Subgroups — C₂×#13114985 #15863184

Order 3328.

730 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	#13114985	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11718148	C₂	1	2	yes	maximal
3	C₂×#11718115	C₂	1	2	yes	maximal
4	C₂×#11718152	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₄×C₁₃×M₁₆	C₄	1	4	yes	complemented
13	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
16	#11718152	C₂²	1	4	yes	—
17	#11718149	C₂²	1	4	yes	—
18	#11718148	C₂²	1	8	yes	—
19	#11718115	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
29	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	—	2	1	no	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	C₂³	1	4	yes	—
52	C₂²×#1055	—	2	2	no	—
53	C₁₃×#97	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
55	C₂×#1841	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	4	2	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂²⋊Dic₁₃	—	2	4	no	—
90	C₂²⋊Dic₁₃	—	4	2	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	4	2	no	—
95	#6099	—	2	4	no	—
96	#6099	—	2	4	no	—
97	#6099	—	4	2	no	—
98	#6099	—	4	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
103	#1841	—	13	8	no	—
104	C₂×#344	—	13	2	no	—
105	C₂×#339	—	13	2	no	—
106	C₂×#338	—	13	1	no	—
107	C₂×H(3,ℤ/4)	—	13	1	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	#117	1	1	yes	characteristic
110	C₂³×C₁₃	#117	1	1	yes	characteristic
111	C₂³×C₁₃	#126	1	2	yes	—
112	C₂³×C₁₃	#126	1	2	yes	—
113	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#117	1	2	yes	—
124	C₂×C₄×C₁₃	#117	1	2	yes	—
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#126	1	2	yes	—
129	C₂×C₄×C₁₃	#126	1	2	yes	—
130	C₂×C₄×C₁₃	#126	1	2	yes	—
131	C₂×C₄×C₁₃	#126	1	2	yes	—
132	C₂×C₄×C₁₃	#126	1	2	yes	—
133	C₂×C₄×C₁₃	#126	1	2	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	8	4	no	—
140	C₈×C₁₃	—	2	4	no	—
141	C₈×C₁₃	—	2	4	no	—
142	#1055	—	4	4	no	—
143	C₂²×M₁₆	—	13	2	no	—
144	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
145	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
146	C₂×#117	—	13	2	no	—
147	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
148	C₄×M₁₆	Dic₁₃	1	4	yes	complemented
149	C₄×M₁₆	C₂×D₁₃	1	4	yes	—
150	C₂×#97	Dic₁₃	1	2	yes	complemented
151	C₂×C₄×C₈	Dic₁₃	1	2	yes	complemented
152	#344	—	13	8	no	—
153	#339	—	13	8	no	—
154	#338	—	13	4	no	—
155	H(3,ℤ/4)	—	13	4	no	—
156	C₂²×C₁₃	—	2	4	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	4	1	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	#383	1	1	yes	characteristic
162	C₂²×C₁₃	#383	1	2	yes	—
163	C₂²×C₁₃	#383	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#126	1	2	yes	—
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	—	4	2	no	—
168	C₄×C₁₃	#383	1	1	yes	characteristic
169	C₄×C₁₃	#383	1	2	yes	—
170	C₄×C₁₃	#383	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#126	1	2	yes	—
174	Dic₁₃	—	8	4	no	—
175	C₂³×C₄	—	2	1	no	—
176	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	13	4	no	—
182	C₂×M₁₆	—	26	2	no	—
183	C₂×M₁₆	—	26	2	no	—
184	C₂×M₁₆	—	26	2	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
188	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
189	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
190	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
191	C₂²×C₈	—	26	2	no	—
192	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
193	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
194	#117	—	13	4	no	—
195	#117	—	26	2	no	—
196	#117	—	26	2	no	—
197	#117	—	26	2	no	—
198	C₂×C₄⋊C₄	—	26	2	no	—
199	C₂×C₂²⋊C₄	—	26	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	26	2	no	—
