Subgroups — C₂×#13112918 #15777772

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal
2	#13112918	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	—	2	4	no	—
8	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×#88052	—	2	2	no	—
11	C₂×#88052	—	2	4	no	—
12	C₂×#88052	C₄	1	8	yes	complemented
13	C₂×#88052	C₂²	1	4	yes	—
14	C₂²×#6090	—	2	2	no	—
15	C₂³×C₄×C₁₃	—	2	1	no	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₈	1	2	yes	—
20	C₂²×C₈×C₁₃	C₈	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	4	yes	—
27	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	—	2	1	no	—
32	C₂×C₄²×C₁₃	C₈	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃	—	2	4	no	—
35	C₄×C₈×C₁₃	—	2	4	no	—
36	#88052	—	2	8	no	—
37	#88052	—	2	4	no	—
38	#88052	C₂×C₄	1	8	yes	complemented
39	C₂×#6090	—	2	8	no	—
40	C₂×#6090	—	2	4	no	—
41	C₂×#1376	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	8	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	—	2	4	no	—
57	C₂×C₈×C₁₃	—	2	4	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₄²	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
64	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	C₂×C₈	1	4	yes	—
76	#6090	—	2	8	no	—
77	C₂²×M₃₂	—	13	2	no	—
78	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
79	#1376	—	13	4	no	—
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	#97	1	2	yes	—
82	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
83	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	16	no	—
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	#95	1	2	yes	—
90	C₂×C₄×C₁₃	#95	1	2	yes	—
91	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₁₃	#97	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₁₃	—	2	4	no	—
108	C₈×C₁₃	#95	1	2	yes	—
109	C₈×C₁₃	#95	1	2	yes	—
110	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
111	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
114	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
115	C₂×M₃₂	—	13	8	no	—
116	C₂×M₃₂	—	13	4	no	—
117	C₂×M₃₂	—	26	2	no	—
118	C₂×M₃₂	—	26	4	no	—
119	C₂²×C₁₆	—	26	2	no	—
120	C₂×C₄×C₈	—	2	4	no	—
121	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
122	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
123	C₂²×C₁₃	—	2	4	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	#336	1	2	yes	—
128	C₂²×C₁₃	#336	1	2	yes	—
129	C₂²×C₁₃	#364	1	4	yes	—
130	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₄×C₁₃	—	2	8	no	—
134	C₄×C₁₃	#336	1	2	yes	—
135	C₄×C₁₃	#336	1	2	yes	—
136	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
137	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	#1055	1	2	yes	—
145	C₂²×C₈	#1055	1	2	yes	—
146	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
147	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
148	C₂²×C₈	C₄×D₁₃	1	4	yes	—
149	C₂²×C₈	D₅₂	1	1	yes	characteristic
150	C₂²×C₈	D₅₂	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
152	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	#1055	1	4	yes	—
158	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
159	M₃₂	—	13	8	no	—
160	M₃₂	—	26	8	no	—
161	M₃₂	—	26	4	no	—
162	C₂×C₁₆	—	26	8	no	—
163	C₂×C₁₆	—	26	4	no	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	—	2	4	no	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	#1376	1	4	yes	complemented
170	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#364	1	2	yes	—
172	C₂⁴	#6099	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
177	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
178	C₂²×C₄	C₂×#1055	1	2	yes	—
179	C₂²×C₄	#6099	1	2	yes	—
180	C₂²×C₄	#6099	1	1	yes	characteristic
181	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
182	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×#1055	1	4	yes	—
191	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
192	C₂×C₈	#6101	1	4	yes	—
193	C₂×C₈	#6101	1	4	yes	—
194	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
195	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
196	C₂×C₈	#6103	1	4	yes	—
197	C₂×C₈	#6103	1	4	yes	—
198	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
199	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
201	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₂×#1055	1	4	yes	—
206	C₁₆	—	26	8	no	—
207	C₁₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	4	no	—
209	C₂³	#88043	1	2	yes	—
210	C₂³	C₄⋊#1055	1	1	yes	characteristic
211	C₂³	C₄⋊#1055	1	1	yes	characteristic
212	C₂³	C₂²⋊#1055	1	2	yes	—
213	C₂³	C₂×#6099	1	1	yes	characteristic
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₄×#1055	1	2	yes	—
218	C₂×C₄	C₄×#1055	1	2	yes	—
219	C₂×C₄	#88043	1	2	yes	—
220	C₂×C₄	C₄⋊#1055	1	2	yes	—
221	C₂×C₄	C₄⋊#1055	1	2	yes	—
222	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
223	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
224	C₂×C₄	C₂²⋊#1055	1	2	yes	—
225	C₂×C₄	C₂²⋊#1055	1	2	yes	—
226	C₂×C₄	C₂²⋊#1055	1	2	yes	—
227	C₂×C₄	#88071	1	2	yes	—
228	C₂×C₄	#88071	1	2	yes	—
229	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
230	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
231	C₂×C₄	C₂×#6099	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	#88071	1	2	yes	—
237	C₈	#88071	1	2	yes	—
238	C₈	C₂×#6101	1	2	yes	—
239	C₈	C₂×#6103	1	2	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	#11718055	1	4	yes	—
245	C₂²	#11718116	1	2	yes	—
246	C₂²	#11718116	1	2	yes	—
247	C₂²	C₂×#88043	1	1	yes	characteristic
248	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	#11718116	1	2	yes	—
252	C₄	#11718116	1	2	yes	—
253	C₄	C₂×C₄×#1055	1	1	yes	characteristic
254	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
255	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
256	C₄	C₂×#88071	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	2	2	no	—
260	C₂	#13112918	1	4	yes	direct product factor
261	C₂	C₂×#11718116	1	1	yes	characteristic
262	C₂	C₂×#11718055	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₅₂
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#1376
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13112918	C₂	direct (N × H)
12. C₂×#88052	C₄	semidirect
38. #88052	C₂×C₄	semidirect
169. C₂×C₁₃	#1376	semidirect
207. C₁₃	C₂×#1376	semidirect
260. C₂	#13112918	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×#88052	2
#13112918	2
C₂²×C₄×C₈×C₁₃	2
C₂×#1376	13

Subgroups — C2×#13112918 #15777772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112918 #15777772