Subgroups — C₂×#13114322 #15845423

Order 3328.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718748	C₂	1	2	yes	maximal
2	C₂×#11718349	C₂	1	1	yes	maximal, characteristic
3	#13114322	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718139	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂²×#6094	—	2	1	no	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₁₃×#494	—	2	2	no	—
13	C₁₃×#494	C₂²	1	4	yes	complemented
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88060	—	2	1	no	—
16	#11718748	—	2	4	no	—
17	#11718748	C₂²	1	8	yes	—
18	C₂×#88049	—	2	2	no	—
19	C₂×#88047	—	2	2	no	—
20	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	#11718349	—	2	2	no	—
23	#11718349	C₂²	1	4	yes	—
24	#11718139	C₂²	1	8	yes	complemented
25	#11718049	C₂²	1	4	yes	complemented
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂²×D₅₂	D₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₃	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	4	no	—
31	C₂×C₁₃×Q₁₆	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	—	4	4	no	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	2	2	no	—
38	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6103	—	2	1	no	—
43	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
44	C₂²×#1055	—	2	1	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₂×#6094	—	2	4	no	—
47	C₂×#6094	—	4	4	no	—
48	C₄×D₅₂	—	2	1	no	—
49	C₄×D₅₂	—	2	1	no	—
50	C₄×D₅₂	D₄	1	2	yes	—
51	C₄×D₅₂	C₂³	1	2	yes	—
52	C₄×C₈×C₁₃	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#88060	—	2	4	no	—
55	#88049	—	2	8	no	—
56	#88047	—	2	8	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	2	1	no	—
59	C₄⋊#1055	C₂³	1	4	yes	—
60	C₄⋊#1055	C₂³	1	2	yes	—
61	C₂×#1689	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	—	2	1	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	4	2	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	2	1	no	—
72	C₂×D₅₂	—	2	4	no	—
73	C₂×D₅₂	—	2	1	no	—
74	C₂×D₅₂	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₁₃	—	2	2	no	—
76	C₂×C₄×D₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	—	4	2	no	—
78	C₂×C₄×D₁₃	—	4	2	no	—
79	C₁₃×Q₁₆	—	4	8	no	—
80	C₁₃×Q₁₆	—	4	8	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	D₈	1	4	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	#41	1	1	yes	characteristic
92	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
93	#6103	—	2	2	no	—
94	#6103	—	2	2	no	—
95	C₄⋊Dic₁₃	—	2	1	no	—
96	C₄⋊Dic₁₃	C₂×D₄	1	2	yes	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	4	no	—
101	#6094	—	4	8	no	—
102	C₂×#494	D₁₃	1	1	yes	characteristic, complemented
103	C₂×#474	—	13	2	no	—
104	C₂×#436	—	13	1	no	—
105	#1689	—	13	8	no	—
106	C₂×#334	—	13	1	no	—
107	C₂×#332	—	13	2	no	—
108	C₂³×C₁₃	#121	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	2	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	—	4	2	no	—
112	C₂²×D₁₃	—	4	2	no	—
113	Q₈×C₁₃	—	4	4	no	—
114	Q₈×C₁₃	—	4	4	no	—
115	Q₈×C₁₃	—	8	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	8	4	no	—
119	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₃	#137	1	4	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	—	4	2	no	—
130	D₅₂	—	2	4	no	—
131	D₅₂	—	4	2	no	—
132	C₄×D₁₃	—	4	2	no	—
133	C₄×D₁₃	—	4	2	no	—
134	C₄×D₁₃	—	4	4	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	C₂×D₈	1	4	yes	—
137	#1055	—	4	2	no	—
138	#1055	—	4	4	no	—
139	C₂²×Q₁₆	—	2	1	no	—
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×M₁₆	—	26	1	no	—
142	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
143	C₂×C₄×D₄	—	13	1	no	—
144	#494	—	2	2	no	—
145	#494	C₂×D₁₃	1	4	yes	complemented
146	#474	—	13	8	no	—
147	#474	—	26	4	no	—
148	#436	—	13	4	no	—
149	#436	—	26	2	no	—
150	C₂×#107	—	26	2	no	—
151	C₂×C₄⋊C₈	—	13	1	no	—
152	C₂×C₄⋊C₈	—	13	2	no	—
153	C₂×#102	—	26	2	no	—
154	C₂×C₂²⋊C₈	—	26	1	no	—
155	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	#334	—	13	4	no	—
157	#332	—	13	8	no	—
158	C₂²×C₁₃	#459	1	2	yes	—
159	C₂²×C₁₃	#470	1	2	yes	—
160	C₂²×C₁₃	#473	1	2	yes	—
161	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
162	C₂×D₁₃	—	4	2	no	—
163	C₂×D₁₃	—	4	2	no	—
164	C₂×D₁₃	—	4	2	no	—
