Subgroups — C₂×#13116664 #16016078

Order 3328.

754 conjugacy classes of subgroups.

The subgroup lattice has 336 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, complemented
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
4	#13116664	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718099	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊#1055	C₄	1	1	yes	characteristic
15	C₈×C₂²⋊D₁₃	C₄	1	4	yes	—
16	C₈×C₂²⋊D₁₃	C₂²	1	4	yes	—
17	C₂×#88060	C₄	1	1	yes	characteristic
18	C₂×#88054	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88052	—	2	4	no	—
21	C₂×#88052	C₂²	1	4	yes	complemented
22	C₂²×#6090	C₂²	1	1	yes	characteristic
23	C₂²×#6090	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×D₁₃	—	2	4	no	—
25	C₂×C₁₆×D₁₃	C₂²	1	4	yes	complemented
26	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
27	#11718099	C₂²	1	4	yes	complemented
28	#11718094	C₂²	1	4	yes	complemented
29	#11718093	C₂²	1	4	yes	complemented
30	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₂²⋊D₁₃	C₈	1	1	yes	characteristic
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₁₃	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₁₃	C₈	1	4	yes	—
41	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
42	C₂×#6103	C₈	1	1	yes	characteristic
43	C₂×#6101	C₈	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	—	2	2	no	—
48	C₂×C₈×D₁₃	—	2	2	no	—
49	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₁₃	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₂×C₁₃×C₁₆	—	2	2	no	—
53	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂²⋊#1055	C₂×C₄	1	4	yes	—
56	#88060	C₂×C₄	1	4	yes	—
57	#88054	C₂×C₄	1	4	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂³	1	2	yes	—
60	#88052	—	2	8	no	—
61	C₂×#6090	—	2	2	no	—
62	C₂×#6090	D₄	1	4	yes	complemented
63	C₂×#6090	C₂³	1	2	yes	—
64	C₂×#6090	C₂³	1	2	yes	—
65	C₁₆×D₁₃	—	2	8	no	—
66	C₂×#2168	—	13	1	no	maximal, Sylow
67	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃	—	2	4	no	—
76	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
77	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	—	2	2	no	—
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
94	#6103	C₂×C₈	1	4	yes	—
95	#6101	C₂×C₈	1	4	yes	—
96	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	C₂²×C₄	1	2	yes	—
102	C₈×D₁₃	—	2	4	no	—
103	C₈×D₁₃	—	2	2	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	—	2	2	no	—
107	#6090	—	2	2	no	—
108	#6090	C₂×D₄	1	4	yes	complemented
109	C₂²×M₃₂	—	13	1	no	—
110	C₂³×C₁₆	—	13	1	no	—
111	C₂×C₈×D₄	—	13	1	no	—
112	#2168	—	13	8	no	—
113	C₂×C₄⋊C₁₆	—	13	1	no	—
114	C₂×C₂²⋊C₁₆	—	13	1	no	—
115	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
116	C₂×#346	—	13	1	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
121	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₁₃	—	2	2	no	—
123	C₂²×D₁₃	—	2	2	no	—
124	C₂²×D₁₃	M₃₂	1	4	yes	complemented
125	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#131	1	2	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	2	8	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
140	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
141	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	C₄×D₄	1	2	yes	—
147	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
149	#1055	—	2	2	no	—
150	#1055	—	2	2	no	—
151	C₂³×C₈	—	13	1	no	—
152	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄×D₄	—	13	1	no	—
154	C₂×M₃₂	—	13	4	no	—
155	C₂×M₃₂	—	26	4	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	C₂²×C₁₆	—	13	1	no	—
158	C₂²×C₁₆	—	13	4	no	—
159	C₂²×C₁₆	—	26	4	no	—
160	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
161	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₈×D₄	—	13	4	no	—
163	C₈×D₄	—	13	4	no	—
164	C₂×C₄⋊C₈	—	13	1	no	—
165	C₂×C₂²⋊C₈	—	13	1	no	—
166	C₂×C₂²⋊C₈	—	13	1	no	—
167	C₂×C₄×C₈	—	13	1	no	—
168	C₄⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	complemented
171	#346	—	13	4	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	C₈×D₄	1	2	yes	—
177	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
178	C₂²×C₁₃	#504	1	2	yes	—
179	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	8	no	—
183	C₂×D₁₃	C₂×M₃₂	1	4	yes	complemented
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	C₈×D₄	1	2	yes	—
