Subgroups — C₂×#13129901 #16789116

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88103	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88101	C₂	1	1	yes	maximal, characteristic
3	#13129901	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	—	2	1	no	—
6	C₂²×#6118	—	2	1	no	—
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88103	—	2	1	no	—
9	C₂×#88103	—	2	2	no	—
10	C₂×#88103	C₄	1	4	yes	complemented
11	C₂×#88103	C₂²	1	2	yes	—
12	C₂×#88101	—	2	1	no	—
13	C₂×#88101	—	2	2	no	—
14	C₂×#88101	C₄	1	4	yes	—
15	C₂×#88101	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
18	C₂×#88038	—	2	2	no	—
19	C₂×#88038	—	2	2	no	—
20	C₂×#88038	C₄	1	2	yes	complemented
21	C₂×#88038	C₂²	1	2	yes	—
22	C₂³×#261	—	4	1	no	—
23	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
24	C₂²×#1057	—	4	1	no	—
25	C₂×C₄⋊#261	—	2	4	no	—
26	C₂×C₄⋊#261	—	2	2	no	—
27	C₂×C₄⋊#261	—	4	2	no	—
28	C₂×C₄⋊#261	—	4	1	no	—
29	C₂×#6118	—	2	4	no	—
30	C₂×#6118	—	2	2	no	—
31	C₂×#6118	—	4	2	no	—
32	C₂×#6118	—	4	1	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	2	yes	—
41	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
43	#88103	—	2	4	no	—
44	#88103	—	2	2	no	—
45	#88103	C₂×C₄	1	4	yes	complemented
46	#88101	—	2	4	no	—
47	#88101	—	2	2	no	—
48	#88101	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
51	C₂×#6090	—	2	2	no	—
52	C₂×#6090	C₂×C₄	1	2	yes	complemented
53	#88038	—	2	4	no	—
54	#88038	—	2	2	no	—
55	#88038	—	2	2	no	—
56	C₂×#1383	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	SD₁₆	1	1	yes	characteristic
58	C₂²×#261	—	4	4	no	—
59	C₂²×#261	—	4	2	no	—
60	C₂²×#261	—	8	2	no	—
61	C₂²×#261	—	8	2	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	SD₁₆	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	D₈	1	2	yes	—
68	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
69	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
70	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×#1057	—	4	4	no	—
73	C₂×#1057	—	4	2	no	—
74	C₄⋊#261	—	2	4	no	—
75	C₄⋊#261	—	4	4	no	—
76	C₄⋊#261	—	4	2	no	—
77	#6118	—	2	4	no	—
78	#6118	—	4	4	no	—
79	#6118	—	4	2	no	—
80	C₂×C₈×C₁₃	C₄²	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
85	C₂×#1055	C₄⋊C₄	1	2	yes	—
86	C₂×#1055	C₄⋊C₄	1	2	yes	—
87	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
88	C₈×D₁₃	C₄²	1	2	yes	—
89	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
90	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
91	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
92	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
94	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
97	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₁₃×C₁₆	—	2	2	no	—
99	#6090	—	2	2	no	—
100	C₂²×M₃₂	—	13	1	no	—
101	C₂²×#108	—	13	1	no	—
102	C₂²×#106	—	13	1	no	—
103	#1383	—	13	4	no	—
104	C₂³×C₁₃	#106	1	1	yes	characteristic
105	C₂²×D₁₃	—	2	2	no	—
106	C₂²×D₁₃	—	2	2	no	—
107	C₂²×D₁₃	#102	1	2	yes	—
108	C₂²×D₁₃	#103	1	2	yes	—
109	C₂²×D₁₃	#106	1	1	yes	characteristic
