Subgroups — C₂×#13131178 #16807012

Order 3328.

698 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	#13131178	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718233	C₂	1	2	yes	maximal, complemented
4	C₂×#11718261	C₂	1	2	yes	maximal, complemented
5	C₂×#11718369	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	D₁₃⋊M₃₂	—	2	4	no	—
11	D₁₃⋊M₃₂	C₂²	1	8	yes	—
12	C₂×#88098	—	2	2	no	—
13	C₂×D₁₃⋊C₁₆	—	2	2	no	—
14	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
15	C₂×#88055	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	#11718369	C₄	1	4	yes	—
19	#11718369	C₂²	1	4	yes	—
20	#11718261	C₂²	1	8	yes	complemented
21	#11718233	C₂²	1	8	yes	complemented
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×#6103	—	2	1	no	—
27	C₂×#6101	—	2	1	no	—
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×#6094	C₈	1	4	yes	—
32	C₂×#6094	C₈	1	4	yes	—
33	C₂×#6094	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₁₃	D₄	1	2	yes	—
36	C₂×C₈×D₁₃	D₄	1	2	yes	—
37	C₂×C₈×D₁₃	Q₈	1	2	yes	—
38	C₂×C₈×D₁₃	Q₈	1	2	yes	—
39	C₂×C₈×D₁₃	C₂³	1	2	yes	—
40	#88126	—	2	2	no	—
41	#88126	C₂×C₄	1	4	yes	—
42	#88106	—	2	4	no	—
43	#88106	C₂³	1	8	yes	—
44	C₂×#6092	—	2	4	no	—
45	#88098	—	2	8	no	—
46	D₁₃⋊C₁₆	—	2	8	no	—
47	C₁₃×#97	C₂×C₄	1	4	yes	—
48	#88055	C₂×C₄	1	4	yes	—
49	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₃	C₂³	1	2	yes	—
51	C₄×#1055	C₂×C₄	1	2	yes	—
52	C₄×#1055	C₂³	1	2	yes	—
53	C₂×#2153	—	13	1	no	maximal, Sylow
54	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	1	no	—
58	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
59	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
62	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
64	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	#6103	—	2	4	no	—
74	#6101	—	2	4	no	—
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂×#1055	C₂×C₈	1	2	yes	—
78	C₂×#1055	C₂×C₈	1	2	yes	—
79	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
80	C₂×#1055	C₂²×C₄	1	2	yes	—
81	C₂×#1055	C₂×D₄	1	1	yes	characteristic
82	C₂×#1055	C₂×D₄	1	1	yes	characteristic
83	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
84	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
85	C₂×#1055	C₂⁴	1	1	yes	characteristic
86	#6094	C₂×C₈	1	16	yes	—
87	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
88	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
89	C₈×D₁₃	C₂×D₄	1	4	yes	—
90	C₈×D₁₃	C₂×Q₈	1	4	yes	—
91	#6092	—	2	8	no	—
92	C₂²×M₃₂	—	13	2	no	—
93	C₂×#405	—	13	1	no	—
94	#2153	—	13	8	no	—
95	C₂×#364	—	13	2	no	—
96	C₂×#364	—	13	2	no	—
97	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₁₃	—	4	2	no	—
99	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
100	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
101	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
113	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
114	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
115	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
116	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
117	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₄×D₁₃	C₂²×C₈	1	4	yes	—
119	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₈×C₁₃	C₂²×C₈	1	4	yes	—
121	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
122	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
123	#1055	C₂²×C₈	1	4	yes	—
124	#1055	C₂²×D₄	1	1	yes	characteristic
125	#1055	C₂²×Q₈	1	1	yes	characteristic
126	C₂²×M₁₆	—	13	1	no	—
127	C₂³×C₈	—	13	1	no	—
128	C₂×#117	—	13	1	no	—
129	C₂×M₃₂	—	13	4	no	—
130	C₂×M₃₂	—	13	8	no	—
131	C₂×M₃₂	—	26	2	no	—
132	C₂×M₃₂	—	26	4	no	—
133	C₂²×C₁₆	—	26	2	no	—
134	#405	—	13	4	no	—
135	#405	—	13	4	no	—
136	C₂×#97	—	13	1	no	—
137	C₂×#97	#261	1	1	yes	characteristic
138	C₂×C₄×C₈	—	13	1	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	#364	—	13	8	no	—
141	#364	—	13	8	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
146	C₂×D₁₃	—	2	4	no	—