204	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
208	#97	C₂×Dic₁₃	1	8	yes	complemented
209	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	#1841	1	2	yes	complemented
215	C₂×C₁₃	C₂×#383	1	1	yes	characteristic
216	C₂⁴	#6127	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	#6127	1	1	yes	characteristic
224	C₂²×C₄	#6127	1	1	yes	characteristic
225	C₂²×C₄	#6127	1	1	yes	characteristic
226	C₂²×C₄	#6127	1	2	yes	—
227	C₂²×C₄	#6127	1	2	yes	—
228	C₂²×C₄	#6127	1	2	yes	—
229	C₂²×C₄	#6127	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	26	4	no	—
238	M₁₆	—	26	4	no	—
239	M₁₆	—	52	2	no	—
240	M₁₆	—	52	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	52	2	no	—
247	C₂×C₈	—	52	2	no	—
248	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
249	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
250	C₄⋊C₄	—	26	4	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₂²⋊C₄	—	26	4	no	—
253	C₂²⋊C₄	—	52	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	26	4	no	—
259	C₄²	—	52	2	no	—
260	C₄²	C₂²×Dic₁₃	1	2	yes	—
261	C₄²	C₂²×Dic₁₃	1	2	yes	—
262	C₁₃	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	4	2	no	—
264	C₂³	#88130	1	2	yes	—
265	C₂³	#88130	1	2	yes	—
266	C₂³	#88180	1	1	yes	characteristic
267	C₂³	#88180	1	1	yes	characteristic
268	C₂³	C₂×#6127	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	52	4	no	—
276	C₂×C₄	—	104	4	no	—
277	C₂×C₄	#88130	1	2	yes	—
278	C₂×C₄	#88130	1	2	yes	—
279	C₂×C₄	#88130	1	2	yes	—
280	C₂×C₄	#88130	1	2	yes	—
281	C₂×C₄	#88130	1	2	yes	—
282	C₂×C₄	#88130	1	2	yes	—
283	C₂×C₄	#88180	1	1	yes	characteristic
284	C₂×C₄	#88180	1	1	yes	characteristic
285	C₂×C₄	#88180	1	1	yes	characteristic
286	C₂×C₄	#88180	1	2	yes	—
287	C₂×C₄	#88180	1	2	yes	—
288	C₂×C₄	#88180	1	1	yes	characteristic
289	C₂×C₄	#88180	1	1	yes	characteristic
290	C₂×C₄	#88180	1	1	yes	characteristic
291	C₂×C₄	C₂×#6127	1	1	yes	characteristic
292	C₂×C₄	C₂×#6127	1	2	yes	—
293	C₂×C₄	C₂×#6127	1	2	yes	—
294	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	52	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#11718598	1	2	yes	—
304	C₂²	#11718598	1	1	yes	characteristic
305	C₂²	#11718598	1	1	yes	characteristic
306	C₂²	C₂×#88130	1	2	yes	—
307	C₂²	C₂×#88180	1	1	yes	characteristic
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	104	4	no	—
311	C₄	#11718598	1	2	yes	—
312	C₄	#11718598	1	1	yes	characteristic
313	C₄	#11718598	1	1	yes	characteristic
314	C₄	C₂×#88130	1	2	yes	—
315	C₄	C₂×#88180	1	1	yes	characteristic
316	C₄	C₂×#88180	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	#13114985	1	2	yes	direct product factor
322	C₂	C₂×#11718598	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	#383
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#1841
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13114985	C₂	direct (N × H)
12. C₄×C₁₃×M₁₆	C₄	semidirect
14. C₂×C₁₃×#97	C₄	semidirect
15. C₂×C₄×C₈×C₁₃	C₄	semidirect
53. C₁₃×#97	C₂×C₄	semidirect
54. C₄×C₈×C₁₃	C₂×C₄	semidirect
148. C₄×M₁₆	Dic₁₃	semidirect
150. C₂×#97	Dic₁₃	semidirect
151. C₂×C₄×C₈	Dic₁₃	semidirect
208. #97	C₂×Dic₁₃	semidirect
209. C₄×C₈	C₂×Dic₁₃	semidirect
214. C₂×C₁₃	#1841	semidirect
262. C₁₃	C₂×#1841	semidirect
321. C₂	#13114985	direct (N × H)

Maximal subgroups

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

subgroup	index
#13114985	2
C₂×#11718148	2
C₂×#11718115	2
C₂×#11718152	2
C₂×#11718149	2
C₂×C₄×C₁₃×M₁₆	2
C₂×#1841	13

Subgroups — C2×#13114985 #15863184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114985 #15863184