Subgroups — C₂×#13113023 #15783378

Order 3328.

302 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	C₂×#11718091	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718097	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113023	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#135	—	2	1	no	—
6	C₁₃×#508	—	2	1	no	—
7	C₁₃×#508	—	2	1	no	—
8	C₁₃×#508	C₄	1	2	yes	complemented
9	C₁₃×#508	C₂²	1	2	yes	—
10	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₃×SD₃₂	—	2	1	no	—
12	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#88058	—	2	1	no	—
15	#11718097	—	2	1	no	—
16	#11718097	C₂²	1	2	yes	complemented
17	#11718091	—	2	1	no	—
18	#11718091	C₂²	1	2	yes	complemented
19	C₂×#88034	—	2	1	no	—
20	C₂×C₁₃×#41	—	4	1	no	—
21	C₁₃×#135	—	2	2	no	—
22	C₁₃×#135	—	2	2	no	—
23	C₁₃×#135	—	4	1	no	—
24	C₁₃×#135	—	4	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×SD₁₆	—	4	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	4	1	no	—
30	C₁₃×Q₃₂	—	2	1	no	—
31	C₁₃×Q₃₂	C₂×C₄	1	2	yes	complemented
32	C₁₃×SD₃₂	—	2	2	no	—
33	C₁₃×SD₃₂	—	2	2	no	—
34	C₁₃×D₁₆	—	2	1	no	—
35	C₁₃×D₁₆	C₂×C₄	1	2	yes	complemented
36	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	#88058	—	2	2	no	—
43	#88058	—	4	1	no	—
44	#88034	—	2	2	no	—
45	C₂×#1418	—	13	1	no	maximal, Sylow
46	C₁₃×#41	—	4	2	no	—
47	C₁₃×#41	—	4	2	no	—
48	C₁₃×#41	—	8	1	no	—
49	C₁₃×#41	—	8	1	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂×D₄×C₁₃	—	4	1	no	—
52	C₂×D₄×C₁₃	—	8	1	no	—
53	C₂²×C₄×C₁₃	—	8	1	no	—
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	8	1	no	—
56	C₁₃×Q₁₆	—	2	2	no	—
57	C₁₃×Q₁₆	—	4	1	no	—
58	C₁₃×SD₁₆	—	4	2	no	—
59	C₁₃×SD₁₆	—	4	2	no	—
60	C₁₃×D₈	—	2	2	no	—
61	C₁₃×D₈	—	4	1	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
65	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊Dic₁₃	—	4	2	no	—
69	C₄⋊Dic₁₃	—	8	1	no	—
70	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
73	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
74	C₂×#508	D₁₃	1	1	yes	characteristic
75	C₂×M₆₄	—	13	1	no	—
76	C₂×#367	—	13	1	no	—
77	#1418	—	13	4	no	—
78	C₂³×C₁₃	—	8	1	no	—
79	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
80	Q₈×C₁₃	—	4	2	no	—
81	Q₈×C₁₃	—	8	1	no	—
82	D₄×C₁₃	—	4	2	no	—
83	D₄×C₁₃	—	8	2	no	—
84	D₄×C₁₃	—	8	2	no	—
85	D₄×C₁₃	—	8	1	no	—
86	C₂×C₄×C₁₃	—	8	1	no	—
87	C₂×C₄×C₁₃	—	8	2	no	—
88	C₂×C₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
93	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×Dic₁₃	—	8	2	no	—
96	C₂×Dic₁₃	—	16	2	no	—
97	C₈×C₁₃	#102	1	1	yes	characteristic
98	C₈×C₁₃	#102	1	1	yes	characteristic
99	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×#135	—	2	1	no	—
102	#508	—	2	1	no	—
103	#508	—	2	1	no	—
104	#508	Dic₁₃	1	2	yes	complemented
105	#508	C₂×D₁₃	1	2	yes	—
106	C₂×Q₃₂	Dic₁₃	1	1	yes	characteristic, complemented
107	C₂×SD₃₂	—	2	1	no	—
108	C₂×D₁₆	Dic₁₃	1	1	yes	characteristic, complemented
109	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
110	C₂×#107	—	26	1	no	—
111	M₆₄	—	13	2	no	—
112	M₆₄	—	26	1	no	—
113	C₂×C₃₂	—	26	1	no	—
114	#367	—	13	2	no	—
115	#367	—	26	1	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	8	2	no	—
118	C₂²×C₁₃	—	16	2	no	—
119	C₂²×C₁₃	#357	1	1	yes	characteristic
120	C₂²×C₁₃	#357	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
122	C₄×C₁₃	—	8	2	no	—
123	C₄×C₁₃	#357	1	1	yes	characteristic
124	C₄×C₁₃	#357	1	1	yes	characteristic
125	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
126	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
127	Dic₁₃	—	16	2	no	—
128	C₂×#41	—	4	1	no	—
129	#135	—	2	2	no	—
130	#135	—	2	2	no	—
131	#135	—	4	1	no	—
