Subgroups — C₂×#13117005 #16017647

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718516	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718857	C₂	1	1	yes	maximal, characteristic
3	#13117005	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718105	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718079	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718158	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×#88202	—	2	1	no	—
11	C₂×#88142	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
14	C₂×#88077	—	2	1	no	—
15	#11718857	—	2	2	no	—
16	#11718857	C₂²	1	4	yes	—
17	C₂×#88059	C₂²	1	1	yes	characteristic
18	C₂×#88057	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88052	C₂²	1	1	yes	characteristic
21	C₂×#88048	—	2	1	no	—
22	#11718516	—	2	2	no	—
23	#11718516	C₂²	1	4	yes	complemented
24	C₁₃×#360	C₂²	1	4	yes	—
25	#11718158	C₂²	1	4	yes	—
26	#11718105	C₂²	1	4	yes	complemented
27	#11718103	C₂²	1	4	yes	complemented
28	#11718079	C₂²	1	4	yes	complemented
29	C₂×C₁₃×#41	—	2	1	no	—
30	C₂²×Dic₂₆	—	4	1	no	—
31	C₁₃×#135	—	2	1	no	—
32	C₁₃×#135	—	2	1	no	—
33	C₁₃×#135	Q₈	1	2	yes	complemented
34	C₁₃×#135	C₂³	1	2	yes	—
35	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
36	C₂×C₁₃×SD₁₆	—	2	1	no	—
37	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	#88202	—	2	4	no	—
41	#88202	—	4	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×#6101	—	4	1	no	—
44	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
45	C₂×#6099	—	2	1	no	—
46	C₂×#6099	—	4	1	no	—
47	C₂²×#1055	—	4	1	no	—
48	C₂²×#1055	D₄	1	1	yes	characteristic
49	#88142	—	2	4	no	—
50	#88142	—	4	2	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₁₃×M₃₂	C₂³	1	2	yes	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×#106	—	2	1	no	—
55	C₁₃×#106	C₂³	1	2	yes	—
56	#88077	—	2	4	no	—
57	#88059	—	2	1	no	—
58	#88059	C₂³	1	2	yes	—
59	#88057	—	2	1	no	—
60	#88057	C₂³	1	2	yes	—
61	C₈×Dic₁₃	D₄	1	2	yes	—
62	C₈×Dic₁₃	C₂³	1	2	yes	—
63	#88052	—	2	1	no	—
64	#88052	C₂³	1	2	yes	—
65	C₂×#6090	—	2	1	no	—
66	#88048	—	2	4	no	—
67	C₂×#2230	—	13	1	no	maximal, Sylow
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	2	1	no	—
73	C₂×D₄×C₁₃	—	2	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	2	1	no	—
79	C₂²×Dic₁₃	—	4	1	no	—
80	C₂×Dic₂₆	—	4	4	no	—
81	C₂×Dic₂₆	—	8	4	no	—
82	C₁₃×Q₁₆	—	2	1	no	—
83	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
84	C₁₃×SD₁₆	—	2	2	no	—
85	C₁₃×SD₁₆	—	2	2	no	—
86	C₁₃×D₈	—	2	1	no	—
87	C₁₃×D₈	C₂×Q₈	1	2	yes	complemented
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₄⋊Dic₁₃	—	2	2	no	—
97	C₄⋊Dic₁₃	—	4	1	no	—
98	#6101	—	4	4	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
101	#6099	—	2	2	no	—
102	#6099	—	4	1	no	—
103	#6099	—	4	4	no	—
104	C₂×#1055	—	4	1	no	—
105	C₂×#1055	—	4	4	no	—
106	C₂×#1055	SD₁₆	1	1	yes	characteristic
107	C₂×#1055	SD₁₆	1	1	yes	characteristic
108	C₂×#1055	SD₁₆	1	2	yes	—
109	C₂×#1055	C₂×D₄	1	2	yes	—
110	C₁₃×C₁₆	—	2	2	no	—
111	#6090	—	2	2	no	—
112	C₂×#498	—	13	1	no	—
113	C₂×#443	—	13	1	no	—
114	#2230	—	13	8	no	—
115	C₂×#365	—	13	1	no	—
116	C₂×#365	—	13	1	no	—
117	C₂×#364	—	13	1	no	—
118	C₂×#360	—	13	1	no	—
119	C₂×#360	D₁₃	1	1	yes	characteristic
120	C₂³×C₁₃	—	4	1	no	—
121	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	2	no	—
123	Q₈×C₁₃	—	4	1	no	—
124	D₄×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	4	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	8	2	no	—
143	Dic₂₆	—	8	8	no	—
144	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	#1055	—	4	2	no	—
149	#1055	C₂×SD₁₆	1	2	yes	—
150	#1055	C₂×SD₁₆	1	2	yes	—
151	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#131	—	26	1	no	—
153	C₂×C₄⋊Q₈	—	26	1	no	—
154	C₂×M₃₂	—	13	1	no	—
155	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
156	#498	—	13	4	no	—
157	#498	—	26	2	no	—
158	#443	—	13	4	no	—
159	#443	—	26	2	no	—
160	C₂×#108	—	13	1	no	—
161	C₂×#106	—	13	1	no	—
162	C₂×#106	—	26	1	no	—
163	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#104	—	26	1	no	—
165	C₂×C₄×C₈	—	13	1	no	—
166	#365	—	13	4	no	—
167	#365	—	13	4	no	—
168	#364	—	13	4	no	—
169	#360	—	13	4	no	—
170	#360	C₂×D₁₃	1	4	yes	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	4	2	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	#476	1	1	yes	characteristic
175	C₂²×C₁₃	#476	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	#476	1	1	yes	characteristic
180	C₄×C₁₃	#476	1	1	yes	characteristic
181	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
