Subgroups — C₂×#13114742 #15861020

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	2	yes	maximal
2	C₂×C₄×#6099	C₂	1	2	yes	maximal
3	#13114742	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718144	C₂	1	2	yes	maximal
5	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	4	yes	—
10	C₂×C₁₃×#97	C₄	1	4	yes	—
11	C₂×C₁₃×#97	C₄	1	4	yes	—
12	C₂×C₁₃×#97	C₂²	1	4	yes	—
13	C₂×#88059	C₄	1	16	yes	complemented
14	C₂×#88059	C₂²	1	8	yes	—
15	C₄×#6099	—	2	4	no	—
16	C₄×#6099	C₂²	1	8	yes	—
17	C₂×#88043	—	2	2	no	—
18	C₂×C₄×#1055	—	2	2	no	—
19	#11718144	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₂×#6099	—	2	4	no	—
37	C₂×#6099	—	2	8	no	—
38	C₂×#6099	C₂×C₄	1	16	yes	—
39	C₂×#6099	C₂³	1	8	yes	—
40	C₂²×#1055	—	2	4	no	—
41	C₁₃×#97	C₂×C₄	1	16	yes	—
42	#88059	C₂×C₄	1	32	yes	complemented
43	#88043	—	2	8	no	—
44	C₄×#1055	—	2	8	no	—
45	C₂×#1778	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	C₄²	1	8	yes	—
63	C₂×C₈×C₁₃	C₄²	1	8	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
70	C₄²×C₁₃	C₂×D₄	1	4	yes	—
71	#6099	—	2	16	no	—
72	#6099	—	2	8	no	—
73	#6099	C₂²×C₄	1	16	yes	—
74	C₂×#1055	—	2	16	no	—
75	C₂×#1055	—	2	8	no	—
76	C₂²×#108	—	13	2	no	—
77	C₂×C₄×M₁₆	—	13	2	no	—
78	C₂²×#97	D₁₃	1	1	yes	characteristic
79	#1778	—	13	8	no	—
80	C₂×#341	—	13	2	no	—
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
85	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#117	1	2	yes	—
95	C₂×C₄×C₁₃	#117	1	2	yes	—
96	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
103	C₈×C₁₃	—	2	4	no	—
104	C₈×C₁₃	C₂×C₄²	1	8	yes	—
105	#1055	—	2	16	no	—
106	C₂²×M₁₆	—	13	4	no	—
107	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
109	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	C₂×#108	—	13	16	no	—
111	C₂×#108	—	13	8	no	—
112	C₄×M₁₆	—	13	8	no	—
113	C₄×M₁₆	—	26	4	no	—
114	C₂×#97	—	26	2	no	—
115	C₂×#97	Dic₁₃	1	4	yes	—
116	C₂×#97	Dic₁₃	1	4	yes	—
117	C₂×#97	C₂×D₁₃	1	4	yes	—
118	C₂×C₄×C₈	—	26	2	no	—
119	#341	—	13	8	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	8	no	—
123	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
124	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₁₃	#430	1	4	yes	—
126	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
128	C₄×C₁₃	—	2	8	no	—
129	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
130	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
131	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
133	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
134	C₂³×C₄	Dic₂₆	1	2	yes	—
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂×M₁₆	—	13	16	no	—
137	C₂×M₁₆	—	13	8	no	—
138	C₂×M₁₆	—	26	4	no	—
139	C₂×M₁₆	—	26	8	no	—
140	C₂²×C₈	—	2	4	no	—
141	C₂²×C₈	—	26	4	no	—
142	C₂²×C₈	C₄×D₁₃	1	8	yes	—
143	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
144	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
145	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
146	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
147	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
148	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
149	C₂×C₄²	Dic₂₆	1	2	yes	—
150	C₂×C₄²	D₅₂	1	2	yes	—
151	C₂×C₄²	D₅₂	1	2	yes	—
152	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
153	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
154	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
155	#108	—	13	32	no	—
156	#97	—	26	8	no	—
157	#97	C₂×Dic₁₃	1	16	yes	—
158	C₄×C₈	—	26	8	no	—
159	C₂×C₁₃	—	2	2	no	—
160	C₂×C₁₃	—	2	2	no	—
161	C₂×C₁₃	#1778	1	4	yes	complemented
162	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#430	1	2	yes	—
164	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
167	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
168	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
169	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
171	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
173	C₂²×C₄	#6127	1	4	yes	—
174	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
177	M₁₆	—	13	16	no	—
178	M₁₆	—	26	16	no	—
179	M₁₆	—	26	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	26	16	no	—
184	C₂×C₈	—	26	8	no	—
185	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
186	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
187	C₂×C₈	C₂×C₄×D₁₃	1	8	yes	—
188	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
189	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
190	C₄²	C₄⋊Dic₁₃	1	4	yes	—
191	C₄²	C₄⋊Dic₁₃	1	4	yes	—
192	C₄²	C₂×D₅₂	1	4	yes	—
193	C₄²	C₂²×Dic₁₃	1	4	yes	—
194	C₁₃	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	2	4	no	—
196	C₂³	C₄×Dic₂₆	1	4	yes	—
197	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
198	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
199	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
204	C₂×C₄	C₄×D₅₂	1	4	yes	—
205	C₂×C₄	C₄×D₅₂	1	4	yes	—
206	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
207	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
211	C₂×C₄	#88180	1	2	yes	—
212	C₂×C₄	#88180	1	2	yes	—
213	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
214	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
215	C₂×C₄	C₂×#6127	1	2	yes	—
216	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
217	C₈	—	2	4	no	—
218	C₈	—	26	16	no	—
219	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
224	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
225	C₂²	#11718786	1	4	yes	—
226	C₂²	C₂×C₄×Dic₂₆	1	2	yes	—
227	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
230	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
231	C₄	C₂×C₄×D₅₂	1	2	yes	—
232	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
233	C₄	C₂×#88180	1	1	yes	characteristic
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	#13114742	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
238	C₂	C₂×#11718786	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	C₂×#1778
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #13114742	C₂	direct (N × H)
13. C₂×#88059	C₄	semidirect
42. #88059	C₂×C₄	semidirect
161. C₂×C₁₃	#1778	semidirect
194. C₁₃	C₂×#1778	semidirect
236. C₂	#13114742	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×#88059	2
C₂×C₄×#6099	2
#13114742	2
C₂×#11718144	2
C₂²×C₁₃×#97	2
C₂×#1778	13

Subgroups — C2×#13114742 #15861020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114742 #15861020