132	#135	—	4	1	no	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×SD₁₆	—	4	1	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	52	1	no	—
138	Q₃₂	—	2	1	no	—
139	Q₃₂	C₂×Dic₁₃	1	2	yes	complemented
140	SD₃₂	—	2	2	no	—
141	SD₃₂	—	2	2	no	—
142	D₁₆	—	2	1	no	—
143	D₁₆	C₂×Dic₁₃	1	2	yes	complemented
144	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
145	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
150	#107	—	26	2	no	—
151	#107	—	52	1	no	—
152	C₃₂	—	26	2	no	—
153	C₂×C₁₃	—	2	1	no	—
154	C₂×C₁₃	—	2	1	no	—
155	C₂×C₁₃	—	16	2	no	—
156	C₂×C₁₃	#1418	1	2	yes	complemented
157	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
158	#41	—	4	2	no	—
159	#41	—	4	2	no	—
160	#41	—	8	1	no	—
161	#41	—	8	1	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×D₄	—	4	1	no	—
164	C₂×D₄	—	8	1	no	—
165	C₂²×C₄	—	8	1	no	—
166	C₂²×C₄	—	104	1	no	—
167	C₂²×C₄	#6104	1	1	yes	characteristic
168	Q₁₆	—	2	2	no	—
169	Q₁₆	—	4	1	no	—
170	SD₁₆	—	4	2	no	—
171	SD₁₆	—	4	2	no	—
172	D₈	—	2	2	no	—
173	D₈	—	4	1	no	—
174	C₂×C₈	#6104	1	1	yes	characteristic
175	C₂×C₈	#6105	1	1	yes	characteristic
176	C₂×C₈	#6105	1	1	yes	characteristic
177	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	52	2	no	—
181	C₄⋊C₄	—	104	1	no	—
182	C₁₆	C₂²⋊Dic₁₃	1	1	yes	characteristic
183	C₁₆	C₂²⋊Dic₁₃	1	1	yes	characteristic
184	C₁₆	C₂²×Dic₁₃	1	1	yes	characteristic
185	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₁₃	C₂×#1418	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	8	1	no	—
188	C₂³	#88066	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	8	1	no	—
191	D₄	—	4	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	2	no	—
194	D₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	104	2	no	—
199	C₂×C₄	—	208	2	no	—
200	C₂×C₄	#88066	1	1	yes	characteristic
201	C₂×C₄	#88067	1	1	yes	characteristic
202	C₂×C₄	#88067	1	1	yes	characteristic
203	C₂×C₄	#88072	1	1	yes	characteristic
204	C₂×C₄	#88072	1	1	yes	characteristic
205	C₂×C₄	C₂×#6104	1	1	yes	characteristic
206	C₈	#88072	1	1	yes	characteristic
207	C₈	#88072	1	1	yes	characteristic
208	C₈	C₂×#6105	1	1	yes	characteristic
209	C₈	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	#11718154	1	1	yes	characteristic
214	C₂²	#11718154	1	1	yes	characteristic
215	C₂²	C₂×#88066	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	208	2	no	—
218	C₄	#11718154	1	1	yes	characteristic
219	C₄	#11718154	1	1	yes	characteristic
220	C₄	C₂×#88067	1	1	yes	characteristic
221	C₄	C₂×#88072	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	16	2	no	—
225	C₂	#13113023	1	2	yes	direct product factor
226	C₂	C₂×#11718154	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718154
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#508	C₂
socle	C₂²×C₁₃	#357
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×#508	D₁₃
hypocenter	C₁₃	C₂×#1418
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#11718091	C₂	semidirect
2. C₂×#11718097	C₂	semidirect
3. #13113023	C₂	direct (N × H)
8. C₁₃×#508	C₄	semidirect
10. C₂×C₁₃×Q₃₂	C₄	semidirect
12. C₂×C₁₃×D₁₆	C₄	semidirect
16. #11718097	C₂²	semidirect
18. #11718091	C₂²	semidirect
31. C₁₃×Q₃₂	C₂×C₄	semidirect
35. C₁₃×D₁₆	C₂×C₄	semidirect
104. #508	Dic₁₃	semidirect
106. C₂×Q₃₂	Dic₁₃	semidirect
108. C₂×D₁₆	Dic₁₃	semidirect
139. Q₃₂	C₂×Dic₁₃	semidirect
143. D₁₆	C₂×Dic₁₃	semidirect
156. C₂×C₁₃	#1418	semidirect
186. C₁₃	C₂×#1418	semidirect
225. C₂	#13113023	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×#11718091	2
C₂×#11718097	2
#13113023	2
C₂×C₁₃×#508	2
C₂×#1418	13

Subgroups — C2×#13113023 #15783378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113023 #15783378