Subgroups — C₂×#13116663 #16016077

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×#88038	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
4	#13116663	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718099	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718099	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88060	C₄	1	1	yes	characteristic
16	C₂×#88060	C₄	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂²×#6090	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
20	C₂×#88038	—	2	4	no	—
21	C₂×#88038	C₂²	1	4	yes	complemented
22	C₂×C₁₆×D₁₃	—	2	4	no	—
23	C₂×C₁₆×D₁₃	C₂²	1	4	yes	complemented
24	C₈×D₅₂	C₄	1	4	yes	—
25	C₈×D₅₂	C₂²	1	4	yes	—
26	C₁₃×#346	C₂²	1	4	yes	complemented
27	#11718099	C₂²	1	4	yes	complemented
28	#11718099	C₂²	1	4	yes	complemented
29	C₄⋊#6090	C₂²	1	4	yes	complemented
30	C₂²×D₅₂	C₈	1	1	yes	characteristic
31	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6103	C₈	1	1	yes	characteristic
37	C₂×#6103	C₈	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₁₃	—	2	2	no	—
42	C₂×C₈×D₁₃	—	2	2	no	—
43	C₂×C₈×D₁₃	—	2	2	no	—
44	C₂×C₈×D₁₃	—	2	2	no	—
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₁₃	C₂³	1	2	yes	—
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	C₄×D₅₂	C₈	1	4	yes	—
50	C₄×D₅₂	C₂×C₄	1	4	yes	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
53	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₃	C₂³	1	2	yes	—
57	#88060	C₂×C₄	1	4	yes	—
58	#88060	C₂×C₄	1	4	yes	—
59	C₂×#6090	—	2	2	no	—
60	C₂×#6090	—	2	2	no	—
61	C₂×#6090	C₂³	1	2	yes	—
62	C₂×#6090	C₂³	1	2	yes	—
63	C₄⋊#1055	C₂×C₄	1	4	yes	—
64	#88038	—	2	8	no	—
65	C₁₆×D₁₃	—	2	8	no	—
66	C₂×#2168	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
74	C₂×D₅₂	—	2	4	no	—
75	C₂×D₅₂	C₂×C₈	1	4	yes	—
76	C₂×C₄×D₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	—	2	2	no	—
78	C₂×C₄×D₁₃	—	2	2	no	—
79	C₂×C₄×D₁₃	—	2	2	no	—
80	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
81	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	#41	1	2	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×C₈	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	#6103	C₂×C₈	1	4	yes	—
93	#6103	C₂×C₈	1	4	yes	—
94	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	C₂²×C₄	1	2	yes	—
98	C₂×#1055	C₂²×C₄	1	2	yes	—
99	C₈×D₁₃	—	2	4	no	—
100	C₈×D₁₃	—	2	4	no	—
101	C₈×D₁₃	—	2	2	no	—
102	C₈×D₁₃	—	2	2	no	—
103	C₈×D₁₃	—	2	2	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₁₃×C₁₆	—	2	2	no	—
106	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
107	#6090	—	2	2	no	—
108	#6090	—	2	2	no	—
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	—	13	1	no	—
116	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂²×D₁₃	—	2	2	no	—
119	C₂²×D₁₃	—	2	2	no	—
120	C₂²×D₁₃	—	2	2	no	—
121	C₂²×D₁₃	—	2	2	no	—
122	C₂²×D₁₃	—	2	2	no	—
123	C₂²×D₁₃	M₃₂	1	4	yes	complemented
124	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
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	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#131	1	2	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
138	D₅₂	—	2	8	no	—
139	C₄×D₁₃	—	2	2	no	—
140	C₄×D₁₃	—	2	2	no	—
141	C₄×D₁₃	—	2	2	no	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	4	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	—	13	1	no	—
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	C₂×D₁₃	1	1	yes	characteristic
168	C₄⋊C₁₆	—	13	4	no	—
169	C₂²⋊C₁₆	—	13	4	no	—
170	C₂²⋊C₁₆	—	13	4	no	—
171	#346	C₂×D₁₃	1	4	yes	complemented
172	C₂²×C₁₃	C₈×D₄	1	2	yes	—
173	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
174	C₂²×C₁₃	#504	1	2	yes	—
175	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
176	C₂×D₁₃	—	2	2	no	—
177	C₂×D₁₃	—	2	2	no	—
178	C₂×D₁₃	—	2	8	no	—
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	4	4	no	—
183	C₂×D₁₃	C₂×M₃₂	1	4	yes	complemented
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	C₈×D₄	1	2	yes	—