182	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
183	Dic₁₃	—	2	4	no	—
184	Dic₁₃	—	8	2	no	—
185	C₂×#41	—	2	1	no	—
186	C₂²×Q₈	—	52	1	no	—
187	#135	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	Dic₂₆	1	2	yes	complemented
190	#135	C₂²×D₁₃	1	2	yes	—
191	C₂×Q₁₆	Dic₂₆	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	2	1	no	—
193	C₂×D₈	Dic₂₆	1	1	yes	characteristic, complemented
194	#131	—	26	4	no	—
195	#131	—	52	2	no	—
196	C₂×M₁₆	—	26	1	no	—
197	C₂×M₁₆	—	52	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	52	1	no	—
200	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	C₄⋊Q₈	—	26	4	no	—
202	C₄⋊Q₈	—	52	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	26	1	no	—
205	C₂×C₄⋊C₄	—	52	1	no	—
206	C₂×C₄²	—	13	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	—	13	2	no	—
209	M₃₂	—	26	1	no	—
210	M₃₂	C₂²×D₁₃	1	2	yes	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	26	1	no	—
213	#108	—	13	2	no	—
214	#108	—	26	1	no	—
215	#106	—	2	1	no	—
216	#106	—	13	2	no	—
217	#106	—	26	4	no	—
218	#106	—	26	1	no	—
219	#106	C₂²×D₁₃	1	2	yes	—
220	#104	—	26	4	no	—
221	C₄×C₈	—	13	2	no	—
222	C₄×C₈	—	13	2	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	8	2	no	—
226	C₂×C₁₃	#2230	1	2	yes	complemented
227	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	52	4	no	—
234	C₂×Q₈	—	104	4	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	26	1	no	—
240	C₂²×C₄	—	52	1	no	—
241	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
242	Q₁₆	—	2	1	no	—
243	Q₁₆	C₂×Dic₂₆	1	2	yes	complemented
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	2	2	no	—
246	D₈	—	2	1	no	—
247	D₈	C₂×Dic₂₆	1	2	yes	complemented
248	M₁₆	—	26	2	no	—
249	M₁₆	—	52	4	no	—
250	M₁₆	—	52	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	52	4	no	—
256	C₂×C₈	—	52	1	no	—
257	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
259	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
260	C₂×C₈	#6129	1	1	yes	characteristic
261	C₂×C₈	#6129	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	26	2	no	—
266	C₄⋊C₄	—	52	4	no	—
267	C₄⋊C₄	—	52	1	no	—
268	C₄²	—	13	2	no	—
269	C₄²	—	26	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	26	2	no	—
272	C₁₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	4	1	no	—
274	C₂³	#88165	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	104	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	26	4	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	52	2	no	—
290	C₂×C₄	—	104	2	no	—
291	C₂×C₄	#88132	1	1	yes	characteristic
292	C₂×C₄	#88132	1	1	yes	characteristic
293	C₂×C₄	#88165	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
297	C₈	—	13	2	no	—
298	C₈	—	13	2	no	—
299	C₈	—	52	2	no	—
300	C₈	#88132	1	1	yes	characteristic
301	C₈	#88132	1	1	yes	characteristic
302	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
303	C₈	C₂×#6129	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#11718422	1	1	yes	characteristic
308	C₂²	#11718422	1	1	yes	characteristic
309	C₂²	C₂×#88165	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	26	4	no	—
313	C₄	—	104	2	no	—
314	C₄	#11718422	1	1	yes	characteristic
315	C₄	#11718422	1	1	yes	characteristic
316	C₄	C₂×#88132	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#13117005	1	2	yes	direct product factor
322	C₂	C₂×#11718422	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718422
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#360	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂×C₁₃×#135	C₂²
hypercenter	C₂×#360	D₁₃
hypocenter	C₁₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#11718516	C₂	semidirect
3. #13117005	C₂	direct (N × H)
4. C₂×#11718105	C₂	semidirect
5. C₂×#11718079	C₂	semidirect
6. C₂×#11718103	C₂	semidirect
23. #11718516	C₂²	semidirect
26. #11718105	C₂²	semidirect
27. #11718103	C₂²	semidirect
28. #11718079	C₂²	semidirect
33. C₁₃×#135	Q₈	semidirect
35. C₂×C₁₃×Q₁₆	Q₈	semidirect
37. C₂×C₁₃×D₈	Q₈	semidirect
83. C₁₃×Q₁₆	C₂×Q₈	semidirect
87. C₁₃×D₈	C₂×Q₈	semidirect
189. #135	Dic₂₆	semidirect
191. C₂×Q₁₆	Dic₂₆	semidirect
193. C₂×D₈	Dic₂₆	semidirect
226. C₂×C₁₃	#2230	semidirect
243. Q₁₆	C₂×Dic₂₆	semidirect
247. D₈	C₂×Dic₂₆	semidirect
272. C₁₃	C₂×#2230	semidirect
321. C₂	#13117005	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×#11718516	2
C₂×#11718857	2
#13117005	2
C₂×#11718105	2
C₂×#11718079	2
C₂×#11718103	2
C₂×#11718158	2
C₂×C₁₃×#360	2
C₂×#2230	13

Subgroups — C2×#13117005 #16017647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117005 #16017647