Subgroups — C₂×#13123214 #16068932

Order 3328.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	2	yes	maximal
2	#13123214	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718590	C₂	1	8	yes	maximal
4	C₂×#11718597	C₂	1	4	yes	maximal
5	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₆	C₂²	1	4	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
11	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
12	C₂×#88123	—	2	16	no	—
13	C₂×#88123	C₂²	1	16	yes	—
14	C₄³×C₁₃	C₂²	1	8	yes	—
15	C₂×#88071	C₂²	1	16	yes	—
16	#11718597	C₂²	1	32	yes	—
17	#11718590	C₂²	1	64	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
21	C₂³×Dic₁₃	C₂³	1	8	yes	—
22	C₂²×Dic₂₆	—	2	16	no	—
23	C₂²×Dic₂₆	—	2	32	no	—
24	C₂²×Dic₂₆	C₂³	1	16	yes	—
25	C₂×C₄²×C₁₃	—	2	16	no	—
26	C₂×C₄²×C₁₃	D₄	1	8	yes	—
27	C₂×C₄²×C₁₃	Q₈	1	16	yes	complemented
28	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
31	C₂×C₄⋊Dic₁₃	—	2	16	no	—
32	C₂×C₄⋊Dic₁₃	—	2	16	no	—
33	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
34	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
35	#88123	—	2	64	no	—
36	#88071	C₂³	1	64	yes	—
37	C₂×#2862	—	13	1	no	maximal, Sylow
38	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	—	2	16	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
43	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
44	C₂²×C₄×C₁₃	#41	1	16	yes	—
45	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
48	C₂²×Dic₁₃	—	2	16	no	—
49	C₂²×Dic₁₃	—	2	32	no	—
50	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
51	C₂×Dic₂₆	—	2	32	no	—
52	C₂×Dic₂₆	—	2	64	no	—
53	C₂×Dic₂₆	—	4	64	no	—
54	C₄²×C₁₃	—	2	32	no	—
55	C₄²×C₁₃	C₂×D₄	1	16	yes	—
56	C₄²×C₁₃	C₂×Q₈	1	32	yes	complemented
57	C₄⋊Dic₁₃	—	2	32	no	—
58	C₄⋊Dic₁₃	—	2	32	no	—
59	C₂²×C₄⋊Q₈	—	13	2	no	—
60	#2862	—	13	16	no	—
61	C₂×#391	—	13	8	no	—
62	C₂×#382	—	13	4	no	—
63	C₂×C₄³	D₁₃	1	1	yes	characteristic
64	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
65	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
66	C₂³×C₁₃	C₂×#41	1	8	yes	—
67	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	32	no	—
69	C₂×C₄×C₁₃	—	2	16	no	—
70	C₂×C₄×C₁₃	#124	1	32	yes	—
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	16	yes	—
72	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
73	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
74	C₂×C₄×C₁₃	C₂²×Q₈	1	16	yes	—
75	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
76	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
77	C₂×Dic₁₃	—	2	16	no	—
78	C₂×Dic₁₃	—	2	32	no	—
79	C₂×Dic₁₃	—	4	32	no	—
80	Dic₂₆	—	4	64	no	—
81	C₂³×Q₈	—	13	4	no	—
82	C₂×C₄⋊Q₈	—	13	16	no	—
83	C₂×C₄⋊Q₈	—	26	16	no	—
84	C₂²×C₄⋊C₄	—	13	4	no	—
85	C₂²×C₄⋊C₄	—	13	4	no	—
86	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
87	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
88	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#391	—	13	64	no	—
90	#382	—	13	32	no	—
91	C₂×#95	—	13	16	no	—
92	C₄³	C₂×D₁₃	1	8	yes	—
93	C₂²×C₁₃	C₂×#124	1	8	yes	—
94	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
95	C₂²×C₁₃	#533	1	16	yes	—
96	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
97	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
98	C₂²×C₁₃	C₂²×#41	1	4	yes	—
99	C₄×C₁₃	—	2	16	no	—
