Subgroups — C₂×#13113625 #15832004

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	#13113625	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718050	C₂	1	2	yes	maximal, complemented
5	C₂×#11718049	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	—	2	1	no	—
9	C₁₃×#431	C₄	1	4	yes	—
10	C₁₃×#431	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊#1055	—	2	1	no	—
15	#11718586	—	2	2	no	—
16	#11718586	C₂²	1	4	yes	—
17	#11718585	—	2	2	no	—
18	#11718585	C₂²	1	4	yes	—
19	C₂×C₄⋊#1055	—	2	1	no	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄×#1055	—	2	1	no	—
24	#11718050	C₂²	1	8	yes	complemented
25	#11718049	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
38	C₂×#6099	—	2	4	no	—
39	C₂×#6099	—	4	4	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	1	no	—
42	C₂²×#1055	—	2	1	no	—
43	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₁₃	C₂³	1	4	yes	—
47	C₂²⋊#1055	—	2	4	no	—
48	C₄⋊#1055	—	2	4	no	—
49	C₄⋊#1055	—	2	2	no	—
50	C₄⋊#1055	—	2	1	no	—
51	C₄⋊#1055	—	2	1	no	—
52	C₄⋊#1055	C₂³	1	2	yes	—
53	C₄⋊#1055	C₂³	1	4	yes	—
54	C₄⋊#1055	C₂³	1	2	yes	—
55	C₄×#1055	—	2	4	no	—
56	C₂×#1574	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₃	—	2	8	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
76	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
86	#6099	—	4	8	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	4	2	no	—
91	C₂×#1055	—	4	4	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#431	—	13	1	no	—
94	C₂×#431	D₁₃	1	1	yes	characteristic
95	C₂×#423	—	13	1	no	—
96	#1574	—	13	8	no	—
97	C₂×#335	—	13	2	no	—
98	C₂×#334	—	13	2	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	#108	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	#131	1	4	yes	—
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₁₃	#137	1	2	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₁₃	—	2	8	no	—
128	#1055	—	4	2	no	—
129	#1055	—	4	4	no	—
130	#1055	—	4	2	no	—
131	C₂²×M₁₆	—	26	1	no	—
132	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
133	#431	—	13	4	no	—
134	#431	—	26	2	no	—
135	#431	Dic₁₃	1	4	yes	—
136	#431	C₂×D₁₃	1	4	yes	—
137	#423	—	13	4	no	—
138	#423	—	26	2	no	—
139	C₂×C₄⋊C₈	—	13	1	no	—
140	C₂×C₄⋊C₈	—	13	2	no	—
141	C₂×C₄⋊C₈	—	13	1	no	—
142	C₂×C₄⋊C₈	—	26	1	no	—
143	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
144	C₂×C₂²⋊C₈	—	26	1	no	—
145	C₂×C₂²⋊C₈	Dic₁₃	1	2	yes	—
146	C₂×C₄×C₈	—	26	1	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
148	#335	—	13	8	no	—
149	#334	—	13	8	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	#424	1	2	yes	—
155	C₂²×C₁₃	#428	1	2	yes	—
156	C₂²×C₁₃	C₂×#108	1	2	yes	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	#424	1	2	yes	—
163	C₄×C₁₃	#428	1	2	yes	—
164	C₄×C₁₃	C₂×#108	1	4	yes	—
165	C₄×C₁₃	C₂×#131	1	2	yes	—
166	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
168	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
169	C₂³×C₄	D₅₂	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	26	4	no	—
172	C₂×M₁₆	—	52	4	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	—	26	1	no	—
175	C₂²×C₈	—	26	1	no	—
176	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
177	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
178	C₂×C₄²	Dic₂₆	1	2	yes	—
179	C₂×C₄²	D₅₂	1	2	yes	—
180	C₂×C₄²	D₅₂	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	C₄⋊C₈	—	13	2	no	—
186	C₄⋊C₈	—	13	4	no	—
187	C₄⋊C₈	—	13	2	no	—
188	C₄⋊C₈	—	26	2	no	—
189	C₄⋊C₈	—	26	1	no	—
190	C₄⋊C₈	—	26	1	no	—
191	C₄⋊C₈	—	26	4	no	—
192	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
193	C₂²⋊C₈	—	26	4	no	—
194	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
195	C₄×C₈	—	26	4	no	—
196	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
197	C₄×C₈	C₂²×D₁₃	1	4	yes	—
198	C₂×C₁₃	—	4	2	no	—
199	C₂×C₁₃	#1574	1	4	yes	complemented
200	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
202	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
203	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
210	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
212	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
216	C₂²×C₄	C₂×D₅₂	1	2	yes	—
217	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
220	M₁₆	—	52	8	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	4	no	—
225	C₂×C₈	—	52	4	no	—
226	C₂×C₈	—	52	2	no	—
227	C₂×C₈	—	52	2	no	—
228	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
229	C₄²	C₄⋊Dic₁₃	1	2	yes	—
230	C₄²	C₄⋊Dic₁₃	1	2	yes	—
231	C₄²	C₄⋊Dic₁₃	1	2	yes	—
232	C₄²	C₂×Dic₂₆	1	2	yes	—
233	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
234	C₄²	C₂×D₅₂	1	2	yes	—
235	C₄²	C₂×D₅₂	1	1	yes	characteristic
236	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
237	C₄²	C₂²×Dic₁₃	1	2	yes	—
238	C₄²	C₂³×D₁₃	1	1	yes	characteristic
239	C₁₃	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
244	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	#88059	1	4	yes	—
256	C₂×C₄	#88162	1	2	yes	—
257	C₂×C₄	#88163	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
264	C₂×C₄	#88202	1	4	yes	—
265	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
267	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	52	2	no	—
270	C₈	—	52	4	no	—
271	C₈	—	52	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#11718660	1	2	yes	—
277	C₂²	C₂×#88059	1	2	yes	—
278	C₂²	#11718782	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#11718660	1	2	yes	—
285	C₄	C₂×#88059	1	4	yes	—
286	C₄	#11718782	1	2	yes	—
287	C₄	C₂×#88162	1	1	yes	characteristic
288	C₄	C₂×#88163	1	1	yes	characteristic
289	C₄	C₂×#88202	1	2	yes	—
290	C₂	—	4	2	no	—
291	C₂	#13113625	1	4	yes	direct product factor
292	C₂	C₂×#11718660	1	1	yes	characteristic
293	C₂	C₂×#11718782	1	1	yes	characteristic
294	C₂	C₂²×#88059	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	C₂×#1574
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13113625	C₂	direct (N × H)
4. C₂×#11718050	C₂	semidirect
5. C₂×#11718049	C₂	semidirect
24. #11718050	C₂²	semidirect
25. #11718049	C₂²	semidirect
199. C₂×C₁₃	#1574	semidirect
239. C₁₃	C₂×#1574	semidirect
291. C₂	#13113625	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×#11718586	2
C₂×#11718585	2
#13113625	2
C₂×#11718050	2
C₂×#11718049	2
C₂×C₁₃×#431	2
C₂×#1574	13

Subgroups — C2×#13113625 #15832004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113625 #15832004