Subgroups — C₂×#13130766 #16798196

Order 3328.

730 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130766	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11718266	C₂	1	2	yes	maximal
3	C₂×#11718234	C₂	1	2	yes	maximal
4	C₂×#11718269	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718267	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#6094	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	C₂²	1	2	yes	—
8	C₂×D₁₃⋊M₁₆	C₂²	1	2	yes	—
9	C₂²×#6094	C₂²	1	1	yes	characteristic
10	C₂²×#6094	C₂²	1	1	yes	characteristic
11	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
13	C₂×#88055	C₄	1	1	yes	characteristic, complemented
14	C₂×#88055	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
16	C₄×#6094	C₄	1	4	yes	complemented
17	C₄×#6094	C₂²	1	4	yes	—
18	#11718269	C₂²	1	4	yes	—
19	#11718267	C₂²	1	4	yes	—
20	#11718266	C₂²	1	8	yes	—
21	#11718234	C₂²	1	8	yes	—
22	C₂²×C₄×D₁₃	—	2	1	no	—
23	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊#261	—	2	2	no	—
25	C₂×#6122	—	2	2	no	—
26	#88237	—	2	2	no	—
27	#88237	—	2	2	no	—
28	#88237	—	2	2	no	—
29	#88237	C₂³	1	4	yes	—
30	C₂×C₄⋊#261	—	2	2	no	—
31	C₂×C₄×#261	—	2	2	no	—
32	D₁₃⋊M₁₆	—	2	2	no	—
33	D₁₃⋊M₁₆	—	2	2	no	—
34	D₁₃⋊M₁₆	—	2	2	no	—
35	D₁₃⋊M₁₆	C₂³	1	4	yes	—
36	C₂×#6118	—	2	2	no	—
37	C₂×D₁₃⋊C₈	—	2	2	no	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	—	2	1	no	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
44	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
45	C₂×#6094	—	2	2	no	—
46	C₂×#6094	—	2	2	no	—
47	C₂×#6094	—	2	2	no	—
48	C₂×#6094	—	2	2	no	—
49	C₂×#6094	C₂×C₄	1	2	yes	—
50	C₂×#6094	C₂×C₄	1	2	yes	—
51	C₂×#6094	C₂³	1	2	yes	—
52	C₂×#6094	C₂³	1	2	yes	—
53	C₄²×D₁₃	—	2	1	no	—
54	C₄²×D₁₃	—	2	1	no	—
55	C₄²×D₁₃	C₂×C₄	1	2	yes	—
56	C₄²×D₁₃	C₂³	1	2	yes	—
57	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
58	#88055	C₂×C₄	1	4	yes	complemented
59	#88055	C₂×C₄	1	4	yes	complemented
60	C₄×#1055	C₂×C₄	1	4	yes	complemented
61	C₂×#1841	—	13	1	no	maximal, Sylow
62	C₂³×D₁₃	#41	1	1	yes	characteristic
63	C₂²×#261	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂²×Dic₁₃	#41	1	1	yes	characteristic
68	C₂×C₄×D₁₃	—	2	2	no	—
69	C₂×C₄×D₁₃	—	2	2	no	—
70	C₂×C₄×D₁₃	—	4	4	no	—
71	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃	#41	1	2	yes	—
73	C₂×C₄×D₁₃	#41	1	2	yes	—
74	C₂×C₄×D₁₃	#41	1	2	yes	—
75	C₂×C₄×D₁₃	#41	1	2	yes	—
76	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
77	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊#261	—	2	4	no	—
80	C₂²⋊#261	—	4	2	no	—
81	#6122	—	2	4	no	—
82	#6122	—	4	2	no	—
83	C₂×#1057	—	4	2	no	—
84	C₄⋊#261	—	2	4	no	—
85	C₄⋊#261	—	4	2	no	—
86	C₄×#261	—	2	4	no	—
87	C₄×#261	—	4	2	no	—
88	#6118	—	2	4	no	—
89	#6118	—	4	2	no	—
90	D₁₃⋊C₈	—	2	4	no	—
91	D₁₃⋊C₈	—	4	2	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₂×C₈×C₁₃	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	1	no	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	—	2	1	no	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
104	C₂×#1055	—	2	1	no	—
105	C₂×#1055	—	2	1	no	—
106	C₂×#1055	—	2	1	no	—
107	C₂×#1055	—	2	1	no	—
108	C₂×#1055	C₂²×C₄	1	2	yes	—
109	C₂×#1055	C₂²×C₄	1	2	yes	—
110	#6094	—	2	4	no	—
111	#6094	—	2	4	no	—
112	#6094	—	2	4	no	—
113	#6094	—	2	4	no	—
114	C₂×C₄×M₁₆	—	13	1	no	—
