Subgroups — C₂×#13131169 #16806993

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88098	C₂	1	2	yes	maximal
2	#13131169	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718260	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#6092	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×#6092	C₂²	1	4	yes	—
10	C₂×#88098	—	2	8	no	—
11	C₂×#88098	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₃	C₄	1	4	yes	—
17	C₄×C₈×D₁₃	C₂²	1	4	yes	—
18	#11718260	C₂²	1	8	yes	complemented
19	C₄⋊#6092	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂×C₈×D₁₃	—	2	4	no	—
32	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₁₃	D₄	1	4	yes	—
35	C₂×C₈×D₁₃	Q₈	1	4	yes	—
36	C₂×C₈×D₁₃	C₂³	1	2	yes	—
37	C₄²×D₁₃	C₈	1	4	yes	—
38	C₄²×D₁₃	C₂×C₄	1	4	yes	—
39	C₂×#6092	—	2	8	no	—
40	C₂×#6092	C₂³	1	8	yes	—
41	#88098	—	2	16	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₁₃	C₂³	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	2	yes	—
47	C₄×#1055	C₂³	1	2	yes	—
48	C₂×#2150	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	C₂×C₈	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	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	—	2	4	no	—
58	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
60	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
61	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
63	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₃	C₂×C₈	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
71	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
72	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
73	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
74	C₂×#1055	—	2	2	no	—
75	C₂×#1055	C₂²×C₄	1	2	yes	—
76	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂×D₄	1	2	yes	—
78	C₂×#1055	C₂×Q₈	1	2	yes	—
79	C₂×#1055	C₂⁴	1	1	yes	characteristic
80	C₈×D₁₃	—	2	8	no	—
81	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
82	C₈×D₁₃	C₂×D₄	1	4	yes	—
83	C₈×D₁₃	C₂×Q₈	1	4	yes	—
84	#6092	—	2	8	no	—
85	C₂²×M₃₂	—	13	2	no	—
86	C₂²×C₄×C₈	—	13	1	no	—
87	#2150	—	13	8	no	—
88	C₂×C₄⋊C₁₆	—	13	2	no	—
89	C₂×C₄⋊C₁₆	—	13	2	no	—
90	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₂²×D₁₃	—	2	2	no	—
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
94	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
105	C₂×Dic₁₃	M₃₂	1	4	yes	complemented
106	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
107	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
108	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
109	C₄×D₁₃	—	2	8	no	—
110	C₄×D₁₃	—	2	8	no	—
111	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
112	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
116	#1055	—	2	2	no	—
117	#1055	C₂×C₄⋊C₄	1	2	yes	—
118	#1055	C₂²×D₄	1	1	yes	characteristic
119	#1055	C₂²×Q₈	1	1	yes	characteristic
120	C₂³×C₈	—	13	1	no	—
121	C₂³×C₈	—	13	1	no	—
122	C₂²×C₄²	—	13	1	no	—
123	C₂×M₃₂	—	13	8	no	—
124	C₂×M₃₂	—	26	8	no	—
125	C₂²×C₁₆	—	13	4	no	—
126	C₂×C₄×C₈	—	13	1	no	—
127	C₂×C₄×C₈	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂×C₄×C₈	—	13	4	no	—
130	C₂×C₄×C₈	—	13	4	no	—
131	C₂×C₄×C₈	#261	1	1	yes	characteristic
132	C₄⋊C₁₆	—	13	8	no	—
133	C₄⋊C₁₆	—	13	8	no	—
134	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
135	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
136	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
137	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
138	C₂×D₁₃	—	2	2	no	—
139	C₂×D₁₃	—	2	2	no	—
140	C₂×D₁₃	—	2	8	no	—
141	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
144	C₄×C₁₃	C₂×M₃₂	1	4	yes	complemented
145	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
149	Dic₁₃	C₂×M₃₂	1	4	yes	complemented
150	C₂³×C₄	—	13	1	no	—
151	C₂³×C₄	—	13	1	no	—
152	C₂³×C₄	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	13	4	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	—	13	2	no	—
