Subgroups — C₂×#13114993 #15863317

Order 3328.

1358 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×#88057	C₂	1	4	yes	maximal
2	C₂×#11718596	C₂	1	2	yes	maximal
3	#13114993	C₂	1	8	yes	maximal, direct product factor
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	—	2	4	no	—
8	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
10	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
11	C₂×#88057	—	2	16	no	—
12	C₂×#88057	C₂²	1	16	yes	—
13	#11718596	—	2	4	no	—
14	#11718596	C₂²	1	8	yes	—
15	C₂³×C₄×C₁₃	D₄	1	2	yes	—
16	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
17	C₂³×Dic₁₃	—	2	4	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₁₃	D₄	1	8	yes	—
20	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
21	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
25	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
26	C₂×C₄⋊Dic₁₃	—	2	16	no	—
27	C₂×C₄⋊Dic₁₃	—	2	16	no	—
28	C₂×C₄⋊Dic₁₃	—	2	4	no	—
29	C₂×C₄⋊Dic₁₃	—	4	16	no	—
30	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
32	#88057	—	2	32	no	—
33	C₂×#1844	—	13	1	no	maximal, Sylow
34	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
36	C₂²×C₄×C₁₃	SD₁₆	1	16	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	8	no	—
45	C₂²×Dic₁₃	—	4	8	no	—
46	C₂²×Dic₁₃	—	4	16	no	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
51	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
52	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
53	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
54	C₄⋊Dic₁₃	—	2	16	no	—
55	C₄⋊Dic₁₃	—	4	8	no	—
56	C₄⋊Dic₁₃	—	4	32	no	—
57	C₂²×#106	—	13	4	no	—
58	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
59	C₂×#384	—	13	2	no	—
60	#1844	—	13	8	no	—
61	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₁₃	#127	1	2	yes	—
63	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
64	C₂³×C₁₃	C₂×SD₁₆	1	8	yes	—
65	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#106	1	32	yes	complemented
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₁₃	#127	1	2	yes	—
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
72	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×SD₁₆	1	16	yes	—
74	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×Dic₁₃	—	4	8	no	—
79	C₂×Dic₁₃	—	4	16	no	—
80	C₂×Dic₁₃	—	8	16	no	—
81	C₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	complemented
82	C₂³×C₈	C₂×D₁₃	1	2	yes	—
83	C₂²×C₄⋊C₄	—	13	4	no	—
84	C₂²×C₄⋊C₄	—	26	4	no	—
85	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
86	C₂×#106	—	13	16	no	—
87	C₂×#106	—	26	16	no	—
88	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
89	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
90	#384	—	13	8	no	—
91	#384	—	26	4	no	—
92	C₂²×C₁₃	#384	1	4	yes	—
93	C₂²×C₁₃	C₂×#106	1	8	yes	—
94	C₂²×C₁₃	#492	1	8	yes	—
95	C₂²×C₁₃	#498	1	8	yes	—
96	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
98	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₁₃	C₂²×SD₁₆	1	4	yes	—
100	C₄×C₁₃	#384	1	4	yes	—
101	C₄×C₁₃	C₂×#106	1	16	yes	complemented
102	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
103	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
104	Dic₁₃	—	8	8	no	—
105	C₂³×C₄	—	26	4	no	—
106	C₂³×C₄	D₅₂	1	2	yes	—
107	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
108	C₂²×C₈	Dic₂₆	1	8	yes	—
109	C₂²×C₈	D₅₂	1	8	yes	—
110	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
111	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
112	C₂×C₄⋊C₄	—	13	8	no	—
113	C₂×C₄⋊C₄	—	26	16	no	—
114	C₂×C₄⋊C₄	—	26	4	no	—
115	C₂×C₄⋊C₄	—	26	16	no	—
116	C₂×C₄⋊C₄	—	52	16	no	—
