Subgroups — C₂×#13114715 #15860925

Order 3328.

1102 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×#88059	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718762	C₂	1	2	yes	maximal
4	#13114715	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718115	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	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	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
14	C₂×#88059	C₄	1	8	yes	complemented
15	C₂×#88059	C₂²	1	4	yes	—
16	#11718767	C₄	1	4	yes	complemented
17	#11718767	C₂²	1	4	yes	—
18	C₂×#88058	C₄	1	2	yes	complemented
19	C₂×#88057	C₄	1	2	yes	complemented
20	#11718762	—	2	4	no	—
21	#11718762	C₂²	1	8	yes	—
22	C₂×#88055	—	2	2	no	—
23	C₂×C₈×Dic₁₃	—	2	2	no	—
24	#11718115	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	D₄	1	2	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	1	no	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
44	#88180	—	2	2	no	—
45	#88180	—	2	2	no	—
46	#88180	C₂×C₄	1	4	yes	—
47	#88180	C₂³	1	4	yes	—
48	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	4	no	—
52	C₂×#6099	C₂×C₄	1	8	yes	—
53	C₂×#6099	C₂³	1	4	yes	—
54	C₂²×#1055	—	2	2	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
56	#88059	C₂×C₄	1	16	yes	complemented
57	#88058	C₂×C₄	1	8	yes	complemented
58	#88057	C₂×C₄	1	8	yes	complemented
59	#88055	—	2	8	no	—
60	C₈×Dic₁₃	—	2	8	no	—
61	C₂×#1772	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	#41	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	C₄²	1	4	yes	—
78	C₂×C₈×C₁₃	C₄²	1	4	yes	—
79	C₂×C₈×C₁₃	C₄²	1	8	yes	—
80	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
94	C₂²⋊Dic₁₃	—	2	4	no	—
95	C₂²⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
98	C₄×Dic₁₃	—	2	4	no	—
99	C₄×Dic₁₃	—	2	4	no	—
100	#6099	—	2	8	no	—
101	#6099	—	2	4	no	—
102	#6099	C₂²×C₄	1	8	yes	—
103	C₂×#1055	—	2	8	no	—
104	C₂×#1055	—	2	4	no	—
105	C₂²×#108	—	13	1	no	—
106	C₂×#427	—	13	1	no	—
107	C₂×#405	—	13	2	no	—
108	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
109	#1772	—	13	8	no	—
110	C₂×#339	—	13	2	no	—
111	C₂³×C₁₃	—	2	4	no	—
112	C₂³×C₁₃	#117	1	2	yes	—
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	16	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	#117	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	2	8	no	—
139	C₂×Dic₁₃	—	4	8	no	—
140	C₈×C₁₃	C₂×C₄²	1	4	yes	—
141	C₈×C₁₃	C₂×C₄²	1	4	yes	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
144	#1055	—	2	8	no	—
145	C₂²×M₁₆	—	13	2	no	—
146	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂×#117	—	13	2	no	—
149	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
150	C₂×#108	—	13	8	no	—
151	C₂×#108	—	13	4	no	—
152	#427	—	13	4	no	—
153	#427	—	13	4	no	—
154	C₂×#107	—	13	2	no	—
155	C₂×#106	—	13	2	no	—
156	#405	—	13	8	no	—
157	#405	—	26	4	no	—
158	C₂×#97	—	26	2	no	—
159	C₂×C₄×C₈	—	26	2	no	—
160	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
161	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
162	#339	—	13	8	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	4	no	—
166	C₂²×C₁₃	—	2	4	no	—
167	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
168	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
169	C₂²×C₁₃	#443	1	4	yes	—
170	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₄×C₁₃	—	2	8	no	—
174	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
175	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
176	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
177	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
180	Dic₁₃	—	4	8	no	—
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×M₁₆	—	13	8	no	—
184	C₂×M₁₆	—	13	4	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	4	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	Dic₂₆	1	4	yes	—
189	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₃	1	2	yes	—
192	C₂²×C₈	C₄×D₁₃	1	4	yes	—
193	C₂²×C₈	D₅₂	1	1	yes	characteristic
194	C₂²×C₈	D₅₂	1	2	yes	—
195	C₂²×C₈	D₅₂	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
197	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
198	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
199	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
200	#117	—	13	4	no	—
201	#117	—	13	4	no	—
202	#117	—	26	2	no	—
203	#117	—	26	2	no	—
204	C₂×C₄⋊C₄	—	13	4	no	—
205	C₂×C₂²⋊C₄	—	26	2	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
