Subgroups — C₂×#13116440 #16015576

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
3	#13116440	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718104	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88180	C₄	1	1	yes	characteristic
8	C₂²×#6099	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₁₃×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₃×C₁₆	C₄	1	2	yes	complemented
14	#11718762	C₄	1	4	yes	—
15	#11718762	C₂²	1	4	yes	—
16	C₂×#88055	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
18	C₂×#88052	C₄	1	2	yes	complemented
19	C₂×#88052	C₄	1	4	yes	complemented
20	C₂×#88052	C₂²	1	2	yes	—
21	C₂×#88052	C₂²	1	4	yes	—
22	C₂²×#6090	C₄	1	2	yes	complemented
23	#11718104	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
36	#88180	—	2	2	no	—
37	#88180	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₂×#6099	—	2	1	no	—
41	C₂×#6099	—	2	1	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	C₂×C₄	1	4	yes	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
46	C₁₃×M₃₂	C₂×C₄	1	8	yes	complemented
47	C₁₃×M₃₂	C₂³	1	4	yes	—
48	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	complemented
50	#88055	C₂×C₄	1	4	yes	—
51	#88055	C₂³	1	4	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
53	#88052	C₂×C₄	1	4	yes	—
54	#88052	C₂×C₄	1	8	yes	complemented
55	#88052	C₂³	1	4	yes	—
56	C₂×#6090	C₂×C₄	1	4	yes	—
57	C₂×#6090	C₂×C₄	1	8	yes	complemented
58	C₂×#2110	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₄²	1	2	yes	—
73	C₂×C₈×C₁₃	C₄²	1	2	yes	—
74	C₂×C₈×C₁₃	C₄²	1	2	yes	—
75	C₂×C₈×C₁₃	C₄²	1	2	yes	—
76	C₂×C₈×C₁₃	C₄²	1	2	yes	—
77	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄×Dic₁₃	—	2	2	no	—
92	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
93	#6099	—	2	4	no	—
94	#6099	—	2	4	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	C₂²×C₄	1	4	yes	—
98	C₁₃×C₁₆	C₂²×C₄	1	8	yes	—
99	#6090	C₂²×C₄	1	8	yes	—
100	C₂²×M₃₂	—	13	1	no	—
101	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
102	C₂×#405	—	13	1	no	—
103	#2110	—	13	8	no	—
104	C₂×#347	—	13	4	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	#97	1	1	yes	characteristic
107	C₂³×C₁₃	#97	1	1	yes	characteristic
108	C₂³×C₁₃	#97	1	2	yes	—
109	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	#97	1	2	yes	—
112	C₂×C₄×C₁₃	#97	1	2	yes	—
113	C₂×C₄×C₁₃	#97	1	2	yes	—
114	C₂×C₄×C₁₃	#97	1	2	yes	—
115	C₂×C₄×C₁₃	#97	1	2	yes	—
116	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₈×C₁₃	C₂×C₄²	1	2	yes	—
132	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₁₃	C₂×C₄²	1	2	yes	—
135	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	#1055	—	2	4	no	—
138	C₂²×M₁₆	—	13	1	no	—
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#117	—	13	1	no	—
141	C₂×M₃₂	—	13	2	no	—
142	C₂×M₃₂	—	13	4	no	—
143	C₂×M₃₂	—	13	2	no	—
144	C₂×M₃₂	—	13	4	no	—
145	C₂×M₃₂	Dic₁₃	1	2	yes	complemented
146	C₂×M₃₂	Dic₁₃	1	4	yes	complemented
147	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
148	C₂×M₃₂	C₂×D₁₃	1	4	yes	—
149	C₂²×C₁₆	—	13	2	no	—
150	C₂²×C₁₆	Dic₁₃	1	2	yes	complemented
151	#405	—	13	4	no	—
152	#405	—	13	4	no	—
153	C₂×#97	—	13	2	no	—
154	C₂×C₄×C₈	—	13	2	no	—
155	#347	—	13	16	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#97	1	2	yes	—
163	C₂²×C₁₃	C₂×#97	1	2	yes	—
164	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
165	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#97	1	2	yes	—
169	C₄×C₁₃	C₂×#97	1	2	yes	—
170	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
171	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
172	Dic₁₃	—	4	4	no	—
173	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
174	C₂×M₁₆	—	13	4	no	—
175	C₂×M₁₆	—	26	1	no	—
176	C₂×M₁₆	—	26	1	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	C₄×D₁₃	1	2	yes	—
180	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₁₃	1	2	yes	—
183	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
186	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
189	#117	—	13	4	no	—
190	#117	—	26	2	no	—
191	C₂×C₄⋊C₄	—	26	1	no	—
192	C₂×C₂²⋊C₄	—	26	1	no	—
193	C₂×C₄²	—	13	2	no	—
194	M₃₂	—	13	4	no	—
195	M₃₂	—	13	8	no	—
196	M₃₂	—	13	4	no	—
197	M₃₂	C₂×Dic₁₃	1	4	yes	—
198	M₃₂	C₂×Dic₁₃	1	8	yes	complemented