110	C₂²×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×#261	—	4	4	no	—
112	C₂×#261	—	8	8	no	—
113	C₂×#261	—	8	2	no	—
114	C₂×#261	—	8	2	no	—
115	C₂×C₄×C₁₃	#95	1	2	yes	—
116	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#107	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	#102	1	2	yes	—
120	C₂×Dic₁₃	#103	1	2	yes	—
121	C₂×Dic₁₃	#106	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₄×D₁₃	#95	1	2	yes	—
124	C₄×D₁₃	#102	1	2	yes	—
125	C₄×D₁₃	#102	1	2	yes	—
126	C₄×D₁₃	#103	1	2	yes	—
127	C₄×D₁₃	#103	1	2	yes	—
128	C₄×D₁₃	#107	1	2	yes	—
129	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
133	#1057	—	4	4	no	—
134	C₈×C₁₃	#95	1	2	yes	—
135	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	#1055	#95	1	2	yes	—
138	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
139	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×M₁₆	—	26	1	no	—
141	C₂³×C₈	—	13	1	no	—
142	C₂²×C₄⋊C₄	—	26	1	no	—
143	C₂×M₃₂	—	13	2	no	—
144	C₂×M₃₂	—	13	2	no	—
145	C₂×M₃₂	—	26	2	no	—
146	C₂×M₃₂	—	26	2	no	—
147	C₂²×C₁₆	—	13	1	no	—
148	C₂²×C₁₆	#261	1	1	yes	characteristic, complemented
149	C₂×#108	—	13	2	no	—
150	C₂×#108	—	13	4	no	—
151	C₂×#108	—	26	1	no	—
152	C₂×#108	—	26	2	no	—
153	C₂×#106	—	13	2	no	—
154	C₂×#106	—	13	4	no	—
155	C₂×#106	—	26	1	no	—
156	C₂×#106	—	26	2	no	—
157	C₂²×C₁₃	#340	1	2	yes	—
158	C₂²×C₁₃	#365	1	2	yes	—
159	C₂²×C₁₃	#365	1	2	yes	—
160	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
161	C₂×D₁₃	—	2	4	no	—
162	C₂×D₁₃	—	2	4	no	—
163	C₂×D₁₃	—	2	1	no	—
164	C₂×D₁₃	—	2	1	no	—
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	#340	1	2	yes	—
168	C₂×D₁₃	C₂×#102	1	1	yes	characteristic
169	C₂×D₁₃	C₂×#103	1	1	yes	characteristic
170	#261	—	8	4	no	—
171	C₄×C₁₃	#340	1	2	yes	—
172	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
174	Dic₁₃	#340	1	2	yes	—
175	Dic₁₃	C₂×#102	1	1	yes	characteristic
176	Dic₁₃	C₂×#103	1	1	yes	characteristic
177	C₂³×C₄	—	13	1	no	—
178	C₂³×C₄	—	52	1	no	—
179	C₂×M₁₆	—	26	4	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	52	1	no	—
182	C₂×M₁₆	—	52	2	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	52	1	no	—
193	C₂²×C₈	C₂×#261	1	1	yes	characteristic
194	C₂×C₄⋊C₄	—	26	4	no	—
195	C₂×C₄⋊C₄	—	26	2	no	—
196	C₂×C₄⋊C₄	—	52	1	no	—
197	C₂×C₄⋊C₄	—	52	2	no	—
198	M₃₂	—	26	2	no	—
199	M₃₂	—	26	2	no	—
200	M₃₂	—	26	4	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	13	2	no	—
203	C₂×C₁₆	—	26	2	no	—
204	C₂×C₁₆	C₂×#261	1	2	yes	complemented
205	#108	—	13	4	no	—
206	#108	—	26	4	no	—
207	#108	—	26	2	no	—
208	#106	—	13	4	no	—
209	#106	—	26	4	no	—
210	#106	—	26	2	no	—
211	C₂×C₁₃	#1383	1	4	yes	complemented
212	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#365	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#365	1	1	yes	characteristic
215	D₁₃	—	2	1	no	—
216	D₁₃	—	2	2	no	—
217	D₁₃	—	2	1	no	—
218	C₂⁴	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	52	4	no	—
229	C₂²×C₄	—	52	2	no	—
230	C₂²×C₄	—	104	2	no	—
231	C₂²×C₄	—	104	2	no	—
232	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
233	M₁₆	—	26	4	no	—
