Subgroups — C₂×#11719227 #13128015

Order 1664.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719227	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88147	C₂	1	2	yes	maximal, complemented
6	C₂×#88141	C₂	1	2	yes	maximal, complemented
7	C₂×#88135	C₂	1	4	yes	maximal, complemented
8	C₂×#88126	C₂	1	2	yes	maximal, complemented
9	C₂×C₄²×D₁₃	C₂	1	2	yes	maximal, complemented
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
12	C₂³×Dic₁₃	C₂²	1	2	yes	—
13	C₂×#6127	C₄	1	16	yes	complemented
14	C₂×#6127	C₂²	1	8	yes	—
15	C₂²×D₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	C₂²	1	4	yes	—
17	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
18	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
19	C₁₃×#117	C₂²	1	8	yes	complemented
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₂×#6103	C₂²	1	4	yes	—
24	C₂×#6101	C₂²	1	4	yes	—
25	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	8	yes	complemented
30	#88147	C₂²	1	16	yes	complemented
31	#88141	C₂²	1	16	yes	complemented
32	#88135	C₂²	1	32	yes	complemented
33	#88126	C₂²	1	16	yes	complemented
34	C₄²×D₁₃	C₂²	1	16	yes	complemented
35	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
36	C₂³×D₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₄	1	8	yes	complemented
38	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₃	C₂×C₄	1	16	yes	complemented
43	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
44	C₂²×Dic₁₃	—	2	8	no	—
45	C₂²×Dic₁₃	C₂³	1	8	yes	—
46	C₂²×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	2	yes	—
48	C₂²×Dic₁₃	C₂³	1	2	yes	—
49	#6127	C₂×C₄	1	64	yes	complemented
50	C₂×D₅₂	C₂×C₄	1	8	yes	complemented
51	C₂×D₅₂	C₂³	1	4	yes	—
52	C₂×C₄×D₁₃	—	2	16	no	—
53	C₂×C₄×D₁₃	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₁₃	C₂³	1	16	yes	—
55	C₂×C₄×D₁₃	C₂³	1	8	yes	—
56	C₂×Dic₂₆	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₂₆	C₂³	1	4	yes	—
58	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
59	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
60	C₄²×C₁₃	C₂³	1	8	yes	—
61	#6103	C₂³	1	16	yes	—
62	#6101	C₂³	1	16	yes	—
63	C₄×Dic₁₃	C₂³	1	4	yes	—
64	C₄×Dic₁₃	C₂³	1	16	yes	—
65	C₄×Dic₁₃	C₂³	1	4	yes	—
66	C₄×Dic₁₃	C₂³	1	8	yes	—
67	C₂×C₄×#41	—	13	1	no	maximal, Sylow
68	C₂³×C₁₃	—	2	4	no	—
69	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂³×C₁₃	C₂⁴	1	2	yes	—
71	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×D₁₃	—	2	8	no	—
73	C₂²×D₁₃	C₂²×C₄	1	8	yes	—
74	C₂²×D₁₃	C₂⁴	1	4	yes	—
75	C₂×C₄×C₁₃	—	2	8	no	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
80	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
81	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
82	C₂²⋊D₁₃	C₂²×C₄	1	32	yes	complemented
83	C₂×Dic₁₃	—	2	4	no	—
84	C₂×Dic₁₃	—	2	16	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	8	yes	—
