Subgroups — C₂×#11718858 #13123732

Order 1664.

464 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	2	yes	maximal, complemented
3	C₂×#88077	C₂	1	2	yes	maximal, complemented
4	#11718858	C₂	1	8	yes	maximal, direct product factor
5	C₂×#88059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88055	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#41	C₂²	1	2	yes	—
8	C₁₃×#135	C₄	1	4	yes	complemented
9	C₁₃×#135	C₂²	1	4	yes	—
10	C₂×C₁₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₃×SD₁₆	C₄	1	2	yes	complemented
12	C₂×C₁₃×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#88202	—	2	4	no	—
15	#88202	C₂²	1	8	yes	complemented
16	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#6099	—	2	2	no	—
18	C₂×#6099	C₂²	1	2	yes	—
19	C₂²×#1055	—	2	2	no	—
20	C₂²×#1055	C₂²	1	1	yes	characteristic
21	#88077	C₂²	1	8	yes	complemented
22	#88059	C₂²	1	4	yes	complemented
23	#88055	C₂²	1	4	yes	complemented
24	C₁₃×#41	—	2	2	no	—
25	C₁₃×#41	—	2	2	no	—
26	C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₁₃×#41	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	2	1	no	—
34	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
35	C₁₃×SD₁₆	C₂×C₄	1	8	yes	complemented
36	C₁₃×D₈	C₂×C₄	1	4	yes	complemented
37	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂³	1	2	yes	—
41	C₄×Dic₁₃	—	2	1	no	—
42	C₄×Dic₁₃	C₂³	1	2	yes	—
43	#6099	—	2	2	no	—
44	#6099	—	2	8	no	—
45	#6099	C₂³	1	4	yes	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	8	no	—
48	C₂×#1055	D₄	1	2	yes	—
49	C₂×#1055	D₄	1	2	yes	—
50	C₂×#1055	C₂³	1	2	yes	—
51	C₂×#444	—	13	1	no	maximal, Sylow
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	#41	1	1	yes	characteristic
54	Q₈×C₁₃	—	2	2	no	—
55	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
56	D₄×C₁₃	—	2	4	no	—
57	D₄×C₁₃	—	2	4	no	—
58	D₄×C₁₃	—	2	2	no	—
59	D₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	4	2	no	—
71	C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	#1055	—	2	4	no	—
74	#1055	C₂×D₄	1	4	yes	—
75	C₂×#135	D₁₃	1	1	yes	characteristic
76	C₂×#131	—	13	2	no	—
77	#444	—	13	8	no	—
78	C₂×#108	—	13	1	no	—
79	C₂×#104	—	13	2	no	—
80	C₂×#97	—	13	1	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	—	2	4	no	—
83	C₂²×C₁₃	—	4	4	no	—
84	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
87	C₄×C₁₃	—	2	4	no	—
88	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
89	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	Dic₁₃	—	4	2	no	—
93	C₂×#41	C₂×D₁₃	1	2	yes	—
94	#135	Dic₁₃	1	4	yes	complemented
95	#135	C₂×D₁₃	1	4	yes	—
96	C₂×Q₁₆	Dic₁₃	1	1	yes	characteristic, complemented
97	C₂×SD₁₆	Dic₁₃	1	2	yes	complemented
98	C₂×D₈	Dic₁₃	1	1	yes	characteristic, complemented
99	#131	—	13	8	no	—
100	#131	—	26	4	no	—
101	C₂×M₁₆	—	13	2	no	—
102	C₂×M₁₆	—	26	2	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂²×C₈	—	26	2	no	—
105	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄²	—	13	1	no	—
107	#108	—	13	4	no	—
108	#104	—	13	8	no	—
109	#97	—	13	4	no	—
110	C₂×C₁₃	—	2	1	no	—
111	C₂×C₁₃	—	2	1	no	—
112	C₂×C₁₃	—	4	4	no	—
113	C₂×C₁₃	#444	1	2	yes	complemented
114	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	#41	C₂×Dic₁₃	1	4	yes	—
118	#41	C₂²×D₁₃	1	4	yes	—
119	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	26	1	no	—
124	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	Q₁₆	C₂×Dic₁₃	1	4	yes	complemented
126	SD₁₆	C₂×Dic₁₃	1	8	yes	complemented
127	D₈	C₂×Dic₁₃	1	4	yes	complemented
128	M₁₆	—	13	4	no	—
129	M₁₆	—	26	8	no	—
130	M₁₆	—	26	2	no	—
131	C₂×C₈	—	13	2	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	26	8	no	—
135	C₂×C₈	—	26	2	no	—
136	C₂×C₈	C₂×Dic₁₃	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₃	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
139	C₂×C₈	C₂²×D₁₃	1	2	yes	—
140	C₄²	—	13	2	no	—
141	C₄²	—	26	1	no	—
142	C₁₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	2	2	no	—
144	C₂³	#6129	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×Dic₁₃	1	4	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×Dic₁₃	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	26	2	no	—
155	C₂×C₄	—	52	2	no	—
156	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
157	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
158	C₂×C₄	#6129	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
161	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
162	C₈	—	13	4	no	—
163	C₈	—	26	4	no	—
164	C₈	C₂²×Dic₁₃	1	2	yes	—
165	C₈	C₂²×Dic₁₃	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
170	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
171	C₂²	C₂×#6129	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	52	2	no	—
174	C₄	D₄×Dic₁₃	1	1	yes	characteristic
175	C₄	D₄×Dic₁₃	1	1	yes	characteristic
176	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
177	C₄	C₂³×Dic₁₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	#11718858	1	2	yes	direct product factor
182	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₄×D₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×#135	D₁₃
hypocenter	C₁₃	C₂×#444
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂×#88077	C₂	semidirect
4. #11718858	C₂	direct (N × H)
5. C₂×#88059	C₂	semidirect
6. C₂×#88055	C₂	semidirect
8. C₁₃×#135	C₄	semidirect
10. C₂×C₁₃×Q₁₆	C₄	semidirect
11. C₂×C₁₃×SD₁₆	C₄	semidirect
12. C₂×C₁₃×D₈	C₄	semidirect
15. #88202	C₂²	semidirect
21. #88077	C₂²	semidirect
22. #88059	C₂²	semidirect
23. #88055	C₂²	semidirect
34. C₁₃×Q₁₆	C₂×C₄	semidirect
35. C₁₃×SD₁₆	C₂×C₄	semidirect
36. C₁₃×D₈	C₂×C₄	semidirect
94. #135	Dic₁₃	semidirect
96. C₂×Q₁₆	Dic₁₃	semidirect
97. C₂×SD₁₆	Dic₁₃	semidirect
98. C₂×D₈	Dic₁₃	semidirect
113. C₂×C₁₃	#444	semidirect
125. Q₁₆	C₂×Dic₁₃	semidirect
126. SD₁₆	C₂×Dic₁₃	semidirect
127. D₈	C₂×Dic₁₃	semidirect
142. C₁₃	C₂×#444	semidirect
181. C₂	#11718858	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#135	2
C₂×#88202	2
C₂×#88077	2
#11718858	2
C₂×#88059	2
C₂×#88055	2
C₂×#444	13

Subgroups — C2×#11718858 #13123732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718858 #13123732