115	#1841	—	13	8	no	—
116	C₂×#344	—	13	2	no	—
117	C₂×#339	—	13	2	no	—
118	C₂×#338	—	13	1	no	—
119	C₂×H(3,ℤ/4)	—	13	1	no	—
120	C₂³×C₁₃	#117	1	1	yes	characteristic
121	C₂²×D₁₃	—	4	2	no	—
122	C₂²×D₁₃	#117	1	1	yes	characteristic
123	C₂²×D₁₃	#126	1	2	yes	—
124	C₂²×D₁₃	#126	1	2	yes	—
125	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×#261	—	4	4	no	—
127	C₂×#261	—	8	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	4	2	no	—
138	C₂×Dic₁₃	#117	1	1	yes	characteristic
139	C₂×Dic₁₃	#126	1	2	yes	—
140	C₂×Dic₁₃	#126	1	2	yes	—
141	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
142	C₄×D₁₃	—	4	4	no	—
143	C₄×D₁₃	—	4	4	no	—
144	C₄×D₁₃	#117	1	2	yes	—
145	C₄×D₁₃	#117	1	2	yes	—
146	C₄×D₁₃	#126	1	2	yes	—
147	C₄×D₁₃	#126	1	2	yes	—
148	C₄×D₁₃	#126	1	2	yes	—
149	C₄×D₁₃	#126	1	2	yes	—
150	C₄×D₁₃	C₂×#41	1	2	yes	—
151	C₄×D₁₃	C₂×#41	1	2	yes	—
152	#1057	—	4	4	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	2	no	—
155	#1055	—	2	2	no	—
156	#1055	—	2	2	no	—
157	C₂²×M₁₆	—	13	1	no	—
158	C₂²×M₁₆	—	13	1	no	—
159	C₂²×M₁₆	—	13	2	no	—
160	C₂×#117	—	13	2	no	—
161	C₂²×C₄²	—	13	1	no	—
162	C₄×M₁₆	—	13	4	no	—
163	C₄×M₁₆	—	13	4	no	—
164	C₂×#97	—	13	1	no	—
165	C₂×#97	—	13	1	no	—
166	C₂×C₄×C₈	—	13	1	no	—
167	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
168	#344	—	13	8	no	—
169	#339	—	13	8	no	—
170	#338	—	13	4	no	—
171	H(3,ℤ/4)	—	13	4	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	#383	1	1	yes	characteristic
174	C₂²×C₁₃	#383	1	1	yes	characteristic
175	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
176	C₂×D₁₃	—	2	4	no	—
177	C₂×D₁₃	—	4	1	no	—
178	C₂×D₁₃	—	4	1	no	—
179	C₂×D₁₃	—	4	2	no	—
180	C₂×D₁₃	#383	1	2	yes	—
181	C₂×D₁₃	C₂×#126	1	2	yes	—
182	#261	—	8	4	no	—
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	#383	1	1	yes	characteristic
185	C₄×C₁₃	#383	1	1	yes	characteristic
186	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
188	Dic₁₃	—	4	2	no	—
189	Dic₁₃	#383	1	2	yes	—
190	Dic₁₃	C₂×#126	1	2	yes	—
191	C₂³×C₄	—	13	1	no	—
192	C₂³×C₄	—	26	1	no	—
193	C₂×M₁₆	—	13	2	no	—
194	C₂×M₁₆	—	13	2	no	—
195	C₂×M₁₆	—	13	2	no	—
196	C₂×M₁₆	—	13	2	no	—
197	C₂×M₁₆	—	13	4	no	—
198	C₂×M₁₆	—	26	2	no	—
199	C₂×M₁₆	—	26	2	no	—
200	C₂×M₁₆	—	26	2	no	—
201	C₂×M₁₆	—	26	2	no	—
202	C₂×M₁₆	—	26	2	no	—
203	C₂×M₁₆	—	26	2	no	—
204	C₂×M₁₆	—	26	2	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂×M₁₆	—	26	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	26	2	no	—
210	C₂²×C₈	C₂×#261	1	1	yes	characteristic
211	C₂²×C₈	C₂×#261	1	1	yes	characteristic
212	#117	—	13	4	no	—
213	#117	—	26	2	no	—
214	#117	—	26	2	no	—
215	#117	—	26	2	no	—
216	C₂×C₄⋊C₄	—	26	2	no	—
217	C₂×C₂²⋊C₄	—	26	2	no	—
218	C₂×C₄²	—	13	1	no	—
219	C₂×C₄²	—	13	2	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	26	1	no	—
222	C₂×C₄²	—	26	1	no	—
223	C₂×C₄²	—	26	1	no	—
224	C₂×C₄²	—	26	2	no	—
225	C₂×C₄²	C₂×#261	1	1	yes	characteristic
226	#97	—	13	4	no	—
227	#97	—	13	4	no	—
228	C₄×C₈	—	13	4	no	—
229	C₄×C₈	C₂×#261	1	4	yes	complemented
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	#1841	1	2	yes	complemented
233	C₂×C₁₃	C₂×#383	1	1	yes	characteristic
234	D₁₃	—	2	2	no	—
235	D₁₃	—	2	2	no	—
236	C₂⁴	—	13	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	13	2	no	—
245	C₂²×C₄	—	13	1	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	26	2	no	—
