Subgroups — C₂×#13113721 #15832647

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718673	C₂	1	2	yes	maximal, complemented
2	#13113721	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊#1055	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×#6099	—	2	2	no	—
10	C₄×C₁₃×D₈	C₄	1	4	yes	—
11	C₄×C₁₃×D₈	C₂²	1	4	yes	—
12	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
13	C₂×C₁₃×#102	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊#1055	—	2	2	no	—
16	#11718673	—	2	4	no	—
17	#11718673	C₂²	1	8	yes	complemented
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	D₄⋊#1055	C₂²	1	8	yes	complemented
22	#11718050	C₂²	1	4	yes	complemented
23	C₈×#1055	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	—	2	2	no	—
26	C₂×C₁₃×D₈	—	2	4	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
33	C₄×D₄×C₁₃	C₂³	1	4	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×#6099	—	2	8	no	—
38	C₂×#6099	—	4	8	no	—
39	C₂²×#1055	—	2	1	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	D₄	1	1	yes	characteristic
42	C₂²×#1055	D₄	1	1	yes	characteristic
43	C₁₃×#107	C₂×C₄	1	4	yes	—
44	C₁₃×#102	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	C₂²⋊#1055	—	2	8	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	C₂³	1	4	yes	—
50	C₄×#1055	—	2	1	no	—
51	C₄×#1055	D₄	1	2	yes	—
52	C₄×#1055	C₂³	1	2	yes	—
53	C₄×#1055	C₂³	1	2	yes	—
54	C₂×#1589	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	—	2	2	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	2	4	no	—
58	C₂×D₄×C₁₃	—	2	4	no	—
59	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	4	4	no	—
64	C₂²×C₄×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
68	C₁₃×D₈	—	2	8	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	M₁₆	1	4	yes	complemented
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
75	C₁₃×C₂²⋊C₄	—	2	4	no	—
76	C₁₃×C₂²⋊C₄	—	2	4	no	—
77	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	#41	1	1	yes	characteristic
80	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
81	#6099	—	4	16	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	4	4	no	—
87	C₂×#1055	D₈	1	4	yes	complemented
88	C₂×#1055	C₂×D₄	1	2	yes	—
89	C₂×#1055	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
91	C₂×#436	—	13	2	no	—
92	#1589	—	13	8	no	—
93	C₂×#335	—	13	1	no	—
94	C₂×D₄⋊C₈	—	13	2	no	—
95	C₂×C₈²	—	13	1	no	—
96	C₂³×C₁₃	—	2	4	no	—
97	C₂³×C₁₃	—	4	4	no	—
98	C₂³×C₁₃	—	4	4	no	—
99	C₂³×C₁₃	—	4	4	no	—
100	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	2	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	4	8	no	—
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₁₃	#131	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃	#135	1	2	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	C₂×M₁₆	1	4	yes	complemented
122	#1055	—	2	4	no	—
123	#1055	—	2	2	no	—
124	#1055	—	4	4	no	—
125	#1055	C₂×D₈	1	4	yes	complemented
126	C₂²×D₈	Dic₁₃	1	1	yes	characteristic
127	C₂²×M₁₆	—	26	2	no	—
128	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
129	C₄×D₈	Dic₁₃	1	4	yes	—
130	C₄×D₈	C₂×D₁₃	1	4	yes	—
131	#436	—	13	8	no	—
132	#436	—	26	4	no	—
133	C₂×#107	Dic₁₃	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	13	2	no	—
135	C₂×#102	Dic₁₃	1	2	yes	—
136	C₂×C₂²⋊C₈	—	26	2	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
140	#335	—	13	4	no	—
141	D₄⋊C₈	—	13	8	no	—
142	C₈²	—	13	4	no	—
143	C₂²×C₁₃	—	4	4	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	8	8	no	—
147	C₂²×C₁₃	#436	1	2	yes	—
148	C₂²×C₁₃	C₄×D₈	1	2	yes	—
149	C₂²×C₁₃	#443	1	2	yes	—
150	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	4	4	no	—
154	C₄×C₁₃	#436	1	2	yes	—
155	C₄×C₁₃	C₄×D₈	1	2	yes	—
156	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