210	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
211	#108	—	13	16	no	—
212	#107	—	13	8	no	—
213	#106	—	13	8	no	—
214	#97	—	26	8	no	—
215	C₄×C₈	—	26	8	no	—
216	C₄×C₈	C₂×Dic₁₃	1	16	yes	complemented
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	#1772	1	4	yes	complemented
221	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#443	1	2	yes	—
223	C₂⁴	#6127	1	1	yes	characteristic
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	26	4	no	—
228	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
232	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
234	C₂²×C₄	#6127	1	1	yes	characteristic
235	C₂²×C₄	#6127	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
237	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
238	M₁₆	—	13	8	no	—
239	M₁₆	—	26	8	no	—
240	M₁₆	—	26	4	no	—
241	C₂×C₈	—	26	8	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
244	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
245	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
246	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
247	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
248	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
249	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
250	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
253	C₂×C₈	C₂×D₅₂	1	2	yes	—
254	C₂×C₈	C₂×D₅₂	1	2	yes	—
255	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
256	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
257	C₄⋊C₄	—	13	8	no	—
258	C₄⋊C₄	—	26	4	no	—
259	C₂²⋊C₄	—	26	4	no	—
260	C₂²⋊C₄	—	26	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	26	4	no	—
264	C₄²	—	26	4	no	—
265	C₄²	C₂²×Dic₁₃	1	4	yes	—
266	C₁₃	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	2	4	no	—
268	C₂³	C₄×Dic₂₆	1	2	yes	—
269	C₂³	C₄×D₅₂	1	1	yes	characteristic
270	C₂³	C₄×D₅₂	1	1	yes	characteristic
271	C₂³	#88180	1	2	yes	—
272	C₂³	C₂×#6127	1	1	yes	characteristic
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	26	8	no	—
277	C₂×C₄	—	52	8	no	—
278	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
279	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
280	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
281	C₂×C₄	C₄×D₅₂	1	2	yes	—
282	C₂×C₄	C₄×D₅₂	1	2	yes	—
283	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
284	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
289	C₂×C₄	#88180	1	2	yes	—
290	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₂×#6127	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
296	C₈	—	26	8	no	—
297	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
298	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
299	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
300	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	#11718361	1	4	yes	—
306	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
307	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
308	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
309	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
310	C₂²	C₂×#88180	1	1	yes	characteristic
311	C₄	—	2	8	no	—
312	C₄	—	52	8	no	—
313	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
314	C₄	C₄×C₄⋊Dic₁₃	1	2	yes	—
315	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
316	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
317	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
318	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	#13114715	1	4	yes	direct product factor
323	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
324	C₂	C₂×#11718361	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#1772
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #13114715	C₂	direct (N × H)
12. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×#88059	C₄	semidirect
16. #11718767	C₄	semidirect
18. C₂×#88058	C₄	semidirect
19. C₂×#88057	C₄	semidirect
55. C₄×C₈×C₁₃	C₂×C₄	semidirect
56. #88059	C₂×C₄	semidirect
57. #88058	C₂×C₄	semidirect
58. #88057	C₂×C₄	semidirect
160. C₂×C₄×C₈	Dic₁₃	semidirect
216. C₄×C₈	C₂×Dic₁₃	semidirect
220. C₂×C₁₃	#1772	semidirect
266. C₁₃	C₂×#1772	semidirect
322. C₂	#13114715	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×#88059	2
C₂×#11718767	2
C₂×#11718762	2
#13114715	2
C₂×#11718115	2
C₂²×C₄×C₈×C₁₃	2
C₂×#1772	13

Subgroups — C2×#13114715 #15860925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114715 #15860925