Subgroups — C₂×#13114723 #15860934

Order 3328.

1062 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718783	C₂	1	2	yes	maximal
3	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
4	#13114723	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718147	C₂	1	2	yes	maximal
6	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₄	1	8	yes	complemented
13	C₂×C₁₃×#108	C₂²	1	4	yes	—
14	#11718783	—	2	4	no	—
15	#11718783	C₂²	1	8	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	C₂×#88055	—	2	2	no	—
21	C₂×#88055	C₄	1	8	yes	complemented
22	C₂×#88055	C₂²	1	4	yes	—
23	C₂×C₈×Dic₁₃	—	2	2	no	—
24	#11718147	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	C₂×C₄	1	4	yes	—
42	#88180	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
49	C₂²×#1055	C₂×C₄	1	4	yes	—
50	C₂²×#1055	D₄	1	2	yes	—
51	C₂²×#1055	D₄	1	2	yes	—
52	C₂²×#1055	Q₈	1	4	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#108	C₂×C₄	1	16	yes	complemented
55	#88058	C₂×C₄	1	8	yes	complemented
56	#88057	C₂×C₄	1	8	yes	complemented
57	#88055	—	2	8	no	—
58	#88055	C₂×C₄	1	16	yes	complemented
59	C₈×Dic₁₃	—	2	8	no	—
60	C₂×#1773	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₂²×Dic₁₃	—	2	2	no	—
75	C₂²×Dic₁₃	—	4	4	no	—
76	C₁₃×M₁₆	—	2	8	no	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₁₃	—	2	8	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	C₄²	1	4	yes	—
82	C₂×C₈×C₁₃	C₄²	1	4	yes	—
83	C₂×C₈×C₁₃	C₄²	1	8	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊Dic₁₃	—	2	4	no	—
89	C₂²⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
92	C₄×Dic₁₃	—	2	4	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	—	2	2	no	—
96	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
97	C₂×#1055	C₄⋊C₄	1	8	yes	—
98	C₂×#1055	C₄⋊C₄	1	8	yes	—
99	C₂×#1055	C₂²×C₄	1	4	yes	—
100	C₂×#1055	C₂×D₄	1	4	yes	—
101	C₂×#1055	C₂×Q₈	1	4	yes	—
102	C₂²×#108	D₁₃	1	1	yes	characteristic
103	C₂×#427	—	13	1	no	—
104	C₂×#405	—	13	2	no	—
105	C₂²×#97	—	13	1	no	—
106	#1773	—	13	8	no	—
107	C₂×#339	—	13	2	no	—
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	#117	1	2	yes	—
110	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
113	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	#117	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₁₃	—	2	4	no	—
133	C₂×Dic₁₃	—	2	8	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	8	no	—
136	C₂×Dic₁₃	—	4	8	no	—
137	C₈×C₁₃	—	2	8	no	—
138	C₈×C₁₃	C₂×C₄²	1	4	yes	—
139	C₈×C₁₃	C₂×C₄²	1	4	yes	—
140	#1055	C₂×C₄⋊C₄	1	8	yes	—
141	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
142	C₂³×C₈	—	13	1	no	—
143	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
144	C₂×#117	—	13	2	no	—
145	C₂²×C₄²	—	13	1	no	—
146	C₂×#108	Dic₁₃	1	8	yes	complemented
147	C₂×#108	C₂×D₁₃	1	4	yes	—
148	#427	—	13	4	no	—
149	#427	—	13	4	no	—
150	C₂×#107	—	13	2	no	—
151	C₂×#106	—	13	2	no	—
152	#405	—	13	8	no	—
153	#405	—	26	4	no	—
154	C₂×#97	—	13	8	no	—
155	C₂×#97	—	13	4	no	—
156	C₂×#97	—	26	2	no	—
157	C₂×C₄×C₈	—	26	2	no	—
158	#339	—	13	8	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	4	no	—
163	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
164	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
165	C₂²×C₁₃	#444	1	4	yes	—
166	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
170	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
171	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
175	Dic₁₃	—	4	4	no	—
176	Dic₁₃	—	4	8	no	—
177	C₂³×C₄	—	26	1	no	—
178	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
182	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	13	4	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	C₄×D₁₃	1	4	yes	—
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₁₃	1	2	yes	—
193	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
194	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
195	#117	—	13	4	no	—
196	#117	—	13	4	no	—
197	#117	—	26	2	no	—
198	#117	—	26	2	no	—
199	C₂×C₄⋊C₄	—	13	4	no	—
200	C₂×C₂²⋊C₄	—	26	2	no	—
201	C₂×C₄²	—	13	4	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	26	2	no	—
