Subgroups — C₂×#13114563 #15860340

Order 3328.

2110 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6099	C₂	1	2	yes	maximal
2	#13114563	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718144	C₂	1	4	yes	maximal, complemented
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×M₁₆	—	2	8	no	—
7	C₂²×C₁₃×M₁₆	C₄	1	4	yes	—
8	C₂²×C₁₃×M₁₆	C₄	1	4	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	4	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×#6099	—	2	2	no	—
12	C₂²×#6099	—	2	4	no	—
13	C₂²×#6099	—	2	8	no	—
14	C₂²×#6099	C₄	1	16	yes	—
15	C₂²×#6099	C₂²	1	4	yes	—
16	C₂²×#6099	C₂²	1	8	yes	—
17	C₂³×#1055	—	2	2	no	—
18	#11718144	C₂²	1	16	yes	complemented
19	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃	D₄	1	2	yes	—
25	C₂³×C₄×C₁₃	D₄	1	8	yes	—
26	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×M₁₆	—	2	16	no	—
31	C₂×C₁₃×M₁₆	—	2	16	no	—
32	C₂×C₁₃×M₁₆	—	2	16	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	16	yes	—
34	C₂²×C₈×C₁₃	—	2	8	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₂×#6099	—	2	16	no	—
39	C₂×#6099	—	2	32	no	—
40	C₂×#6099	—	2	8	no	—
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	—	2	8	no	—
43	C₂×#6099	—	2	16	no	—
44	C₂×#6099	C₂×C₄	1	16	yes	—
45	C₂×#6099	C₂×C₄	1	32	yes	—
46	C₂×#6099	C₂³	1	8	yes	—
47	C₂²×#1055	—	2	16	no	—
48	C₂²×#1055	—	2	8	no	—
49	C₂²×#1055	—	2	4	no	—
50	C₂×#1738	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	—	4	4	no	—
52	C₂⁴×C₁₃	C₄⋊C₄	1	8	yes	—
53	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂⁴×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
58	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	16	yes	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
65	C₂²×C₄×C₁₃	C₄⋊C₄	1	16	yes	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₁₃×M₁₆	—	2	32	no	—
84	C₂×C₈×C₁₃	—	2	8	no	—
85	C₂×C₈×C₁₃	—	2	8	no	—
86	C₂×C₈×C₁₃	—	2	8	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
88	#6099	—	2	32	no	—
89	#6099	—	2	16	no	—
90	#6099	—	2	8	no	—
91	#6099	C₂²×C₄	1	16	yes	—
92	C₂×#1055	—	2	32	no	—
93	C₂×#1055	—	2	16	no	—
94	C₂×#1055	—	2	8	no	—
95	C₂³×M₁₆	—	13	2	no	—
96	C₂³×M₁₆	D₁₃	1	1	yes	characteristic
97	#1738	—	13	8	no	—
98	C₂×#341	—	13	4	no	—
99	C₂³×C₁₃	—	2	8	no	—
100	C₂³×C₁₃	—	2	4	no	—
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	4	16	no	—
103	C₂³×C₁₃	—	4	4	no	—
104	C₂³×C₁₃	#95	1	8	yes	—
105	C₂³×C₁₃	#95	1	8	yes	—
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂³×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
110	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#95	1	16	yes	—
113	C₂×C₄×C₁₃	#95	1	8	yes	—
114	C₂×C₄×C₁₃	#95	1	16	yes	—
115	C₂×C₄×C₁₃	#95	1	8	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄²	1	8	yes	—
118	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
135	C₈×C₁₃	—	2	8	no	—
136	#1055	—	2	16	no	—
137	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×M₁₆	—	2	8	no	—
139	C₂²×M₁₆	—	13	16	no	—
140	C₂²×M₁₆	—	13	4	no	—
141	C₂²×M₁₆	—	13	8	no	—
142	C₂²×M₁₆	—	26	4	no	—
143	C₂²×M₁₆	—	26	2	no	—
144	C₂²×M₁₆	—	26	8	no	—
145	C₂²×M₁₆	Dic₁₃	1	4	yes	—
146	C₂²×M₁₆	Dic₁₃	1	4	yes	—
147	C₂²×M₁₆	C₂×D₁₃	1	4	yes	—
148	C₂³×C₈	—	26	2	no	—
149	C₂³×C₈	C₂×D₁₃	1	2	yes	—
150	#341	—	13	16	no	—
151	C₂²×C₁₃	—	2	8	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	16	no	—
155	C₂²×C₁₃	—	2	8	no	—
156	C₂²×C₁₃	—	4	8	no	—
157	C₂²×C₁₃	—	4	8	no	—
158	C₂²×C₁₃	C₂×#95	1	8	yes	—
159	C₂²×C₁₃	C₂×#95	1	2	yes	—
160	C₂²×C₁₃	C₂×#95	1	2	yes	—
161	C₂²×C₁₃	#430	1	4	yes	—
162	C₂²×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
163	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#95	1	2	yes	—
165	C₄×C₁₃	C₂×#95	1	8	yes	—
166	C₄×C₁₃	C₂×#95	1	2	yes	—
167	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
168	C₄×C₁₃	C₂²×C₂²⋊C₄	1	2	yes	—
169	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
170	C₂⁵	Dic₂₆	1	1	yes	characteristic
171	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
172	C₂³×C₄	Dic₂₆	1	2	yes	—
173	C₂³×C₄	C₄×D₁₃	1	8	yes	—
174	C₂³×C₄	D₅₂	1	2	yes	—
175	C₂³×C₄	D₅₂	1	2	yes	—
176	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
177	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
178	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
179	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
180	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×M₁₆	—	2	16	no	—
182	C₂×M₁₆	—	2	16	no	—
183	C₂×M₁₆	—	2	16	no	—
184	C₂×M₁₆	—	13	16	no	—
185	C₂×M₁₆	—	13	32	no	—
186	C₂×M₁₆	—	13	8	no	—