187	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
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	—	13	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	2	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	—	26	2	no	—
203	C₂²×C₈	—	26	2	no	—
204	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
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	C₄×D₁₃	1	1	yes	characteristic
212	M₃₂	—	26	8	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	2	no	—
216	C₂×C₁₆	—	26	2	no	—
217	C₂×C₁₆	—	26	2	no	—
218	C₂×C₁₆	—	26	8	no	—
219	C₂×C₁₆	D₅₂	1	4	yes	complemented
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	C₄×D₁₃	1	2	yes	—
226	C₄×C₈	C₂²×D₁₃	1	2	yes	—
227	C₂×C₁₃	#2168	1	4	yes	complemented
228	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
229	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
230	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
231	D₁₃	—	2	4	no	—
232	D₁₃	—	4	2	no	—
233	C₂⁴	—	13	1	no	—
234	C₂⁴	—	13	1	no	—
235	C₂×D₄	—	13	4	no	—
236	C₂×D₄	—	26	4	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	4	no	—
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	2	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	C₂×D₅₂	1	2	yes	—
264	C₂×C₈	#6127	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	C₈×D₁₃	1	2	yes	—
270	C₄²	C₂×C₄×D₁₃	1	2	yes	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	C₂×D₅₂	1	4	yes	complemented
275	C₁₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	2	no	—
277	C₂³	—	13	4	no	—
278	C₂³	—	13	2	no	—
279	C₂³	—	26	2	no	—
280	C₂³	—	26	2	no	—
281	C₂³	—	26	2	no	—
282	C₂³	—	26	2	no	—
283	C₂³	—	26	2	no	—
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	—	13	2	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	4	no	—
297	C₂×C₄	—	26	4	no	—
298	C₂×C₄	C₄×D₅₂	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₁₃	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	#88155	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	C₄×D₅₂	1	2	yes	—
308	C₈	C₂²×D₅₂	1	1	yes	characteristic
309	C₈	C₂×#6127	1	1	yes	characteristic
310	C₂²	—	13	4	no	—
311	C₂²	—	26	2	no	—
312	C₂²	—	26	2	no	—
313	C₂²	—	26	8	no	—
314	C₂²	—	26	2	no	—
315	C₂²	—	26	2	no	—
316	C₂²	—	26	2	no	—
317	C₂²	—	52	4	no	—
318	C₂²	C₈×D₅₂	1	2	yes	—
319	C₂²	D₁₃×M₃₂	1	2	yes	—
320	C₂²	#11718568	1	2	yes	—
321	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	26	2	no	—
325	C₄	—	26	2	no	—
326	C₄	C₈×D₅₂	1	2	yes	—
327	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
328	C₄	C₂×#88155	1	1	yes	characteristic
329	C₂	—	26	4	no	—
330	C₂	—	52	2	no	—
331	C₂	#13116663	1	4	yes	direct product factor
332	C₂	C₂×C₈×D₅₂	1	1	yes	characteristic
333	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic
334	C₂	C₂×#11718568	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×#346	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₅₂	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	C₂×#2168
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂²×#88038	C₂	semidirect
2. C₂²×C₁₆×D₁₃	C₂	semidirect
4. #13116663	C₂	direct (N × H)
5. C₂×C₄⋊#6090	C₂	semidirect
6. C₂×#11718099	C₂	semidirect
7. C₂×#11718099	C₂	semidirect
8. C₂×C₁₃×#346	C₂	semidirect
21. C₂×#88038	C₂²	semidirect
23. C₂×C₁₆×D₁₃	C₂²	semidirect
26. C₁₃×#346	C₂²	semidirect
27. #11718099	C₂²	semidirect
28. #11718099	C₂²	semidirect
29. C₄⋊#6090	C₂²	semidirect
52. C₂×C₁₃×C₁₆	D₄	semidirect
106. C₁₃×C₁₆	C₂×D₄	semidirect
116. C₂×#346	D₁₃	semidirect
123. C₂²×D₁₃	M₃₂	semidirect
171. #346	C₂×D₁₃	semidirect
183. C₂×D₁₃	C₂×M₃₂	semidirect
219. C₂×C₁₆	D₅₂	semidirect
227. C₂×C₁₃	#2168	semidirect
274. C₁₆	C₂×D₅₂	semidirect
275. C₁₃	C₂×#2168	semidirect
331. C₂	#13116663	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×#88038	2
C₂²×C₁₆×D₁₃	2
C₂×C₈×D₅₂	2
#13116663	2
C₂×C₄⋊#6090	2
C₂×#11718099	2
C₂×C₁₃×#346	2
C₂×#2168	13

Subgroups — C2×#13116663 #16016077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116663 #16016077