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	—	4	2	no	—
149	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
155	Dic₁₃	—	4	2	no	—
156	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
157	C₂³×C₄	—	13	1	no	—
158	C₂×M₁₆	—	13	4	no	—
159	C₂×M₁₆	—	13	4	no	—
160	C₂×M₁₆	—	13	4	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	—	13	2	no	—
166	C₂²×C₈	—	13	2	no	—
167	C₂²×C₈	—	13	2	no	—
168	C₂²×C₈	—	13	2	no	—
169	C₂²×C₈	C₂×#261	1	1	yes	characteristic
170	C₂²×C₈	C₂×#261	1	1	yes	characteristic
171	#117	—	13	4	no	—
172	#117	—	26	2	no	—
173	C₂×C₄⋊C₄	—	26	1	no	—
174	C₂×C₂²⋊C₄	—	26	1	no	—
175	C₂×C₄²	—	13	1	no	—
176	C₂×C₄²	C₂×#261	1	1	yes	characteristic
177	M₃₂	—	13	8	no	—
178	M₃₂	—	26	8	no	—
179	M₃₂	—	26	4	no	—
180	C₂×C₁₆	—	26	8	no	—
181	C₂×C₁₆	—	26	4	no	—
182	#97	—	13	4	no	—
183	#97	C₂×#261	1	4	yes	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
186	C₄×C₈	—	13	2	no	—
187	C₄×C₈	—	13	2	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	#2153	1	2	yes	complemented
191	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
192	D₁₃	—	2	4	no	—
193	C₂⁴	—	13	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	2	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	2	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	26	1	no	—
203	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
204	M₁₆	—	13	16	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	1	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	—	13	4	no	—
216	C₂×C₈	—	13	4	no	—
217	C₂×C₈	—	13	4	no	—
218	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
219	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
220	C₂×C₈	C₄⋊#261	1	2	yes	—
221	C₂×C₈	C₄⋊#261	1	2	yes	—
222	C₂×C₈	C₂²×#261	1	2	yes	—
223	C₂×C₈	C₂²×#261	1	2	yes	—
224	C₄⋊C₄	—	26	4	no	—
225	C₂²⋊C₄	—	26	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	13	2	no	—
228	C₄²	—	26	1	no	—
229	C₄²	C₂²×#261	1	2	yes	—
230	C₁₆	—	26	8	no	—
231	C₁₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	13	2	no	—
233	C₂³	—	13	2	no	—
234	C₂³	—	13	2	no	—
235	C₂³	—	52	2	no	—
236	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	2	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	13	4	no	—
244	C₂×C₄	—	13	4	no	—
245	C₂×C₄	—	13	4	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	52	2	no	—
248	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
250	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
253	C₂×C₄	C₂³×#261	1	1	yes	characteristic
254	C₈	—	13	4	no	—
255	C₈	—	13	1	no	—
256	C₈	—	13	1	no	—
257	C₈	—	13	1	no	—
258	C₈	—	13	1	no	—
259	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
260	C₈	C₂×C₄⋊#261	1	4	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	13	4	no	—
263	C₂²	—	26	4	no	—
264	C₂²	—	52	2	no	—
265	C₂²	—	52	2	no	—
266	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
267	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
268	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	13	4	no	—
271	C₄	—	52	2	no	—
272	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
273	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
274	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
275	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	26	4	no	—
279	C₂	#13131178	1	2	yes	direct product factor
280	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#97	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#97	#261
hypocenter	C₁₃	C₂×#2153
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #13131178	C₂	direct (N × H)
3. C₂×#11718233	C₂	semidirect
4. C₂×#11718261	C₂	semidirect
20. #11718261	C₂²	semidirect
21. #11718233	C₂²	semidirect
190. C₂×C₁₃	#2153	semidirect
231. C₁₃	C₂×#2153	semidirect
279. C₂	#13131178	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₃⋊M₃₂	2
#13131178	2
C₂×#11718233	2
C₂×#11718261	2
C₂×#11718369	2
C₂×#2153	13

Subgroups — C2×#13131178 #16807012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131178 #16807012