199	M₃₂	C₂²×D₁₃	1	4	yes	—
200	C₂×C₁₆	—	13	4	no	—
201	C₂×C₁₆	—	13	8	no	—
202	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
203	C₂×C₁₆	C₂×Dic₁₃	1	8	yes	complemented
204	#97	—	13	4	no	—
205	#97	—	13	4	no	—
206	C₄×C₈	—	13	8	no	—
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	2	no	—
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	#2110	1	2	yes	complemented
212	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
213	C₂⁴	#6094	1	1	yes	characteristic
214	C₂²×C₄	—	26	2	no	—
215	C₂²×C₄	#6094	1	1	yes	characteristic
216	C₂²×C₄	#6094	1	1	yes	characteristic
217	C₂²×C₄	#6094	1	2	yes	—
218	C₂²×C₄	#6094	1	1	yes	characteristic
219	C₂²×C₄	#6094	1	2	yes	—
220	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
221	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
225	M₁₆	—	26	4	no	—
226	M₁₆	—	26	4	no	—
227	C₂×C₈	—	13	4	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
231	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
232	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
234	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
235	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
236	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
243	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
245	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
247	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
248	C₄⋊C₄	—	26	4	no	—
249	C₂²⋊C₄	—	26	4	no	—
250	C₄²	—	13	4	no	—
251	C₄²	—	26	2	no	—
252	C₁₆	—	13	8	no	—
253	C₁₆	C₂²×Dic₁₃	1	8	yes	—
254	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	#88055	1	1	yes	characteristic
257	C₂³	#88055	1	1	yes	characteristic
258	C₂³	#88055	1	2	yes	—
259	C₂³	C₂×#6094	1	1	yes	characteristic
260	C₂³	C₂×#6094	1	2	yes	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	#88055	1	2	yes	—
264	C₂×C₄	#88055	1	2	yes	—
265	C₂×C₄	#88055	1	2	yes	—
266	C₂×C₄	#88055	1	2	yes	—
267	C₂×C₄	#88055	1	2	yes	—
268	C₂×C₄	#88055	1	1	yes	characteristic
269	C₂×C₄	#88055	1	1	yes	characteristic
270	C₂×C₄	C₂×#6094	1	1	yes	characteristic
271	C₂×C₄	C₂×#6094	1	1	yes	characteristic
272	C₂×C₄	C₂×#6094	1	2	yes	—
273	C₂×C₄	C₂×#6094	1	2	yes	—
274	C₂×C₄	C₂×#6094	1	1	yes	characteristic
275	C₂×C₄	C₂×#6094	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
278	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
281	C₈	—	26	4	no	—
282	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
283	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
284	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
285	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
286	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
287	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#88055	1	1	yes	characteristic
294	C₂²	C₂×#88055	1	2	yes	—
295	C₂²	C₂×#88055	1	2	yes	—
296	C₂²	C₂×#88055	1	1	yes	characteristic
297	C₂²	C₂²×#6094	1	1	yes	characteristic
298	C₄	—	52	4	no	—
299	C₄	C₂×#88055	1	1	yes	characteristic
300	C₄	C₂×#88055	1	1	yes	characteristic
301	C₄	C₂×#88055	1	2	yes	—
302	C₄	C₂×#88055	1	2	yes	—
303	C₄	C₂²×#6094	1	1	yes	characteristic
304	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#13116440	1	2	yes	direct product factor
310	C₂	C₂²×#88055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	C₂×#2110
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13116440	C₂	direct (N × H)
4. C₂×#11718104	C₂	semidirect
9. C₂×C₁₃×M₃₂	C₄	semidirect
10. C₂×C₁₃×M₃₂	C₄	semidirect
13. C₂²×C₁₃×C₁₆	C₄	semidirect
18. C₂×#88052	C₄	semidirect
19. C₂×#88052	C₄	semidirect
22. C₂²×#6090	C₄	semidirect
23. #11718104	C₂²	semidirect
46. C₁₃×M₃₂	C₂×C₄	semidirect
49. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
54. #88052	C₂×C₄	semidirect
57. C₂×#6090	C₂×C₄	semidirect
145. C₂×M₃₂	Dic₁₃	semidirect
146. C₂×M₃₂	Dic₁₃	semidirect
150. C₂²×C₁₆	Dic₁₃	semidirect
198. M₃₂	C₂×Dic₁₃	semidirect
203. C₂×C₁₆	C₂×Dic₁₃	semidirect
211. C₂×C₁₃	#2110	semidirect
254. C₁₃	C₂×#2110	semidirect
309. C₂	#13116440	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×#88052	2
C₂×#11718762	2
#13116440	2
C₂×#11718104	2
C₂²×C₁₃×M₃₂	2
C₂×#2110	13

Subgroups — C2×#13116440 #16015576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116440 #16015576