165	C₂×D₁₃	—	8	4	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	8	4	no	—
169	C₄×C₁₃	#459	1	2	yes	—
170	C₄×C₁₃	#470	1	2	yes	—
171	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#137	1	2	yes	—
174	Dic₁₃	—	4	2	no	—
175	C₂²×Q₈	—	2	2	no	—
176	C₂²×D₄	—	13	1	no	—
177	C₂³×C₄	—	26	1	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	4	4	no	—
181	C₂×Q₁₆	—	4	4	no	—
182	C₂×M₁₆	—	26	4	no	—
183	C₂×M₁₆	—	52	4	no	—
184	C₂²×C₈	—	26	2	no	—
185	C₂²×C₈	—	26	1	no	—
186	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
192	C₄×D₄	—	13	2	no	—
193	C₄×D₄	—	13	2	no	—
194	C₄×D₄	—	26	1	no	—
195	C₄×D₄	—	26	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	13	1	no	—
199	C₂×C₂²⋊C₄	—	26	1	no	—
200	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
201	#107	—	26	8	no	—
202	C₄⋊C₈	—	13	2	no	—
203	C₄⋊C₈	—	13	4	no	—
204	C₄⋊C₈	—	26	2	no	—
205	C₄⋊C₈	—	26	1	no	—
206	#102	—	26	8	no	—
207	C₂²⋊C₈	—	26	4	no	—
208	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
209	C₄×C₈	C₂²×D₁₃	1	2	yes	—
210	C₂×C₁₃	#1689	1	4	yes	complemented
211	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#470	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
214	D₁₃	—	8	2	no	—
215	C₂⁴	—	26	1	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×D₄	—	13	2	no	—
221	C₂×D₄	—	26	4	no	—
222	C₂×D₄	—	26	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	26	1	no	—
225	C₂²×C₄	—	26	2	no	—
226	C₂²×C₄	—	26	2	no	—
227	C₂²×C₄	—	52	2	no	—
228	C₂²×C₄	—	52	2	no	—
229	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	#6129	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	Q₁₆	—	4	8	no	—
233	Q₁₆	—	4	8	no	—
234	M₁₆	—	52	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	2	no	—
238	C₂×C₈	—	52	4	no	—
239	C₂×C₈	—	52	2	no	—
240	C₂×C₈	#6104	1	4	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	13	2	no	—
248	C₄⋊C₄	—	26	1	no	—
249	C₂²⋊C₄	—	26	2	no	—
250	C₂²⋊C₄	—	26	2	no	—
251	C₄²	D₄×D₁₃	1	1	yes	characteristic
252	C₄²	#6129	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
254	C₄²	C₂³×D₁₃	1	1	yes	characteristic
255	C₁₃	C₂×#1689	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	26	2	no	—
257	C₂³	—	52	2	no	—
258	C₂³	—	52	2	no	—
259	C₂³	—	52	2	no	—
260	C₂³	#88192	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	26	4	no	—
265	D₄	—	52	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	52	4	no	—
274	C₂×C₄	#88172	1	4	yes	—
275	C₂×C₄	C₂×#6104	1	2	yes	—
276	C₂×C₄	#88186	1	2	yes	—
277	C₂×C₄	#88192	1	1	yes	characteristic
278	C₂×C₄	#88192	1	1	yes	characteristic
279	C₂×C₄	#88192	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#6129	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	52	4	no	—
285	C₈	—	52	2	no	—
286	C₈	C₂×#6104	1	4	yes	—
287	C₂²	—	52	2	no	—
288	C₂²	—	52	2	no	—
289	C₂²	—	52	2	no	—
290	C₂²	—	104	4	no	—
291	C₂²	#11718733	1	2	yes	—
292	C₂²	#11718850	1	2	yes	—
293	C₂²	#11718853	1	2	yes	—
294	C₂²	C₂×#88192	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	4	no	—
298	C₄	—	52	2	no	—
299	C₄	#11718733	1	2	yes	—
300	C₄	#11718850	1	2	yes	—
301	C₄	C₂×#88172	1	2	yes	—
302	C₄	C₂²×#6104	1	1	yes	characteristic
303	C₄	C₂×#88186	1	1	yes	characteristic
304	C₂	—	104	2	no	—
305	C₂	#13114322	1	4	yes	direct product factor
306	C₂	C₂×#11718733	1	1	yes	characteristic
307	C₂	C₂×#11718850	1	1	yes	characteristic
308	C₂	C₂×#11718853	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂²×D₅₂	D₄
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	C₂×#1689
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13114322	C₂	direct (N × H)
4. C₂×#11718049	C₂	semidirect
5. C₂×#11718139	C₂	semidirect
6. C₂×C₁₃×#494	C₂	semidirect
13. C₁₃×#494	C₂²	semidirect
24. #11718139	C₂²	semidirect
25. #11718049	C₂²	semidirect
102. C₂×#494	D₁₃	semidirect
145. #494	C₂×D₁₃	semidirect
210. C₂×C₁₃	#1689	semidirect
255. C₁₃	C₂×#1689	semidirect
305. C₂	#13114322	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×#11718748	2
C₂×#11718349	2
#13114322	2
C₂×#11718049	2
C₂×#11718139	2
C₂×C₁₃×#494	2
C₂×#1689	13

Subgroups — C2×#13114322 #15845423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114322 #15845423