186	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
188	Dic₁₃	—	2	2	no	—
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	C₂²×D₄	—	13	1	no	—
192	C₂³×C₄	—	13	1	no	—
193	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	26	2	no	—
201	C₂²×C₈	—	26	2	no	—
202	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₁₃	1	2	yes	—
204	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
205	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
206	C₄×D₄	—	13	4	no	—
207	C₄×D₄	—	13	4	no	—
208	C₂×C₄⋊C₄	—	13	1	no	—
209	C₂×C₂²⋊C₄	—	13	1	no	—
210	C₂×C₂²⋊C₄	—	13	1	no	—
211	C₂×C₄²	—	13	1	no	—
212	M₃₂	—	26	8	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	—	13	4	no	—
216	C₂×C₁₆	—	13	2	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	26	8	no	—
219	C₂×C₁₆	—	26	2	no	—
220	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
221	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
222	C₄⋊C₈	—	13	4	no	—
223	C₂²⋊C₈	—	13	4	no	—
224	C₂²⋊C₈	—	13	4	no	—
225	C₄×C₈	—	13	2	no	—
226	C₄×C₈	—	13	2	no	—
227	C₂×C₁₃	—	4	2	no	—
228	C₂×C₁₃	#2168	1	4	yes	complemented
229	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
230	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
232	D₁₃	—	2	4	no	—
233	C₂⁴	—	13	1	no	—
234	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
235	C₂×D₄	—	13	4	no	—
236	C₂×D₄	—	26	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₈×D₁₃	1	2	yes	—
248	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	4	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
262	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₈	D₄×D₁₃	1	2	yes	—
264	C₂×C₈	#6129	1	2	yes	—
265	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
266	C₄⋊C₄	—	13	4	no	—
267	C₂²⋊C₄	—	13	4	no	—
268	C₂²⋊C₄	—	13	4	no	—
269	C₄²	—	13	2	no	—
270	C₄²	—	13	2	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	13	4	no	—
274	C₁₆	—	26	2	no	—
275	C₁₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	13	2	no	—
280	C₂³	—	13	4	no	—
281	C₂³	—	26	2	no	—
282	C₂³	—	26	2	no	—
283	C₂³	C₂×C₈×D₁₃	1	2	yes	—
284	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
285	D₄	—	26	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	4	no	—
299	C₂×C₄	#88135	1	2	yes	—
300	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	#88161	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	26	2	no	—
306	C₈	—	26	2	no	—
307	C₈	#88135	1	2	yes	—
308	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
309	C₈	C₂×#6129	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	13	4	no	—
315	C₂²	—	26	2	no	—
316	C₂²	—	26	8	no	—
317	C₂²	—	26	2	no	—
318	C₂²	#11718386	1	2	yes	—
319	C₂²	#11718562	1	2	yes	—
320	C₂²	D₁₃×M₃₂	1	2	yes	—
321	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	26	2	no	—
324	C₄	—	26	2	no	—
325	C₄	—	26	2	no	—
326	C₄	#11718386	1	2	yes	—
327	C₄	C₂×#88135	1	1	yes	characteristic
328	C₄	C₂×#88161	1	1	yes	characteristic
329	C₂	—	4	2	no	—
330	C₂	—	26	4	no	—
331	C₂	#13116664	1	4	yes	direct product factor
332	C₂	C₂×#11718386	1	1	yes	characteristic
333	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
334	C₂	C₂×#11718562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×C₂²⋊D₁₃	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2168
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88052	C₂	semidirect
2. C₂²×C₁₆×D₁₃	C₂	semidirect
4. #13116664	C₂	direct (N × H)
5. C₂×#11718093	C₂	semidirect
6. C₂×#11718099	C₂	semidirect
7. C₂×#11718094	C₂	semidirect
8. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
21. C₂×#88052	C₂²	semidirect
25. C₂×C₁₆×D₁₃	C₂²	semidirect
26. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
27. #11718099	C₂²	semidirect
28. #11718094	C₂²	semidirect
29. #11718093	C₂²	semidirect
62. C₂×#6090	D₄	semidirect
108. #6090	C₂×D₄	semidirect
115. C₂×C₂²⋊C₁₆	D₁₃	semidirect
124. C₂²×D₁₃	M₃₂	semidirect
170. C₂²⋊C₁₆	C₂×D₁₃	semidirect
183. C₂×D₁₃	C₂×M₃₂	semidirect
228. C₂×C₁₃	#2168	semidirect
275. C₁₃	C₂×#2168	semidirect
331. C₂	#13116664	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
C₂²×C₁₆×D₁₃	2
C₂×C₈×C₂²⋊D₁₃	2
#13116664	2
C₂×#11718093	2
C₂×#11718099	2
C₂×#11718094	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2168	13

Subgroups — C2×#13116664 #16016078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116664 #16016078