187	C₂×M₁₆	—	26	32	no	—
188	C₂×M₁₆	—	26	16	no	—
189	C₂×M₁₆	—	26	8	no	—
190	C₂×M₁₆	—	26	4	no	—
191	C₂×M₁₆	—	26	8	no	—
192	C₂×M₁₆	—	26	16	no	—
193	C₂×M₁₆	C₂×Dic₁₃	1	16	yes	—
194	C₂²×C₈	—	2	8	no	—
195	C₂²×C₈	—	26	16	no	—
196	C₂²×C₈	—	26	8	no	—
197	C₂²×C₈	—	26	4	no	—
198	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
199	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
200	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	8	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	#1738	1	4	yes	complemented
205	C₂×C₁₃	C₂²×#95	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#430	1	2	yes	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	#6101	1	8	yes	—
209	C₂⁴	C₄⋊Dic₁₃	1	2	yes	—
210	C₂⁴	C₄⋊Dic₁₃	1	2	yes	—
211	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
212	C₂⁴	C₂×Dic₂₆	1	2	yes	—
213	C₂²×C₄	C₄×Dic₁₃	1	8	yes	—
214	C₂²×C₄	C₄×Dic₁₃	1	8	yes	—
215	C₂²×C₄	#6101	1	8	yes	—
216	C₂²×C₄	#6101	1	16	yes	—
217	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
218	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
219	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
221	C₂²×C₄	#6103	1	8	yes	—
222	C₂²×C₄	#6103	1	16	yes	—
223	C₂²×C₄	#6103	1	8	yes	—
224	C₂²×C₄	C₂²⋊Dic₁₃	1	8	yes	—
225	C₂²×C₄	C₂²⋊Dic₁₃	1	8	yes	—
226	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
227	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
229	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
230	C₂²×C₄	C₂×D₅₂	1	4	yes	—
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₁₃	1	8	yes	—
238	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
239	M₁₆	—	2	32	no	—
240	M₁₆	—	13	16	no	—
241	M₁₆	—	26	32	no	—
242	M₁₆	—	26	16	no	—
243	M₁₆	—	26	8	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	—	26	32	no	—
248	C₂×C₈	—	26	16	no	—
249	C₂×C₈	—	26	8	no	—
250	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
251	C₁₃	C₂×#1738	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	2	8	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	4	16	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	#88071	1	8	yes	—
258	C₂³	#88071	1	8	yes	—
259	C₂³	C₂×#6101	1	4	yes	—
260	C₂³	C₂×#6101	1	8	yes	—
261	C₂³	C₂×C₄⋊Dic₁₃	1	4	yes	—
262	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
263	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
264	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
265	C₂×C₄	#88071	1	16	yes	—
266	C₂×C₄	#88071	1	8	yes	—
267	C₂×C₄	#88071	1	16	yes	—
268	C₂×C₄	#88071	1	8	yes	—
269	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₁₃	1	8	yes	—
271	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
272	C₂×C₄	C₂×#6101	1	4	yes	—
273	C₂×C₄	C₂×#6101	1	8	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#6103	1	8	yes	—
278	C₂×C₄	C₂×#6103	1	4	yes	—
279	C₂×C₄	C₂×#6103	1	8	yes	—
280	C₂×C₄	C₂×#6103	1	4	yes	—
281	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
283	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	8	yes	—
284	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
285	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
286	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
288	C₈	—	2	8	no	—
289	C₈	—	26	16	no	—
290	C₂²	—	2	8	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	16	no	—
294	C₂²	—	2	8	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	8	no	—
297	C₂²	#11718786	1	4	yes	—
298	C₂²	C₂×#88071	1	8	yes	—
299	C₂²	C₂×#88071	1	2	yes	—
300	C₂²	C₂×#88071	1	2	yes	—
301	C₂²	C₂²×#6101	1	2	yes	—
302	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
303	C₄	C₂×#88071	1	8	yes	—
304	C₄	C₂×#88071	1	2	yes	—
305	C₄	C₂×#88071	1	2	yes	—
306	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
307	C₄	C₂²×#6103	1	2	yes	—
308	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	—	2	8	no	—
311	C₂	—	2	2	no	—
312	C₂	#13114563	1	4	yes	direct product factor
313	C₂	C₂²×#88071	1	1	yes	characteristic
314	C₂	C₂×#11718786	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₃	Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#1738
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13114563	C₂	direct (N × H)
3. C₂×#11718144	C₂	semidirect
18. #11718144	C₂²	semidirect
204. C₂×C₁₃	#1738	semidirect
251. C₁₃	C₂×#1738	semidirect
312. C₂	#13114563	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
#13114563	2
C₂×#11718144	2
C₂³×C₁₃×M₁₆	2
C₂×#1738	13

Subgroups — C2×#13114563 #15860340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114563 #15860340