
107	#390	—	13	48	no	—
108	#384	—	13	8	no	—
109	C₄³	C₂×D₁₃	1	4	yes	—
110	C₄³	C₂×D₁₃	1	3	yes	—
111	C₄³	C₂×D₁₃	1	1	yes	characteristic
112	C₂²×C₁₃	—	2	12	no	—
113	C₂²×C₁₃	C₂×#127	1	4	yes	—
114	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
115	C₂²×C₁₃	C₂²×#41	1	4	yes	—
116	C₂×D₁₃	—	4	8	no	—
117	C₂×D₁₃	—	4	48	no	—
118	C₄×C₁₃	—	2	12	no	—
119	C₄×C₁₃	C₂×#127	1	4	yes	—
120	C₄×C₁₃	C₂×#127	1	4	yes	—
121	C₄×C₁₃	#532	1	24	yes	complemented
122	Dic₁₃	—	4	8	no	—
123	C₂×#41	—	26	48	no	—
124	C₂²×Q₈	—	13	4	no	—
125	C₂²×D₄	—	13	8	no	—
126	C₂²×D₄	—	13	24	no	—
127	C₂²×D₄	—	13	4	no	—
128	C₂³×C₄	—	13	8	no	—
129	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
130	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	26	24	no	—
132	#127	—	26	24	no	—
133	#127	—	26	32	no	—
134	#124	—	26	48	no	—
135	#121	—	26	96	no	—
136	C₄×D₄	—	26	96	no	—
137	C₂×C₄⋊C₄	—	13	24	no	—
138	C₂×C₂²⋊C₄	—	13	48	no	—
139	C₂×C₄²	—	2	16	no	—
140	C₂×C₄²	D₅₂	1	24	yes	complemented
141	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
142	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
143	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
144	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
145	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
146	C₂×C₁₃	—	2	4	no	—
147	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#532	1	6	yes	—
149	D₁₃	—	8	8	no	—
150	C₂⁴	—	13	8	no	—
151	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
152	#41	—	52	64	no	—
153	C₂×Q₈	—	26	24	no	—
154	C₂×D₄	—	26	48	no	—
155	C₂×D₄	—	26	48	no	—
156	C₂×D₄	—	26	24	no	—
157	C₂×D₄	—	26	96	no	—
158	C₂²×C₄	—	2	24	no	—
159	C₂²×C₄	—	13	8	no	—
160	C₂²×C₄	—	26	48	no	—
161	C₂²×C₄	C₂×D₅₂	1	24	yes	—
162	C₂²×C₄	C₂×D₅₂	1	12	yes	—
163	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
164	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
166	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	26	48	no	—
168	C₂²⋊C₄	—	26	96	no	—
169	C₄²	—	2	24	no	—
170	C₄²	C₂×D₅₂	1	24	yes	—
171	C₄²	C₂×D₅₂	1	12	yes	—
172	C₄²	C₂×D₅₂	1	12	yes	—
173	C₄²	#6127	1	16	yes	complemented
174	C₁₃	#36644	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	2	4	no	—
176	C₂³	—	26	24	no	—
177	C₂³	—	26	32	no	—
178	C₂³	C₂²×D₅₂	1	6	yes	—
179	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
180	Q₈	—	52	16	no	—
181	D₄	—	52	32	no	—
182	D₄	—	52	16	no	—
183	D₄	—	52	96	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	—	26	24	no	—
187	C₂×C₄	—	52	32	no	—
188	C₂×C₄	#88128	1	16	yes	complemented
189	C₂×C₄	C₂²×D₅₂	1	12	yes	—
190	C₂×C₄	C₂²×D₅₂	1	6	yes	—
191	C₂×C₄	C₂²×D₅₂	1	6	yes	—
192	C₂×C₄	C₂²×D₅₂	1	12	yes	—
193	C₂×C₄	C₂×#6127	1	24	yes	—
194	C₂²	—	2	12	no	—
195	C₂²	—	52	8	no	—
196	C₂²	—	52	48	no	—
197	C₂²	C₂×#88128	1	4	yes	—
198	C₂²	C₂³×D₅₂	1	3	yes	—
199	C₂²	C₂²×#6127	1	4	yes	—
200	C₄	—	2	12	no	—
201	C₄	—	52	8	no	—
202	C₄	C₂×#88128	1	4	yes	—
203	C₄	C₂×#88128	1	4	yes	—
204	C₄	#11719182	1	24	yes	complemented
205	C₂	—	2	4	no	—
206	C₂	—	104	8	no	—
207	C₂	C₂²×#88128	1	1	yes	characteristic
208	C₂	C₂×#11719182	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	#36644
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂×#11719182	C₂	semidirect
2. #13123234	C₂	semidirect
3. #13123231	C₂	semidirect
4. #13123193	C₂	semidirect
5. #13123181	C₂	semidirect
6. C₄×#88128	C₂	semidirect
7. C₂×C₄³×C₁₃	C₂	semidirect
32. C₂×C₄²×C₁₃	D₄	semidirect
68. C₄²×C₁₃	#41	semidirect
77. C₂×C₄³	D₁₃	semidirect
85. C₂×C₄×C₁₃	#127	semidirect
121. C₄×C₁₃	#532	semidirect
140. C₂×C₄²	D₅₂	semidirect
173. C₄²	#6127	semidirect
174. C₁₃	#36644	semidirect
188. C₂×C₄	#88128	semidirect
204. C₄	#11719182	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719182	2
#13123234	2
#13123231	2
#13123193	2
#13123181	2
C₄×#88128	2
C₂×C₄³×C₁₃	2
#36644	13