Subgroups — C₂²×#11718539 #16732512

Order 3328.

1966 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	C₂²×#88144	C₂	1	2	yes	maximal
2	C₂×#11718539	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88048	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×#6101	—	2	2	no	—
12	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂³×#1055	C₂²	1	1	yes	characteristic
14	C₂×#88144	—	2	12	no	—
15	C₂×#88144	C₂²	1	24	yes	—
16	C₂×C₁₃×#107	C₂²	1	12	yes	—
17	C₂×#88058	C₂²	1	12	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	12	yes	—
19	C₂×#88048	C₂²	1	24	yes	—
20	#11718539	C₂²	1	64	yes	direct product factor
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	2	no	—
24	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
25	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
30	C₂×C₄⋊Dic₁₃	—	2	6	no	—
31	C₂×C₄⋊Dic₁₃	—	2	24	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
33	C₂×#6101	—	2	24	no	—
34	C₂×C₄×Dic₁₃	D₄	1	6	yes	—
35	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
36	C₂²×#1055	Q₈	1	8	yes	—
37	C₂²×#1055	C₂³	1	6	yes	—
38	#88144	—	2	48	no	—
39	#88144	C₂³	1	32	yes	—
40	C₁₃×#107	C₂³	1	16	yes	—
41	#88058	C₂³	1	16	yes	—
42	C₈×Dic₁₃	C₂³	1	16	yes	—
43	#88048	C₂³	1	32	yes	—
44	C₂²×#499	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	12	no	—
47	C₂²×C₄×C₁₃	—	4	8	no	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
52	C₂²×Dic₁₃	—	2	4	no	—
53	C₂²×Dic₁₃	—	2	4	no	—
54	C₂²×Dic₁₃	—	2	12	no	—
55	C₂²×Dic₁₃	—	4	8	no	—
56	C₂²×Dic₁₃	C₂×D₄	1	6	yes	—
57	C₂²×Dic₁₃	C₂×D₄	1	6	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
60	C₁₃×C₄⋊C₄	—	2	12	no	—
61	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
62	C₄⋊Dic₁₃	—	2	12	no	—
63	C₄⋊Dic₁₃	—	2	32	no	—
64	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
65	#6101	—	2	32	no	—
66	C₄×Dic₁₃	C₂×D₄	1	12	yes	—
67	C₄×Dic₁₃	C₂⁴	1	4	yes	—
68	C₂×#1055	C₂×Q₈	1	24	yes	—
69	C₂×#1055	C₂⁴	1	4	yes	—
70	C₂²×#125	—	13	2	no	—
71	C₂×#499	—	13	24	no	—
72	C₂²×#107	—	13	1	no	—
73	C₂²×#107	D₁₃	1	1	yes	characteristic
74	C₂²×#106	—	13	2	no	—
75	C₂²×C₄×C₈	—	13	1	no	—
76	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
77	C₂³×C₁₃	#135	1	4	yes	—
78	C₂³×C₁₃	#135	1	4	yes	—
79	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	4	24	no	—
82	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
83	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
84	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
86	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	12	no	—
88	C₂×Dic₁₃	—	2	12	no	—
89	C₂×Dic₁₃	—	2	8	no	—
90	C₂×Dic₁₃	—	4	24	no	—
91	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
92	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
93	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
94	#1055	C₂²×Q₈	1	8	yes	—
95	C₂³×C₈	—	13	1	no	—
96	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
97	C₂×#125	—	13	24	no	—
98	C₂×#125	—	26	12	no	—
99	C₂²×C₄⋊C₄	—	13	2	no	—
100	C₂²×C₄⋊C₄	—	26	2	no	—
101	C₂²×C₄⋊C₄	—	26	2	no	—
102	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
103	C₂²×C₄²	—	13	1	no	—
104	#499	—	13	64	no	—
105	C₂×#107	—	13	12	no	—
106	C₂×#107	C₂×D₁₃	1	12	yes	—
107	C₂×#106	—	13	24	no	—
108	C₂×C₄×C₈	—	13	12	no	—
109	C₂²×C₁₃	#499	1	16	yes	complemented
110	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
111	C₂²×C₁₃	C₂×#135	1	6	yes	—
112	C₂²×C₁₃	C₂×#135	1	6	yes	—
113	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
114	C₄×C₁₃	—	4	8	no	—
115	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
116	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