86	C₂×Dic₁₃	#41	1	8	yes	complemented
87	C₂×Dic₁₃	C₂⁴	1	4	yes	—
88	C₂×Dic₁₃	C₂⁴	1	4	yes	—
89	C₂×Dic₁₃	C₂⁴	1	4	yes	—
90	D₅₂	C₂²×C₄	1	16	yes	complemented
91	C₄×D₁₃	—	2	32	no	—
92	C₄×D₁₃	C₂²×C₄	1	32	yes	complemented
93	Dic₂₆	C₂²×C₄	1	16	yes	complemented
94	C₂²×#41	—	13	1	no	—
95	C₄×#41	—	13	16	no	—
96	C₂×C₄×Q₈	—	13	2	no	—
97	C₂×C₄×D₄	—	13	2	no	—
98	C₂×C₄×D₄	—	13	4	no	—
99	C₂×#117	—	13	2	no	—
100	C₂×#117	D₁₃	1	1	yes	characteristic, complemented
101	C₂²×C₄²	—	13	1	no	—
102	C₂²×C₄²	—	13	2	no	—
103	C₂²×C₁₃	—	2	8	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	C₂³×C₄	1	4	yes	—
106	C₂²×C₁₃	C₂³×C₄	1	2	yes	—
107	C₂²×C₁₃	C₂×#41	1	4	yes	—
108	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
109	C₂×D₁₃	—	2	16	no	—
110	C₂×D₁₃	—	2	8	no	—
111	C₂×D₁₃	C₂³×C₄	1	8	yes	—
112	C₄×C₁₃	—	2	8	no	—
113	C₄×C₁₃	C₂³×C₄	1	4	yes	—
114	C₄×C₁₃	C₂³×C₄	1	4	yes	—
115	Dic₁₃	—	2	4	no	—
116	Dic₁₃	C₂³×C₄	1	8	yes	—
117	Dic₁₃	C₂×#41	1	8	yes	complemented
118	C₂×#41	—	13	16	no	—
119	C₂×#41	—	13	8	no	—
120	C₂²×Q₈	—	13	1	no	—
121	C₂²×D₄	—	13	2	no	—
122	C₂²×D₄	—	13	1	no	—
123	C₂³×C₄	—	13	2	no	—
124	C₂³×C₄	—	13	2	no	—
125	C₂³×C₄	—	13	4	no	—
126	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
127	C₄×Q₈	—	13	16	no	—
128	C₄×D₄	—	13	16	no	—
129	C₄×D₄	—	13	32	no	—
130	#117	—	13	16	no	—
131	#117	C₂×D₁₃	1	8	yes	complemented
132	C₂×C₄⋊C₄	—	13	4	no	—
133	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
134	C₂×C₂²⋊C₄	—	13	4	no	—
135	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
136	C₂×C₄²	—	13	4	no	—
137	C₂×C₄²	—	13	2	no	—
138	C₂×C₄²	—	13	8	no	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	16	no	—
142	C₂×C₄²	C₂×D₁₃	1	2	yes	—
143	C₂×C₁₃	—	2	4	no	—
144	C₂×C₁₃	C₄×#41	1	4	yes	complemented
145	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
146	C₂×C₁₃	C₂²×#41	1	2	yes	—
147	D₁₃	—	2	8	no	—
148	C₂⁴	—	13	2	no	—
149	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
150	#41	—	13	64	no	—
151	C₂×Q₈	—	13	8	no	—
152	C₂×Q₈	—	13	4	no	—
153	C₂×D₄	—	13	16	no	—
154	C₂×D₄	—	13	8	no	—
155	C₂×D₄	—	13	8	no	—
156	C₂×D₄	—	13	4	no	—
157	C₂²×C₄	—	13	2	no	—
158	C₂²×C₄	—	13	2	no	—
159	C₂²×C₄	—	13	16	no	—
160	C₂²×C₄	—	13	8	no	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	—	13	8	no	—
163	C₂²×C₄	—	13	16	no	—
164	C₂²×C₄	—	26	16	no	—
165	C₂²×C₄	—	26	8	no	—
166	C₂²×C₄	C₄×D₁₃	1	8	yes	complemented
167	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
168	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
170	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	13	16	no	—
172	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
173	C₂²⋊C₄	—	13	16	no	—
174	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
175	C₄²	—	13	4	no	—
176	C₄²	—	13	4	no	—
177	C₄²	—	13	16	no	—
178	C₄²	—	13	8	no	—
179	C₄²	C₂²×D₁₃	1	8	yes	—