158	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
160	C₂²×D₄	C₂×Dic₁₃	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
164	C₂×M₁₆	—	26	8	no	—
165	C₂×M₁₆	—	52	8	no	—
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	26	1	no	—
169	C₂²×C₈	—	26	2	no	—
170	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	C₂×Dic₁₃	1	4	yes	—
175	C₄×D₄	C₂²×D₁₃	1	4	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
179	#107	C₂×Dic₁₃	1	4	yes	—
180	C₄⋊C₈	—	13	4	no	—
181	C₄⋊C₈	—	26	2	no	—
182	#102	C₂×Dic₁₃	1	8	yes	—
183	C₂²⋊C₈	—	26	8	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
186	C₄×C₈	—	13	2	no	—
187	C₄×C₈	—	26	1	no	—
188	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
189	C₄×C₈	C₂²×D₁₃	1	2	yes	—
190	C₂×C₁₃	—	8	4	no	—
191	C₂×C₁₃	#1589	1	4	yes	complemented
192	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
193	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
194	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
195	C₂⁴	—	2	2	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	#6129	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
208	D₈	—	2	8	no	—
209	M₁₆	—	52	16	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	13	4	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	26	2	no	—
216	C₂×C₈	—	26	2	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	52	4	no	—
219	C₂×C₈	#6099	1	4	yes	complemented
220	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
221	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₄²	D₄×D₁₃	1	1	yes	characteristic
227	C₄²	#6129	1	1	yes	characteristic
228	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
229	C₄²	C₂³×D₁₃	1	1	yes	characteristic
230	C₁₃	C₂×#1589	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	D₈×D₁₃	1	2	yes	—
246	C₂×C₄	#88166	1	2	yes	—
247	C₂×C₄	C₂×#6099	1	2	yes	—
248	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
251	C₂×C₄	#88202	1	2	yes	—
252	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#6129	1	1	yes	characteristic
254	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	13	4	no	—
257	C₈	—	26	4	no	—
258	C₈	—	26	2	no	—
259	C₈	—	52	4	no	—
260	C₈	C₂×#6099	1	4	yes	complemented
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	#11718673	1	2	yes	—
266	C₂²	D₈×Dic₁₃	1	2	yes	—
267	C₂²	#11718857	1	2	yes	—
268	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	#11718673	1	2	yes	—
273	C₄	D₈×Dic₁₃	1	2	yes	—
274	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
275	C₄	C₂×#88166	1	1	yes	characteristic
276	C₄	C₂²×#6099	1	1	yes	characteristic
277	C₄	C₂×#88202	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	#13113721	1	4	yes	direct product factor
280	C₂	C₂×#11718673	1	1	yes	characteristic
281	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
282	C₂	C₂×#11718857	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×C₁₃×D₈	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	C₂×#1589
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11718673	C₂	semidirect
2. #13113721	C₂	direct (N × H)
3. C₂×#11718050	C₂	semidirect
4. C₂×D₄⋊#1055	C₂	semidirect
5. C₂×C₈×#1055	C₂	semidirect
17. #11718673	C₂²	semidirect
21. D₄⋊#1055	C₂²	semidirect
22. #11718050	C₂²	semidirect
23. C₈×#1055	C₂²	semidirect
70. C₂×C₈×C₁₃	M₁₆	semidirect
87. C₂×#1055	D₈	semidirect
121. C₈×C₁₃	C₂×M₁₆	semidirect
125. #1055	C₂×D₈	semidirect
191. C₂×C₁₃	#1589	semidirect
219. C₂×C₈	#6099	semidirect
230. C₁₃	C₂×#1589	semidirect
260. C₈	C₂×#6099	semidirect
279. C₂	#13113721	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×#11718673	2
#13113721	2
C₂×#11718050	2
C₂×D₄⋊#1055	2
C₂×C₈×#1055	2
C₂×C₄×C₁₃×D₈	2
C₂×#1589	13

Subgroups — C2×#13113721 #15832647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113721 #15832647