117	C₂×C₄²	Dic₂₆	1	4	yes	—
118	C₂×C₄²	D₅₂	1	2	yes	—
119	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
120	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
121	#106	—	26	32	no	—
122	C₄×C₈	C₂×Dic₁₃	1	16	yes	complemented
123	C₂×C₁₃	#1844	1	8	yes	complemented
124	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
125	C₂×C₁₃	C₂²×#106	1	2	yes	—
126	C₂×C₁₃	C₂×#492	1	2	yes	—
127	C₂×C₁₃	C₂×#498	1	2	yes	—
128	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
129	C₂²×C₄	—	26	8	no	—
130	C₂²×C₄	—	52	8	no	—
131	C₂²×C₄	—	52	16	no	—
132	C₂²×C₄	#6095	1	16	yes	—
133	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
134	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
135	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
136	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₅₂	1	4	yes	—
138	C₂²×C₄	C₂×D₅₂	1	4	yes	—
139	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
140	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₁₃	1	32	yes	complemented
142	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
143	C₂×C₈	C₂×D₅₂	1	8	yes	—
144	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
145	C₄⋊C₄	—	26	16	no	—
146	C₄⋊C₄	—	52	8	no	—
147	C₄⋊C₄	—	52	32	no	—
148	C₄²	C₄⋊Dic₁₃	1	8	yes	—
149	C₄²	C₂×Dic₂₆	1	4	yes	—
150	C₄²	C₂²×Dic₁₃	1	4	yes	—
151	C₁₃	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
152	C₂³	#88123	1	2	yes	—
153	C₂³	#88128	1	2	yes	—
154	C₂³	C₂×#6095	1	8	yes	—
155	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
156	C₂³	C₂²×D₅₂	1	1	yes	characteristic
157	C₂×C₄	—	52	8	no	—
158	C₂×C₄	—	52	16	no	—
159	C₂×C₄	—	104	16	no	—
160	C₂×C₄	#88057	1	32	yes	complemented
161	C₂×C₄	#88123	1	8	yes	—
162	C₂×C₄	#88123	1	2	yes	—
163	C₂×C₄	#88128	1	2	yes	—
164	C₂×C₄	C₂×#6095	1	16	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
167	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
168	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
169	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
170	C₂×C₄	C₂²×D₅₂	1	2	yes	—
171	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₁₃	1	16	yes	complemented
173	C₂²	#11718341	1	8	yes	—
174	C₂²	#11718354	1	8	yes	—
175	C₂²	#11718596	1	4	yes	—
176	C₂²	C₂×#88057	1	8	yes	—
177	C₂²	C₂×#88123	1	1	yes	characteristic
178	C₂²	C₂×#88128	1	1	yes	characteristic
179	C₂²	C₂²×#6095	1	4	yes	—
180	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
181	C₄	—	104	8	no	—
182	C₄	#11718596	1	4	yes	—
183	C₄	C₂×#88057	1	16	yes	complemented
184	C₄	C₂×#88123	1	2	yes	—
185	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
186	C₂	#13114993	1	8	yes	direct product factor
187	C₂	C₂×#11718596	1	1	yes	characteristic
188	C₂	C₂²×#88057	1	2	yes	—
189	C₂	C₂×#11718354	1	2	yes	—
190	C₂	C₂×#11718341	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13114993	C₂	direct (N × H)
9. C₂×C₄×C₈×C₁₃	C₄	semidirect
31. C₄×C₈×C₁₃	C₂×C₄	semidirect
47. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
66. C₂×C₄×C₁₃	#106	semidirect
81. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
88. C₂×C₄×C₈	Dic₁₃	semidirect
101. C₄×C₁₃	C₂×#106	semidirect
122. C₄×C₈	C₂×Dic₁₃	semidirect
123. C₂×C₁₃	#1844	semidirect
141. C₂×C₈	C₄⋊Dic₁₃	semidirect
151. C₁₃	C₂×#1844	semidirect
160. C₂×C₄	#88057	semidirect
172. C₈	C₂×C₄⋊Dic₁₃	semidirect
183. C₄	C₂×#88057	semidirect
186. C₂	#13114993	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×#88057	2
C₂×#11718596	2
#13114993	2
C₂²×C₄×C₈×C₁₃	2
C₂×#1844	13

Subgroups — C2×#13114993 #15863317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114993 #15863317