180	C₁₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
181	C₂³	—	2	4	no	—
182	C₂³	—	13	8	no	—
183	C₂³	—	13	4	no	—
184	C₂³	—	26	8	no	—
185	C₂³	C₂×C₄×D₁₃	1	4	yes	—
186	C₂³	C₂³×D₁₃	1	2	yes	—
187	C₂³	C₂³×D₁₃	1	1	yes	characteristic
188	Q₈	—	13	16	no	—
189	D₄	—	13	32	no	—
190	D₄	—	13	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	13	8	no	—
193	C₂×C₄	—	13	8	no	—
194	C₂×C₄	—	13	4	no	—
195	C₂×C₄	—	13	4	no	—
196	C₂×C₄	—	13	32	no	—
197	C₂×C₄	—	13	4	no	—
198	C₂×C₄	—	26	32	no	—
199	C₂×C₄	—	26	16	no	—
200	C₂×C₄	—	26	4	no	—
201	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
202	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
203	C₂×C₄	C₂×C₄×D₁₃	1	16	yes	complemented
204	C₂×C₄	C₂³×D₁₃	1	8	yes	—
205	C₂×C₄	C₂³×D₁₃	1	2	yes	—
206	C₂×C₄	C₂³×D₁₃	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	13	8	no	—
210	C₂²	—	26	16	no	—
211	C₂²	—	26	8	no	—
212	C₂²	C₂²×C₄×D₁₃	1	4	yes	—
213	C₂²	C₂²×C₄×D₁₃	1	2	yes	—
214	C₂²	#41×D₁₃	1	4	yes	—
215	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
216	C₄	—	2	8	no	—
217	C₄	—	13	8	no	—
218	C₄	—	13	8	no	—
219	C₄	—	26	4	no	—
220	C₄	C₂²×C₄×D₁₃	1	4	yes	—
221	C₄	C₂²×C₄×D₁₃	1	4	yes	—
222	C₂	—	2	4	no	—
223	C₂	—	26	8	no	—
224	C₂	#11719227	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₁₃	1	1	yes	characteristic
226	C₂	C₂×#41×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×#6127	C₂
cosocle	C₂²×C₁₃	C₂⁵
hypercenter	C₂×#117	D₁₃
hypocenter	C₁₃	C₂×C₄×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#117	C₂	semidirect
3. C₂²×C₄×Dic₁₃	C₂	semidirect
4. #11719227	C₂	direct (N × H)
5. C₂×#88147	C₂	semidirect
6. C₂×#88141	C₂	semidirect
7. C₂×#88135	C₂	semidirect
8. C₂×#88126	C₂	semidirect
9. C₂×C₄²×D₁₃	C₂	semidirect
13. C₂×#6127	C₄	semidirect
19. C₁₃×#117	C₂²	semidirect
29. C₂×C₄×Dic₁₃	C₂²	semidirect
30. #88147	C₂²	semidirect
31. #88141	C₂²	semidirect
32. #88135	C₂²	semidirect
33. #88126	C₂²	semidirect
34. C₄²×D₁₃	C₂²	semidirect
37. C₂²×C₄×C₁₃	C₂×C₄	semidirect
42. C₂×C₂²⋊D₁₃	C₂×C₄	semidirect
49. #6127	C₂×C₄	semidirect
50. C₂×D₅₂	C₂×C₄	semidirect
53. C₂×C₄×D₁₃	C₂×C₄	semidirect
56. C₂×Dic₂₆	C₂×C₄	semidirect
78. C₂×C₄×C₁₃	C₂²×C₄	semidirect
82. C₂²⋊D₁₃	C₂²×C₄	semidirect
86. C₂×Dic₁₃	#41	semidirect
90. D₅₂	C₂²×C₄	semidirect
92. C₄×D₁₃	C₂²×C₄	semidirect
93. Dic₂₆	C₂²×C₄	semidirect
100. C₂×#117	D₁₃	semidirect
117. Dic₁₃	C₂×#41	semidirect
131. #117	C₂×D₁₃	semidirect
144. C₂×C₁₃	C₄×#41	semidirect
166. C₂²×C₄	C₄×D₁₃	semidirect
180. C₁₃	C₂×C₄×#41	semidirect
203. C₂×C₄	C₂×C₄×D₁₃	semidirect
224. C₂	#11719227	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×#6127	2
C₂×C₁₃×#117	2
C₂²×C₄×Dic₁₃	2
#11719227	2
C₂×#88147	2
C₂×#88141	2
C₂×#88135	2
C₂×#88126	2
C₂×C₄²×D₁₃	2
C₂×C₄×#41	13

Subgroups — C2×#11719227 #13128015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719227 #13128015