Subgroups — C₂²×#11718519 #16732511

Order 3328.

1966 conjugacy classes of subgroups.

The subgroup lattice has 192 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×#11718519	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88047	C₂	1	2	yes	maximal
4	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#106	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×#106	C₂²	1	12	yes	—
17	C₂×#88057	C₂²	1	12	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	12	yes	—
19	C₂×#88047	C₂²	1	24	yes	—
20	#11718519	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	2	yes	—
25	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
29	C₂×C₄⋊Dic₁₃	—	2	6	no	—
30	C₂×C₄⋊Dic₁₃	—	2	24	no	—
31	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
32	C₂×#6101	—	2	24	no	—
33	C₂×C₄×Dic₁₃	D₄	1	6	yes	—
34	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
35	C₂²×#1055	Q₈	1	8	yes	—
36	C₂²×#1055	C₂³	1	6	yes	—
37	#88144	—	2	48	no	—
38	#88144	C₂³	1	32	yes	—
39	C₁₃×#106	C₂³	1	16	yes	—
40	#88057	C₂³	1	16	yes	—
41	C₈×Dic₁₃	C₂³	1	16	yes	—
42	#88047	C₂³	1	32	yes	—
43	C₂²×#499	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	2	12	no	—
46	C₂²×C₄×C₁₃	—	4	8	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
51	C₂²×Dic₁₃	—	2	8	no	—
52	C₂²×Dic₁₃	—	2	12	no	—
53	C₂²×Dic₁₃	—	4	8	no	—
54	C₂²×Dic₁₃	C₂×D₄	1	12	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	—
56	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
57	C₁₃×C₄⋊C₄	—	2	12	no	—
58	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
59	C₄⋊Dic₁₃	—	2	12	no	—
60	C₄⋊Dic₁₃	—	2	32	no	—
61	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
62	#6101	—	2	32	no	—
63	C₄×Dic₁₃	C₂×D₄	1	12	yes	—
64	C₄×Dic₁₃	C₂⁴	1	4	yes	—
65	C₂×#1055	C₂×Q₈	1	24	yes	—
66	C₂×#1055	C₂⁴	1	4	yes	—
67	C₂²×#125	—	13	2	no	—
68	C₂×#499	—	13	24	no	—
69	C₂²×#107	—	13	2	no	—
70	C₂²×#106	—	13	1	no	—
71	C₂²×#106	D₁₃	1	1	yes	characteristic
72	C₂²×C₄×C₈	—	13	1	no	—
73	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
74	C₂³×C₁₃	#135	1	8	yes	—
75	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
76	C₂×C₄×C₁₃	—	2	8	no	—
77	C₂×C₄×C₁₃	—	4	24	no	—
78	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
79	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
80	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
81	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
82	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	24	no	—
84	C₂×Dic₁₃	—	2	8	no	—
85	C₂×Dic₁₃	—	4	24	no	—
86	C₂×Dic₁₃	C₂²×D₄	1	8	yes	—
87	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
88	#1055	C₂²×Q₈	1	8	yes	—
89	C₂³×C₈	—	13	1	no	—
90	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
91	C₂×#125	—	13	24	no	—
92	C₂×#125	—	26	12	no	—
93	C₂²×C₄⋊C₄	—	13	2	no	—
94	C₂²×C₄⋊C₄	—	26	2	no	—
95	C₂²×C₄⋊C₄	—	26	2	no	—
96	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
97	C₂²×C₄²	—	13	1	no	—
98	#499	—	13	64	no	—
99	C₂×#107	—	13	24	no	—
100	C₂×#106	—	13	12	no	—
101	C₂×#106	C₂×D₁₃	1	12	yes	—
102	C₂×C₄×C₈	—	13	12	no	—
103	C₂²×C₁₃	#499	1	16	yes	complemented
104	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
105	C₂²×C₁₃	C₂×#135	1	12	yes	—
106	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
107	C₄×C₁₃	—	4	8	no	—
108	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
