Subgroups — C₂×#13112914 #15777756

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6090	C₂	1	2	yes	maximal
2	#13112914	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	C₂²×#6090	—	2	4	no	—
11	C₂²×#6090	—	2	4	no	—
12	C₂²×#6090	C₄	1	8	yes	complemented
13	C₂²×#6090	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃	C₈	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₈	1	4	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	4	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₈	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	—	2	4	no	—
31	C₄×C₈×C₁₃	—	2	4	no	—
32	C₂×#6090	—	2	16	no	—
33	C₂×#6090	—	2	4	no	—
34	C₂×#6090	—	2	4	no	—
35	C₂×#6090	C₂×C₄	1	8	yes	complemented
36	C₂×#1374	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	—	2	4	no	—
40	C₂²×C₄×C₁₃	C₁₆	1	8	yes	complemented
41	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	C₄²	1	2	yes	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
58	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₃	—	2	4	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	#6090	—	2	8	no	—
67	C₂³×C₁₆	—	13	2	no	—
68	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
69	#1374	—	13	4	no	—
70	C₂³×C₁₃	C₄×C₈	1	2	yes	—
71	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
72	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
73	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
74	C₂³×C₁₃	M₃₂	1	4	yes	—
75	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	16	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	#95	1	4	yes	—
80	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	#97	1	4	yes	—
82	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×C₁₆	1	8	yes	complemented
87	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₁₃	—	2	4	no	—
94	C₈×C₁₃	#95	1	4	yes	—
95	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
98	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
99	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
100	C₂²×C₁₆	—	13	8	no	—
101	C₂²×C₁₆	—	13	4	no	—
102	C₂²×C₁₆	—	26	4	no	—
103	C₂²×C₁₆	—	26	4	no	—
104	C₂×C₄×C₈	—	2	4	no	—
105	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
106	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
107	C₂²×C₁₃	#336	1	4	yes	—
108	C₂²×C₁₃	C₄×C₁₆	1	4	yes	—
109	C₂²×C₁₃	#346	1	4	yes	—
110	C₂²×C₁₃	C₂²⋊C₁₆	1	8	yes	—
111	C₂²×C₁₃	C₄⋊C₁₆	1	8	yes	—
112	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
116	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
117	C₄×C₁₃	—	2	8	no	—
118	C₄×C₁₃	#336	1	4	yes	—
119	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
120	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
121	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂³×C₄	#1055	1	2	yes	—
124	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	#1055	1	4	yes	—
128	C₂²×C₈	Dic₂₆	1	2	yes	—
129	C₂²×C₈	C₄×D₁₃	1	4	yes	—
130	C₂²×C₈	D₅₂	1	2	yes	—
131	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
132	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
133	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
134	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	#1055	1	2	yes	—
138	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
139	C₂×C₁₆	—	13	8	no	—
140	C₂×C₁₆	—	26	16	no	—
141	C₂×C₁₆	—	26	4	no	—
142	C₂×C₁₆	—	26	4	no	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	2	4	no	—
145	C₂×C₁₃	#1374	1	8	yes	complemented
146	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
147	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
149	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	2	yes	—
150	C₂×C₁₃	C₂×C₄⋊C₁₆	1	2	yes	—
151	C₂⁴	C₂×#1055	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	#6090	1	8	yes	complemented
155	C₂²×C₄	C₂×#1055	1	4	yes	—
156	C₂²×C₄	C₂×#1055	1	2	yes	—
157	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
158	C₂²×C₄	#6099	1	2	yes	—
159	C₂²×C₄	#6099	1	2	yes	—
160	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
161	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
162	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂×#1055	1	4	yes	—
168	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
169	C₂×C₈	#6101	1	8	yes	—
170	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
171	C₂×C₈	#6103	1	8	yes	—
172	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
173	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
175	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₁₆	—	26	8	no	—
181	C₁₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
182	C₂³	C₄×#1055	1	2	yes	—
183	C₂³	C₄⋊#1055	1	2	yes	—
184	C₂³	C₂×#6090	1	4	yes	—
185	C₂³	#88052	1	4	yes	—
186	C₂³	C₂²⋊#1055	1	2	yes	—
187	C₂³	C₂²×#1055	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	16	no	—
191	C₂×C₄	C₄×#1055	1	2	yes	—
192	C₂×C₄	#88043	1	4	yes	—
193	C₂×C₄	C₄⋊#1055	1	4	yes	—
194	C₂×C₄	C₄⋊#1055	1	2	yes	—
195	C₂×C₄	C₂×#6090	1	8	yes	complemented
196	C₂×C₄	C₂²⋊#1055	1	4	yes	—
197	C₂×C₄	C₂²⋊#1055	1	2	yes	—
198	C₂×C₄	#88071	1	4	yes	—
199	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
200	C₂×C₄	C₂×#6099	1	1	yes	characteristic
201	C₂×C₄	C₂×#6099	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	#88071	1	4	yes	—
207	C₈	C₂×#6101	1	2	yes	—
208	C₈	C₂×#6103	1	2	yes	—
209	C₂²	C₄×#6090	1	4	yes	—
210	C₂²	#11718053	1	4	yes	—
211	C₂²	C₄⋊#6090	1	8	yes	—
212	C₂²	#11718116	1	4	yes	—
213	C₂²	C₂²⋊#6090	1	8	yes	—
214	C₂²	C₂×C₄×#1055	1	1	yes	characteristic
215	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
216	C₂²	C₂²×#6090	1	2	yes	—
217	C₂²	C₂×#88052	1	2	yes	—
218	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
219	C₄	—	2	8	no	—
220	C₄	#11718116	1	4	yes	—
221	C₄	C₂×#88043	1	1	yes	characteristic
222	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
223	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
224	C₄	C₂×#88071	1	1	yes	characteristic
225	C₂	#13112914	1	8	yes	direct product factor
226	C₂	C₂×#11718116	1	1	yes	characteristic
227	C₂	C₂×C₄×#6090	1	1	yes	characteristic
228	C₂	C₂×#11718053	1	1	yes	characteristic
229	C₂	C₂×C₂²⋊#6090	1	2	yes	—
230	C₂	C₂×C₄⋊#6090	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#1374
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13112914	C₂	direct (N × H)
12. C₂²×#6090	C₄	semidirect
35. C₂×#6090	C₂×C₄	semidirect
40. C₂²×C₄×C₁₃	C₁₆	semidirect
86. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
145. C₂×C₁₃	#1374	semidirect
154. C₂²×C₄	#6090	semidirect
181. C₁₃	C₂×#1374	semidirect
195. C₂×C₄	C₂×#6090	semidirect
225. C₂	#13112914	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#6090	2
#13112914	2
C₂²×C₄×C₈×C₁₃	2
C₂×#1374	13

Subgroups — C2×#13112914 #15777756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112914 #15777756