234	M₁₆	—	52	4	no	—
235	M₁₆	—	52	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	2	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	13	2	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	52	4	no	—
251	C₂×C₈	—	52	2	no	—
252	C₂×C₈	C₄×#261	1	2	yes	—
253	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊#261	1	2	yes	—
255	C₂×C₈	C₂²×#261	1	1	yes	characteristic
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	52	4	no	—
258	C₄⋊C₄	—	52	2	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	26	2	no	—
261	C₁₃	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	13	2	no	—
263	C₂³	—	13	2	no	—
264	C₂³	—	13	1	no	—
265	C₂³	—	13	1	no	—
266	C₂³	—	26	2	no	—
267	C₂³	—	26	2	no	—
268	C₂³	#88101	1	1	yes	characteristic
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	1	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	2	no	—
276	C₂×C₄	—	13	2	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	1	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	—	104	2	no	—
285	C₂×C₄	—	104	8	no	—
286	C₂×C₄	—	104	2	no	—
287	C₂×C₄	#88101	1	1	yes	characteristic
288	C₂×C₄	#88102	1	2	yes	—
289	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
291	C₈	—	13	2	no	—
292	C₈	—	13	1	no	—
293	C₈	—	13	1	no	—
294	C₈	—	52	4	no	—
295	C₈	(C₂×#261)⋊C₄	1	2	yes	—
296	C₈	C₂×C₄×#261	1	1	yes	characteristic
297	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
298	C₂²	—	13	2	no	—
299	C₂²	—	13	1	no	—
300	C₂²	—	13	1	no	—
301	C₂²	—	26	1	no	—
302	C₂²	—	26	1	no	—
303	C₂²	—	26	4	no	—
304	C₂²	—	26	4	no	—
305	C₂²	—	26	1	no	—
306	C₂²	—	26	1	no	—
307	C₂²	#11718219	1	2	yes	—
308	C₂²	#11718220	1	2	yes	—
309	C₂²	#11718265	1	2	yes	—
310	C₂²	C₂×#88101	1	1	yes	characteristic
311	C₄	—	13	2	no	—
312	C₄	—	13	1	no	—
313	C₄	—	13	1	no	—
314	C₄	—	104	4	no	—
315	C₄	#11718265	1	2	yes	—
316	C₄	C₂×#88102	1	1	yes	characteristic
317	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
318	C₂	—	26	1	no	—
319	C₂	—	26	2	no	—
320	C₂	—	26	1	no	—
321	C₂	#13129901	1	4	yes	direct product factor
322	C₂	C₂×#11718265	1	1	yes	characteristic
323	C₂	C₂×#11718219	1	1	yes	characteristic
324	C₂	C₂×#11718220	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88101
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂³×C₁₃	#106
prime radical	C₂³×D₁₃	SD₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	C₂×#1383
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #13129901	C₂	direct (N × H)
10. C₂×#88103	C₄	semidirect
16. C₂²×C₁₃×C₁₆	C₄	semidirect
17. C₂²×#6090	C₄	semidirect
20. C₂×#88038	C₄	semidirect
45. #88103	C₂×C₄	semidirect
50. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
52. C₂×#6090	C₂×C₄	semidirect
148. C₂²×C₁₆	#261	semidirect
204. C₂×C₁₆	C₂×#261	semidirect
211. C₂×C₁₃	#1383	semidirect
261. C₁₃	C₂×#1383	semidirect
321. C₂	#13129901	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88103	2
C₂²×#88101	2
#13129901	2
C₂²×#88038	2
C₂×#1383	13

Subgroups — C2×#13129901 #16789116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129901 #16789116