109	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
110	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
111	Dic₁₃	—	2	8	no	—
112	Dic₁₃	—	4	8	no	—
113	C₂³×C₄	—	2	2	no	—
114	C₂³×C₄	—	13	2	no	—
115	C₂³×C₄	—	26	2	no	—
116	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
117	C₂²×C₈	—	13	8	no	—
118	C₂²×C₈	—	13	6	no	—
119	C₂²×C₈	Dic₂₆	1	8	yes	—
120	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
121	#125	—	13	32	no	—
122	#125	—	26	48	no	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄⋊C₄	—	13	12	no	—
125	C₂×C₄⋊C₄	—	26	24	no	—
126	C₂×C₄⋊C₄	—	26	24	no	—
127	C₂×C₄⋊C₄	—	26	6	no	—
128	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
129	C₂×C₄²	—	13	6	no	—
130	C₂×C₄²	—	13	6	no	—
131	#107	—	13	32	no	—
132	#106	—	13	16	no	—
133	#106	C₂²×D₁₃	1	16	yes	—
134	C₄×C₈	—	13	16	no	—
135	C₂×C₁₃	C₂×#499	1	12	yes	complemented
136	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₁₃	C₂²×#135	1	2	yes	—
138	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	—	2	12	no	—
140	C₂²×C₄	—	4	8	no	—
141	C₂²×C₄	—	13	12	no	—
142	C₂²×C₄	—	26	8	no	—
143	C₂²×C₄	—	26	12	no	—
144	C₂²×C₄	—	52	8	no	—
145	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
146	C₂²×C₄	D₄×D₁₃	1	3	yes	—
147	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
148	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
149	C₂×C₈	—	13	24	no	—
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	C₂×Dic₂₆	1	24	yes	—
152	C₂×C₈	C₂³×D₁₃	1	4	yes	—
153	C₄⋊C₄	—	2	12	no	—
154	C₄⋊C₄	—	13	8	no	—
155	C₄⋊C₄	—	26	32	no	—
156	C₄⋊C₄	—	26	32	no	—
157	C₄⋊C₄	—	26	12	no	—
158	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
159	C₄²	—	13	12	no	—
160	C₄²	—	13	4	no	—
161	C₁₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
162	C₂³	#88142	1	4	yes	—
163	C₂³	#88170	1	8	yes	—
164	C₂³	C₂×D₄×D₁₃	1	3	yes	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	24	no	—
167	C₂×C₄	—	13	8	no	—
168	C₂×C₄	—	26	24	no	—
169	C₂×C₄	—	26	8	no	—
170	C₂×C₄	—	52	24	no	—
171	C₂×C₄	#88142	1	12	yes	—
172	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
173	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
174	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
175	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
176	C₈	—	13	8	no	—
177	C₈	C₂²×Dic₂₆	1	8	yes	—
178	C₂²	#11718519	1	16	yes	direct product factor
179	C₂²	C₂×#88142	1	6	yes	—
180	C₂²	C₂×#88170	1	12	yes	—
181	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
182	C₄	—	4	8	no	—
183	C₄	—	26	8	no	—
184	C₄	—	52	8	no	—
185	C₄	C₂×#88142	1	6	yes	—
186	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
187	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
188	C₂	C₂×#11718519	1	12	yes	direct product factor
189	C₂	C₂²×#88142	1	1	yes	characteristic
190	C₂	C₂²×#88170	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#11718519	C₂	direct (N × H)
20. #11718519	C₂²	direct (N × H)
103. C₂²×C₁₃	#499	semidirect
135. C₂×C₁₃	C₂×#499	semidirect
161. C₁₃	C₂²×#499	semidirect
178. C₂²	#11718519	direct (N × H)
188. C₂	C₂×#11718519	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×#11718519	2
C₂²×#88047	2
C₂²×#88057	2
C₂²×C₈×Dic₁₃	2
C₂²×C₁₃×#106	2
C₂²×#499	13

Subgroups — C22×#11718519 #16732511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718519 #16732511