Subgroups — C₂×#13131055 #16806749

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719089	C₂	1	1	yes	maximal, characteristic
3	#13131055	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718217	C₂	1	2	yes	maximal, complemented
5	C₂×#11718218	C₂	1	2	yes	maximal, complemented
6	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
7	C₂×#88237	C₄	1	1	yes	characteristic
8	C₂×D₁₃⋊M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₄	1	2	yes	complemented
11	#11719089	C₄	1	4	yes	—
12	#11719089	C₂²	1	4	yes	—
13	C₂×#88100	C₂²	1	2	yes	—
14	C₂×C₈×#261	C₄	1	2	yes	—
15	D₁₃⋊M₃₂	C₄	1	4	yes	complemented
16	D₁₃⋊M₃₂	C₂²	1	4	yes	—
17	C₂×#88098	C₂²	1	2	yes	—
18	C₂×D₁₃⋊C₁₆	C₄	1	2	yes	complemented
19	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
21	C₂×#88038	C₄	1	1	yes	characteristic, complemented
22	C₂×#88038	C₄	1	1	yes	characteristic, complemented
23	C₂×#88038	C₄	1	4	yes	complemented
24	C₂×#88038	C₂²	1	1	yes	characteristic
25	C₂×#88038	C₂²	1	1	yes	characteristic
26	C₂×#88038	C₂²	1	4	yes	—
27	#11718218	C₂²	1	8	yes	complemented
28	#11718217	C₂²	1	8	yes	complemented
29	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊#261	—	2	1	no	—
31	C₂×#6122	—	2	1	no	—
32	#88237	—	2	2	no	—
33	#88237	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊#261	—	2	1	no	—
35	C₂×C₄×#261	C₂×C₄	1	2	yes	—
36	D₁₃⋊M₁₆	—	2	2	no	—
37	D₁₃⋊M₁₆	C₂×C₄	1	4	yes	—
38	C₂×#6118	—	2	1	no	—
39	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂³	1	2	yes	—
50	#88106	C₂×C₄	1	8	yes	complemented
51	C₂×#6092	C₂×C₄	1	4	yes	—
52	#88100	C₂×C₄	1	4	yes	—
53	#88100	C₂³	1	4	yes	—
54	C₈×#261	C₂×C₄	1	8	yes	—
55	#88098	C₂×C₄	1	4	yes	—
56	#88098	C₂³	1	4	yes	—
57	D₁₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
58	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
61	C₂×#6090	C₂×C₄	1	1	yes	characteristic
62	C₂×#6090	C₂×C₄	1	1	yes	characteristic
63	C₂×#6090	C₂×C₄	1	4	yes	complemented
64	#88038	C₂×C₄	1	2	yes	—
65	#88038	C₂×C₄	1	2	yes	—
66	#88038	C₂×C₄	1	4	yes	complemented
67	#88038	C₂×C₄	1	4	yes	complemented
68	#88038	C₂³	1	2	yes	—
69	#88038	C₂³	1	2	yes	—
70	C₂×#2110	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
72	C₂²×#261	—	2	2	no	—
73	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
74	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₄²	1	2	yes	—
77	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
78	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
79	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
83	C₂²⋊#261	—	2	4	no	—
84	#6122	—	2	4	no	—
85	C₂×#1057	—	2	2	no	—
86	C₄⋊#261	—	2	4	no	—
87	C₄×#261	—	2	2	no	—
88	C₄×#261	C₂²×C₄	1	4	yes	—
89	#6118	—	2	4	no	—
90	D₁₃⋊C₈	—	2	2	no	—
91	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₄²	1	2	yes	—
93	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₄²	1	1	yes	characteristic
97	C₂×#1055	C₄²	1	2	yes	—
98	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₈×D₁₃	C₄²	1	2	yes	—
101	C₈×D₁₃	C₄²	1	2	yes	—
102	C₈×D₁₃	C₄²	1	2	yes	—
103	C₈×D₁₃	C₂²×C₄	1	2	yes	—
104	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₈×D₁₃	C₂²×C₄	1	2	yes	—
107	C₈×D₁₃	C₂²×C₄	1	2	yes	—
108	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
110	#6092	C₂²×C₄	1	8	yes	—
111	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
112	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
113	#6090	C₂²×C₄	1	2	yes	—
114	#6090	C₂²×C₄	1	2	yes	—
115	C₂²×M₃₂	—	13	1	no	—
116	C₂²×M₃₂	—	13	1	no	—
117	C₂×#405	—	13	1	no	—
118	#2110	—	13	8	no	—
119	C₂×#347	—	13	2	no	—
120	C₂×#347	—	13	2	no	—
121	C₂³×C₁₃	#97	1	1	yes	characteristic
122	C₂²×D₁₃	—	4	2	no	—
123	C₂²×D₁₃	#97	1	1	yes	characteristic
124	C₂²×D₁₃	#97	1	2	yes	—
125	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
127	C₂×#261	—	2	4	no	—
128	C₂×#261	—	4	4	no	—
129	C₂×C₄×C₁₃	#97	1	2	yes	—
