Subgroups — C₂×#13114992 #15863316

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88048	C₂	1	4	yes	maximal
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718596	C₂	1	1	yes	maximal, characteristic
4	#13114992	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	2	yes	—
12	C₁₃×#384	—	2	2	no	—
13	C₁₃×#384	C₂²	1	4	yes	—
14	C₂×#88048	—	2	16	no	—
15	C₂×#88048	C₂²	1	16	yes	—
16	#11718596	—	2	2	no	—
17	#11718596	C₂²	1	4	yes	—
18	C₂×C₄×#1055	C₄	1	8	yes	complemented
19	C₂×C₄×#1055	C₂²	1	4	yes	—
20	C₂³×C₄×C₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
27	C₂×C₁₃×C₄⋊C₄	—	4	8	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	—	2	8	no	—
34	C₂×C₄⋊Dic₁₃	—	2	8	no	—
35	C₂×C₄⋊Dic₁₃	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	—	4	8	no	—
37	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
38	C₂²×#1055	C₂×C₄	1	8	yes	—
39	C₂²×#1055	D₄	1	8	yes	—
40	C₂²×#1055	Q₈	1	8	yes	—
41	C₂²×#1055	C₂³	1	4	yes	—
42	#88048	—	2	32	no	—
43	C₄×#1055	C₂×C₄	1	16	yes	complemented
44	C₂×#1844	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	8	no	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
51	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	4	no	—
60	C₂²×Dic₁₃	—	4	4	no	—
61	C₂²×Dic₁₃	—	4	8	no	—
62	C₁₃×C₄⋊C₄	—	2	8	no	—
63	C₁₃×C₄⋊C₄	—	4	4	no	—
64	C₁₃×C₄⋊C₄	—	4	16	no	—
65	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
68	C₄⋊Dic₁₃	—	2	8	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	C₄⋊Dic₁₃	—	4	16	no	—
71	C₂×#1055	C₄⋊C₄	1	32	yes	complemented
72	C₂×#1055	C₂²×C₄	1	8	yes	—
73	C₂×#1055	C₂×D₄	1	8	yes	—
74	C₂×#1055	C₂×Q₈	1	8	yes	—
75	C₂²×#106	—	13	4	no	—
76	C₂²×C₄×C₈	—	13	1	no	—
77	C₂×#384	—	13	1	no	—
78	C₂×#384	D₁₃	1	1	yes	characteristic
79	#1844	—	13	8	no	—
80	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₃	#127	1	2	yes	—
82	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
83	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
84	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	—	4	8	no	—
88	C₂×C₄×C₁₃	—	8	8	no	—
89	C₂×C₄×C₁₃	#106	1	32	yes	complemented
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
93	C₂×C₄×C₁₃	#127	1	2	yes	—
94	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
95	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	8	yes	—
97	C₂×C₄×C₁₃	C₂×SD₁₆	1	8	yes	—
98	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×Dic₁₃	—	4	4	no	—
103	C₂×Dic₁₃	—	4	8	no	—
104	C₂×Dic₁₃	—	8	8	no	—
105	#1055	C₂×C₄⋊C₄	1	16	yes	complemented
106	C₂³×C₈	—	13	2	no	—
107	C₂²×C₄⋊C₄	—	2	2	no	—
108	C₂²×C₄⋊C₄	—	13	2	no	—
109	C₂²×C₄⋊C₄	—	26	2	no	—
110	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
111	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
112	C₂×#106	—	13	16	no	—
113	C₂×#106	—	26	16	no	—
114	C₂×C₄×C₈	—	13	8	no	—
115	C₂×C₄×C₈	—	13	4	no	—
116	#384	—	2	2	no	—
117	#384	—	13	4	no	—
118	#384	—	26	2	no	—
119	#384	C₂×D₁₃	1	4	yes	—
120	C₂²×C₁₃	#384	1	4	yes	—
121	C₂²×C₁₃	C₂×#106	1	8	yes	—
122	C₂²×C₁₃	#492	1	4	yes	—
123	C₂²×C₁₃	#492	1	4	yes	—
124	C₂²×C₁₃	#498	1	8	yes	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
129	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
130	C₄×C₁₃	—	8	4	no	—
131	C₄×C₁₃	#384	1	4	yes	—
132	C₄×C₁₃	C₂×#106	1	16	yes	complemented
133	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
134	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
135	Dic₁₃	—	8	4	no	—
136	C₂³×C₄	—	2	2	no	—
137	C₂³×C₄	—	26	2	no	—
138	C₂³×C₄	C₂²⋊D₁₃	1	2	yes	—
139	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
140	C₂²×C₈	—	13	8	no	—
141	C₂²×C₈	—	13	8	no	—
142	C₂²×C₈	—	13	8	no	—
143	C₂²×C₈	—	13	4	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	4	8	no	—