117	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
118	Dic₁₃	—	2	4	no	—
119	Dic₁₃	—	2	4	no	—
120	Dic₁₃	—	4	8	no	—
121	C₂³×C₄	—	2	2	no	—
122	C₂³×C₄	—	13	1	no	—
123	C₂³×C₄	—	13	1	no	—
124	C₂³×C₄	—	26	2	no	—
125	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	C₂²×C₈	—	13	8	no	—
127	C₂²×C₈	—	13	6	no	—
128	C₂²×C₈	Dic₂₆	1	8	yes	—
129	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
130	#125	—	13	32	no	—
131	#125	—	26	48	no	—
132	C₂×C₄⋊C₄	—	2	6	no	—
133	C₂×C₄⋊C₄	—	13	12	no	—
134	C₂×C₄⋊C₄	—	26	24	no	—
135	C₂×C₄⋊C₄	—	26	24	no	—
136	C₂×C₄⋊C₄	—	26	6	no	—
137	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
138	C₂×C₄²	—	13	6	no	—
139	C₂×C₄²	—	13	6	no	—
140	#107	—	13	16	no	—
141	#107	C₂²×D₁₃	1	16	yes	—
142	#106	—	13	32	no	—
143	C₄×C₈	—	13	16	no	—
144	C₂×C₁₃	C₂×#499	1	12	yes	complemented
145	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
147	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
148	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
149	C₂²×C₄	—	2	12	no	—
150	C₂²×C₄	—	4	8	no	—
151	C₂²×C₄	—	13	6	no	—
152	C₂²×C₄	—	13	6	no	—
153	C₂²×C₄	—	26	4	no	—
154	C₂²×C₄	—	26	4	no	—
155	C₂²×C₄	—	26	12	no	—
156	C₂²×C₄	—	52	8	no	—
157	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
158	C₂²×C₄	D₄×D₁₃	1	3	yes	—
159	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
160	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
161	C₂×C₈	—	13	24	no	—
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	C₂×Dic₂₆	1	24	yes	—
164	C₂×C₈	C₂³×D₁₃	1	4	yes	—
165	C₄⋊C₄	—	2	12	no	—
166	C₄⋊C₄	—	13	8	no	—
167	C₄⋊C₄	—	26	32	no	—
168	C₄⋊C₄	—	26	32	no	—
169	C₄⋊C₄	—	26	12	no	—
170	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
171	C₄²	—	13	12	no	—
172	C₄²	—	13	4	no	—
173	C₁₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88142	1	4	yes	—
175	C₂³	#88166	1	4	yes	—
176	C₂³	#88173	1	4	yes	—
177	C₂³	C₂×D₄×D₁₃	1	3	yes	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	—	13	4	no	—
181	C₂×C₄	—	13	4	no	—
182	C₂×C₄	—	26	12	no	—
183	C₂×C₄	—	26	12	no	—
184	C₂×C₄	—	26	8	no	—
185	C₂×C₄	—	52	24	no	—
186	C₂×C₄	#88142	1	12	yes	—
187	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
188	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
189	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
190	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
191	C₈	—	13	8	no	—
192	C₈	C₂²×Dic₂₆	1	8	yes	—
193	C₂²	#11718539	1	16	yes	direct product factor
194	C₂²	C₂×#88142	1	6	yes	—
195	C₂²	C₂×#88166	1	6	yes	—
196	C₂²	C₂×#88173	1	6	yes	—
197	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	26	4	no	—
200	C₄	—	26	4	no	—
201	C₄	—	52	8	no	—
202	C₄	C₂×#88142	1	6	yes	—
203	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
204	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
205	C₂	C₂×#11718539	1	12	yes	direct product factor
206	C₂	C₂²×#88142	1	1	yes	characteristic
207	C₂	C₂²×#88166	1	1	yes	characteristic
208	C₂	C₂²×#88173	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#107	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#107	D₁₃
hypocenter	C₁₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#11718539	C₂	direct (N × H)
20. #11718539	C₂²	direct (N × H)
109. C₂²×C₁₃	#499	semidirect
144. C₂×C₁₃	C₂×#499	semidirect
173. C₁₃	C₂²×#499	semidirect
193. C₂²	#11718539	direct (N × H)
205. C₂	C₂×#11718539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88144	2
C₂×#11718539	2
C₂²×#88058	2
C₂²×#88048	2
C₂²×C₈×Dic₁₃	2
C₂²×C₁₃×#107	2
C₂²×#499	13

Subgroups — C22×#11718539 #16732512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718539 #16732512