248	C₂²×C₄	—	26	2	no	—
249	C₂²×C₄	—	26	1	no	—
250	C₂²×C₄	—	52	4	no	—
251	C₂²×C₄	—	52	2	no	—
252	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
253	M₁₆	—	26	4	no	—
254	M₁₆	—	26	4	no	—
255	M₁₆	—	26	4	no	—
256	M₁₆	—	26	4	no	—
257	M₁₆	—	26	4	no	—
258	M₁₆	—	26	4	no	—
259	M₁₆	—	52	2	no	—
260	M₁₆	—	52	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	26	1	no	—
268	C₂×C₈	—	26	1	no	—
269	C₂×C₈	—	26	1	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	—	26	4	no	—
272	C₂×C₈	—	52	2	no	—
273	C₂×C₈	—	52	2	no	—
274	C₂×C₈	C₂²×#261	1	2	yes	—
275	C₂×C₈	C₂²×#261	1	2	yes	—
276	C₄⋊C₄	—	26	4	no	—
277	C₄⋊C₄	—	52	2	no	—
278	C₂²⋊C₄	—	26	4	no	—
279	C₂²⋊C₄	—	52	2	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	26	2	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	4	no	—
285	C₄²	—	26	1	no	—
286	C₄²	—	52	2	no	—
287	C₄²	C₂²×#261	1	2	yes	—
288	C₁₃	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	13	2	no	—
290	C₂³	—	13	2	no	—
291	C₂³	—	13	1	no	—
292	C₂³	—	13	1	no	—
293	C₂³	—	52	2	no	—
294	C₂³	#88237	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	2	no	—
305	C₂×C₄	—	13	2	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	26	2	no	—
310	C₂×C₄	—	52	4	no	—
311	C₂×C₄	—	52	4	no	—
312	C₂×C₄	—	52	2	no	—
313	C₂×C₄	—	52	4	no	—
314	C₂×C₄	—	104	4	no	—
315	C₂×C₄	#88237	1	1	yes	characteristic
316	C₂×C₄	#88237	1	1	yes	characteristic
317	C₂×C₄	#88237	1	1	yes	characteristic
318	C₂×C₄	#88237	1	1	yes	characteristic
319	C₂×C₄	#88237	1	1	yes	characteristic
320	C₂×C₄	C₂³×#261	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	—	26	2	no	—
324	C₈	—	26	2	no	—
325	C₈	—	52	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	13	2	no	—
328	C₂²	—	13	2	no	—
329	C₂²	—	26	4	no	—
330	C₂²	—	52	1	no	—
331	C₂²	—	52	1	no	—
332	C₂²	—	52	2	no	—
333	C₂²	#11719061	1	1	yes	characteristic
334	C₂²	#11719061	1	1	yes	characteristic
335	C₂²	C₂×#88237	1	1	yes	characteristic
336	C₄	—	2	4	no	—
337	C₄	—	13	2	no	—
338	C₄	—	13	2	no	—
339	C₄	—	52	2	no	—
340	C₄	—	104	4	no	—
341	C₄	#11719061	1	1	yes	characteristic
342	C₄	#11719061	1	1	yes	characteristic
343	C₄	C₂×#88237	1	1	yes	characteristic
344	C₄	C₂×#88237	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	26	2	no	—
348	C₂	—	26	2	no	—
349	C₂	#13130766	1	2	yes	direct product factor
350	C₂	C₂×#11719061	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#383
prime radical	C₂³×D₁₃	#41
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1841
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #13130766	C₂	direct (N × H)
12. C₂×C₄×C₈×C₁₃	C₄	semidirect
13. C₂×#88055	C₄	semidirect
14. C₂×#88055	C₄	semidirect
15. C₂×C₄×#1055	C₄	semidirect
16. C₄×#6094	C₄	semidirect
57. C₄×C₈×C₁₃	C₂×C₄	semidirect
58. #88055	C₂×C₄	semidirect
59. #88055	C₂×C₄	semidirect
60. C₄×#1055	C₂×C₄	semidirect
167. C₂×C₄×C₈	#261	semidirect
229. C₄×C₈	C₂×#261	semidirect
232. C₂×C₁₃	#1841	semidirect
288. C₁₃	C₂×#1841	semidirect
349. C₂	#13130766	direct (N × H)

Maximal subgroups

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

subgroup	index
#13130766	2
C₂×#11718266	2
C₂×#11718234	2
C₂×#11718269	2
C₂×#11718267	2
C₂×C₄×#6094	2
C₂×#1841	13

Subgroups — C2×#13130766 #16798196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130766 #16798196