100	C₄×C₁₃	C₂×#124	1	16	yes	—
101	C₄×C₁₃	C₂×C₄⋊Q₈	1	8	yes	—
102	Dic₁₃	—	4	16	no	—
103	C₂²×Q₈	—	13	16	no	—
104	C₂²×Q₈	—	26	16	no	—
105	C₂²×Q₈	—	26	32	no	—
106	C₂³×C₄	—	13	8	no	—
107	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
108	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
109	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
110	C₄⋊Q₈	—	26	64	no	—
111	C₂×C₄⋊C₄	—	13	16	no	—
112	C₂×C₄⋊C₄	—	13	16	no	—
113	C₂×C₄⋊C₄	—	26	16	no	—
114	C₂×C₄⋊C₄	—	26	16	no	—
115	C₂×C₄²	—	2	16	no	—
116	C₂×C₄²	Dic₂₆	1	16	yes	complemented
117	C₂×C₄²	D₅₂	1	8	yes	—
118	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
119	C₂×C₄²	C₂²×D₁₃	1	8	yes	—
120	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
121	#95	—	13	64	no	—
122	C₂×C₁₃	#2862	1	8	yes	complemented
123	C₂×C₁₃	C₂²×#124	1	2	yes	—
124	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₁₃	C₂×#533	1	4	yes	—
126	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
127	C₂×Q₈	—	26	32	no	—
128	C₂×Q₈	—	26	64	no	—
129	C₂×Q₈	—	52	64	no	—
130	C₂²×C₄	—	2	16	no	—
131	C₂²×C₄	—	13	16	no	—
132	C₂²×C₄	—	26	16	no	—
133	C₂²×C₄	—	26	32	no	—
134	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
135	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
136	C₂²×C₄	C₂×D₅₂	1	4	yes	—
137	C₂²×C₄	C₂×D₅₂	1	8	yes	—
138	C₂²×C₄	#6127	1	16	yes	—
139	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
140	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
141	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
142	C₄⋊C₄	—	26	32	no	—
143	C₄⋊C₄	—	26	32	no	—
144	C₄²	—	2	32	no	—
145	C₄²	C₂×Dic₂₆	1	32	yes	complemented
146	C₄²	C₂×D₅₂	1	16	yes	—
147	C₁₃	C₂×#2862	1	1	yes	characteristic, complemented, Sylow
148	C₂³	C₂²×Dic₂₆	1	4	yes	—
149	C₂³	C₂²×D₅₂	1	2	yes	—
150	C₂³	C₂×#6127	1	8	yes	—
151	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
152	Q₈	—	52	64	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	26	16	no	—
156	C₂×C₄	—	26	32	no	—
157	C₂×C₄	—	52	32	no	—
158	C₂×C₄	#88123	1	16	yes	—
159	C₂×C₄	#88129	1	32	yes	—
160	C₂×C₄	C₂²×Dic₂₆	1	16	yes	—
161	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
162	C₂×C₄	C₂²×D₅₂	1	4	yes	—
163	C₂×C₄	C₂²×D₅₂	1	8	yes	—
164	C₂×C₄	C₂×#6127	1	16	yes	—
165	C₂²	C₂×#88123	1	4	yes	—
166	C₂²	#11719175	1	16	yes	—
167	C₂²	C₂×#88129	1	8	yes	—
168	C₂²	C₂³×Dic₂₆	1	2	yes	—
169	C₂²	C₂³×D₅₂	1	1	yes	characteristic
170	C₂²	C₂²×#6127	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	52	16	no	—
173	C₄	C₂×#88123	1	8	yes	—
174	C₄	C₂×#88129	1	16	yes	—
175	C₂	#13123214	1	8	yes	direct product factor
176	C₂	C₂²×#88129	1	2	yes	—
177	C₂	C₂²×#88123	1	1	yes	characteristic
178	C₂	C₂×#11719175	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	C₂×#2862
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #13123214	C₂	direct (N × H)
27. C₂×C₄²×C₁₃	Q₈	semidirect
56. C₄²×C₁₃	C₂×Q₈	semidirect
116. C₂×C₄²	Dic₂₆	semidirect
122. C₂×C₁₃	#2862	semidirect
145. C₄²	C₂×Dic₂₆	semidirect
147. C₁₃	C₂×#2862	semidirect
175. C₂	#13123214	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×#88123	2
#13123214	2
C₂×#11718590	2
C₂×#11718597	2
C₂×C₄³×C₁₃	2
C₂×#2862	13

Subgroups — C2×#13123214 #16068932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123214 #16068932