204	C₂×C₄²	—	26	2	no	—
205	C₂×C₄²	—	26	1	no	—
206	#108	C₂×Dic₁₃	1	16	yes	complemented
207	#107	—	13	8	no	—
208	#106	—	13	8	no	—
209	#97	—	13	16	no	—
210	#97	—	26	8	no	—
211	C₄×C₈	—	26	8	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	#1773	1	4	yes	complemented
216	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#444	1	2	yes	—
218	C₂⁴	#6131	1	1	yes	characteristic
219	C₂²×C₄	—	26	4	no	—
220	C₂²×C₄	—	26	4	no	—
221	C₂²×C₄	—	26	2	no	—
222	C₂²×C₄	—	52	4	no	—
223	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
226	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
228	C₂²×C₄	#6129	1	2	yes	—
229	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
230	C₂²×C₄	#6131	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	C₂²×Dic₁₃	1	8	yes	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	13	8	no	—
239	C₂×C₈	—	13	8	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	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	C₂×C₄×D₁₃	1	4	yes	—
247	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
249	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
250	C₄⋊C₄	—	13	8	no	—
251	C₄⋊C₄	—	26	4	no	—
252	C₂²⋊C₄	—	26	4	no	—
253	C₂²⋊C₄	—	26	4	no	—
254	C₄²	—	13	4	no	—
255	C₄²	—	26	4	no	—
256	C₄²	—	26	4	no	—
257	C₄²	—	26	4	no	—
258	C₄²	—	26	2	no	—
259	C₁₃	C₂×#1773	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	2	4	no	—
261	C₂³	#88146	1	2	yes	—
262	C₂³	#88147	1	1	yes	characteristic
263	C₂³	#88147	1	1	yes	characteristic
264	C₂³	Q₈×Dic₁₃	1	2	yes	—
265	C₂³	C₂×#6131	1	1	yes	characteristic
266	C₂×C₄	—	26	4	no	—
267	C₂×C₄	—	26	8	no	—
268	C₂×C₄	—	52	8	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	52	8	no	—
271	C₂×C₄	#88141	1	2	yes	—
272	C₂×C₄	#88141	1	2	yes	—
273	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
275	C₂×C₄	#88146	1	2	yes	—
276	C₂×C₄	#88147	1	1	yes	characteristic
277	C₂×C₄	#88147	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
280	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
281	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
282	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
283	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#6129	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
289	C₈	—	2	8	no	—
290	C₈	—	13	8	no	—
291	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
292	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	#11718555	1	4	yes	—
298	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
299	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
300	C₂²	C₂×#88146	1	1	yes	characteristic
301	C₂²	C₂×#88147	1	1	yes	characteristic
302	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
303	C₄	—	52	4	no	—
304	C₄	—	52	8	no	—
305	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
306	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
307	C₄	C₂×#88141	1	1	yes	characteristic
308	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
309	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
310	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	2	no	—
314	C₂	#13114723	1	4	yes	direct product factor
315	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
316	C₂	C₂×#11718555	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#1773
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #13114723	C₂	direct (N × H)
12. C₂×C₁₃×#108	C₄	semidirect
16. #11718767	C₄	semidirect
18. C₂×#88058	C₄	semidirect
19. C₂×#88057	C₄	semidirect
21. C₂×#88055	C₄	semidirect
54. C₁₃×#108	C₂×C₄	semidirect
55. #88058	C₂×C₄	semidirect
56. #88057	C₂×C₄	semidirect
58. #88055	C₂×C₄	semidirect
146. C₂×#108	Dic₁₃	semidirect
206. #108	C₂×Dic₁₃	semidirect
215. C₂×C₁₃	#1773	semidirect
259. C₁₃	C₂×#1773	semidirect
314. C₂	#13114723	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×#11718767	2
C₂×#11718783	2
C₂²×#88055	2
#13114723	2
C₂×#11718147	2
C₂²×C₁₃×#108	2
C₂×#1773	13

Subgroups — C2×#13114723 #15860934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114723 #15860934