157	C₂²×C₈	—	13	2	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	—	26	4	no	—
161	C₂²×C₈	C₂×#261	1	1	yes	characteristic
162	C₂²×C₈	C₂×#261	1	1	yes	characteristic
163	C₂×C₄²	—	13	1	no	—
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₂×C₄²	—	13	4	no	—
167	C₂×C₄²	—	13	4	no	—
168	C₂×C₄²	C₂×#261	1	1	yes	characteristic
169	M₃₂	—	26	16	no	—
170	C₂×C₁₆	—	13	8	no	—
171	C₂×C₁₆	—	26	8	no	—
172	C₄×C₈	—	13	2	no	—
173	C₄×C₈	—	13	2	no	—
174	C₄×C₈	—	13	4	no	—
175	C₄×C₈	—	13	2	no	—
176	C₄×C₈	—	13	2	no	—
177	C₄×C₈	C₂×#261	1	4	yes	—
178	C₂×C₁₃	#2150	1	4	yes	complemented
179	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
180	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
181	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
182	D₁₃	—	2	4	no	—
183	C₂⁴	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	13	4	no	—
188	C₂²×C₄	—	13	4	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	4	no	—
192	C₂²×C₄	—	13	4	no	—
193	C₂²×C₄	—	26	4	no	—
194	C₂²×C₄	—	26	4	no	—
195	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
196	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	8	no	—
205	C₂×C₈	—	13	4	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	26	8	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	C₄⋊#261	1	4	yes	—
210	C₂×C₈	C₂²×#261	1	2	yes	—
211	C₂×C₈	C₂²×#261	1	2	yes	—
212	C₄²	—	13	4	no	—
213	C₄²	—	13	4	no	—
214	C₄²	—	13	2	no	—
215	C₄²	—	13	2	no	—
216	C₄²	D₁₃⋊C₈	1	2	yes	—
217	C₄²	C₂²×#261	1	2	yes	—
218	C₁₆	—	26	8	no	—
219	C₁₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	13	4	no	—
221	C₂³	—	13	2	no	—
222	C₂³	—	26	2	no	—
223	C₂³	—	26	2	no	—
224	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	13	4	no	—
227	C₂×C₄	—	13	4	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	13	8	no	—
231	C₂×C₄	—	13	4	no	—
232	C₂×C₄	—	13	4	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	26	8	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	26	8	no	—
237	C₂×C₄	#88098	1	4	yes	complemented
238	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
239	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
241	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
243	C₂×C₄	C₂³×#261	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	13	1	no	—
246	C₈	—	13	2	no	—
247	C₈	—	13	1	no	—
248	C₈	—	26	2	no	—
249	C₈	C₂×C₄⋊#261	1	4	yes	—
250	C₂²	—	13	4	no	—
251	C₂²	—	26	2	no	—
252	C₂²	—	26	2	no	—
253	C₂²	—	26	8	no	—
254	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
255	C₂²	C₂×#88098	1	2	yes	—
256	C₂²	D₁₃⋊M₃₂	1	2	yes	—
257	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	13	4	no	—
260	C₄	—	13	4	no	—
261	C₄	—	26	2	no	—
262	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
263	C₄	C₂×#88098	1	4	yes	complemented
264	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
265	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
266	C₂	—	26	4	no	—
267	C₂	#13131169	1	4	yes	direct product factor
268	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
269	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
270	C₂	C₂²×#88098	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#2150
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13131169	C₂	direct (N × H)
3. C₂×#11718260	C₂	semidirect
4. C₂×C₄⋊#6092	C₂	semidirect
18. #11718260	C₂²	semidirect
19. C₄⋊#6092	C₂²	semidirect
96. C₂×C₄×C₁₃	M₃₂	semidirect
105. C₂×Dic₁₃	M₃₂	semidirect
144. C₄×C₁₃	C₂×M₃₂	semidirect
149. Dic₁₃	C₂×M₃₂	semidirect
178. C₂×C₁₃	#2150	semidirect
219. C₁₃	C₂×#2150	semidirect
237. C₂×C₄	#88098	semidirect
263. C₄	C₂×#88098	semidirect
267. C₂	#13131169	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×#88098	2
#13131169	2
C₂×#11718260	2
C₂×C₄⋊#6092	2
C₂×C₄×C₈×D₁₃	2
C₂×#2150	13

Subgroups — C2×#13131169 #16806993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131169 #16806993