148	C₂×C₄⋊C₄	—	13	4	no	—
149	C₂×C₄⋊C₄	—	26	8	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄⋊C₄	—	26	8	no	—
152	C₂×C₄⋊C₄	—	52	8	no	—
153	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
154	C₂×C₄²	Dic₂₆	1	4	yes	—
155	C₂×C₄²	C₄×D₁₃	1	4	yes	—
156	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
157	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
158	#106	—	26	32	no	—
159	C₄×C₈	—	13	16	no	—
160	C₂×C₁₃	#1844	1	8	yes	complemented
161	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
162	C₂×C₁₃	C₂²×#106	1	2	yes	—
163	C₂×C₁₃	C₂×#492	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#492	1	1	yes	characteristic
165	C₂×C₁₃	C₂×#498	1	2	yes	—
166	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	4	8	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	26	4	no	—
171	C₂²×C₄	—	52	4	no	—
172	C₂²×C₄	—	52	8	no	—
173	C₂²×C₄	#6101	1	8	yes	—
174	C₂²×C₄	#6105	1	8	yes	—
175	C₂²×C₄	#6106	1	8	yes	—
176	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
178	C₂²×C₄	D₄×D₁₃	1	4	yes	—
179	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
183	C₂×C₈	—	13	32	no	—
184	C₂×C₈	—	13	8	no	—
185	C₂×C₈	—	13	8	no	—
186	C₂×C₈	—	13	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	4	16	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	26	8	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	16	no	—
193	C₄²	#6101	1	8	yes	—
194	C₄²	C₂×Dic₂₆	1	4	yes	—
195	C₄²	C₂×C₄×D₁₃	1	4	yes	—
196	C₁₃	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×#6101	1	2	yes	—
198	C₂³	C₂×#6105	1	4	yes	—
199	C₂³	#88194	1	2	yes	—
200	C₂³	C₂×#6106	1	4	yes	—
201	C₂³	#88198	1	2	yes	—
202	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	—	52	4	no	—
207	C₂×C₄	—	52	8	no	—
208	C₂×C₄	—	104	8	no	—
209	C₂×C₄	#88048	1	32	yes	complemented
210	C₂×C₄	#88142	1	8	yes	—
211	C₂×C₄	C₄⋊C₄×D₁₃	1	8	yes	—
212	C₂×C₄	C₂×#6101	1	8	yes	—
213	C₂×C₄	C₂×#6101	1	2	yes	—
214	C₂×C₄	C₂×#6105	1	8	yes	—
215	C₂×C₄	#88194	1	2	yes	—
216	C₂×C₄	C₂×#6106	1	8	yes	—
217	C₂×C₄	#88198	1	2	yes	—
218	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
221	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
222	C₈	—	13	16	no	—
223	C₂²	C₂×#88048	1	8	yes	—
224	C₂²	#11718637	1	4	yes	—
225	C₂²	#11718737	1	4	yes	—
226	C₂²	#11718740	1	8	yes	—
227	C₂²	#11718746	1	4	yes	—
228	C₂²	C₂²×#6101	1	1	yes	characteristic
229	C₂²	C₂²×#6105	1	2	yes	—
230	C₂²	C₂×#88194	1	1	yes	characteristic
231	C₂²	C₂²×#6106	1	2	yes	—
232	C₂²	C₂×#88198	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	104	4	no	—
235	C₄	C₂×#88048	1	16	yes	complemented
236	C₄	#11718637	1	4	yes	—
237	C₄	C₂×#88142	1	2	yes	—
238	C₄	C₂×C₄⋊C₄×D₁₃	1	2	yes	—
239	C₂	#13114992	1	8	yes	direct product factor
240	C₂	C₂×#11718637	1	1	yes	characteristic
241	C₂	C₂²×#88048	1	2	yes	—
242	C₂	C₂×#11718737	1	1	yes	characteristic
243	C₂	C₂×#11718746	1	1	yes	characteristic
244	C₂	C₂×#11718740	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#384	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#384	D₁₃
hypocenter	C₁₃	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
4. #13114992	C₂	direct (N × H)
18. C₂×C₄×#1055	C₄	semidirect
43. C₄×#1055	C₂×C₄	semidirect
71. C₂×#1055	C₄⋊C₄	semidirect
89. C₂×C₄×C₁₃	#106	semidirect
105. #1055	C₂×C₄⋊C₄	semidirect
132. C₄×C₁₃	C₂×#106	semidirect
160. C₂×C₁₃	#1844	semidirect
196. C₁₃	C₂×#1844	semidirect
209. C₂×C₄	#88048	semidirect
235. C₄	C₂×#88048	semidirect
239. C₂	#13114992	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×#88048	2
C₂²×C₄×#1055	2
C₂×#11718596	2
#13114992	2
C₂×C₁₃×#384	2
C₂×#1844	13

Subgroups — C2×#13114992 #15863316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114992 #15863316