130	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×Dic₁₃	#97	1	2	yes	—
134	C₂×Dic₁₃	#97	1	1	yes	characteristic
135	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
137	C₄×D₁₃	#97	1	2	yes	—
138	C₄×D₁₃	#97	1	2	yes	—
139	C₄×D₁₃	#97	1	2	yes	—
140	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
141	C₄×D₁₃	C₂×C₄²	1	2	yes	—
142	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
143	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
144	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
146	C₄×D₁₃	C₂³×C₄	1	1	yes	characteristic
147	#1057	—	2	4	no	—
148	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
149	C₈×C₁₃	C₂×C₄²	1	2	yes	—
150	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	#1055	C₂×C₄²	1	2	yes	—
152	#1055	C₂×C₄²	1	1	yes	characteristic
153	#1055	C₂³×C₄	1	1	yes	characteristic
154	C₂²×M₁₆	—	13	1	no	—
155	C₂³×C₈	—	13	1	no	—
156	C₂×#117	—	13	1	no	—
157	C₂×M₃₂	—	13	2	no	—
158	C₂×M₃₂	—	13	4	no	—
159	C₂×M₃₂	—	13	4	no	—
160	C₂×M₃₂	—	13	2	no	—
161	C₂×M₃₂	—	13	1	no	—
162	C₂×M₃₂	—	13	1	no	—
163	C₂×M₃₂	—	13	1	no	—
164	C₂×M₃₂	—	13	1	no	—
165	C₂×M₃₂	—	13	4	no	—
166	C₂×M₃₂	—	13	4	no	—
167	C₂²×C₁₆	—	13	2	no	—
168	C₂²×C₁₆	—	13	1	no	—
169	C₂²×C₁₆	#261	1	1	yes	characteristic, complemented
170	#405	—	13	4	no	—
171	#405	—	13	4	no	—
172	C₂×#97	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	#347	—	13	8	no	—
175	#347	—	13	8	no	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	C₂×#97	1	2	yes	—
178	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	2	2	no	—
181	C₂×D₁₃	—	4	2	no	—
182	C₂×D₁₃	—	4	2	no	—
183	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
184	C₂×D₁₃	C₂×#97	1	2	yes	—
185	C₂×D₁₃	C₂²×M₁₆	1	1	yes	characteristic
186	#261	—	4	4	no	—
187	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
188	C₄×C₁₃	C₂×#97	1	2	yes	—
189	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
190	Dic₁₃	C₂×#97	1	1	yes	characteristic
191	Dic₁₃	C₂×#97	1	2	yes	—
192	Dic₁₃	C₂²×M₁₆	1	1	yes	characteristic
193	C₂³×C₄	—	13	1	no	—
194	C₂×M₁₆	—	13	4	no	—
195	C₂×M₁₆	—	26	1	no	—
196	C₂×M₁₆	—	26	2	no	—
197	C₂×M₁₆	—	26	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	13	1	no	—
202	C₂²×C₈	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	C₂×#261	1	1	yes	characteristic
209	#117	—	13	4	no	—
210	#117	—	26	2	no	—
211	C₂×C₄⋊C₄	—	26	1	no	—
212	C₂×C₂²⋊C₄	—	26	1	no	—
213	C₂×C₄²	—	13	2	no	—
214	M₃₂	—	13	4	no	—
215	M₃₂	—	13	4	no	—
216	M₃₂	—	13	8	no	—
217	M₃₂	—	13	2	no	—
218	M₃₂	—	13	2	no	—
219	M₃₂	—	13	2	no	—
220	M₃₂	—	13	2	no	—
221	M₃₂	—	13	4	no	—
222	M₃₂	—	13	4	no	—
223	C₂×C₁₆	—	13	8	no	—
224	C₂×C₁₆	—	13	4	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	1	no	—
227	C₂×C₁₆	—	13	4	no	—
228	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
229	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
230	C₂×C₁₆	C₂×#261	1	4	yes	complemented
231	#97	—	13	4	no	—
232	#97	—	13	4	no	—
233	C₄×C₈	—	13	8	no	—
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	#2110	1	2	yes	complemented
236	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
237	D₁₃	—	2	1	no	—
238	D₁₃	—	2	1	no	—
239	D₁₃	—	2	2	no	—
240	C₂⁴	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	2	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	1	no	—
247	C₂²×C₄	—	13	2	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	C₄×#261	1	1	yes	characteristic
252	M₁₆	—	26	4	no	—
253	M₁₆	—	26	4	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	13	2	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	1	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	4	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	2	no	—
271	C₂×C₈	C₄×#261	1	2	yes	—
272	C₂×C₈	C₄×#261	1	1	yes	characteristic
273	C₂×C₈	C₂²×#261	1	1	yes	characteristic
274	C₂×C₈	C₂²×#261	1	2	yes	—
275	C₄⋊C₄	—	26	4	no	—
276	C₂²⋊C₄	—	26	4	no	—
277	C₄²	—	13	4	no	—
278	C₄²	—	26	2	no	—
279	C₁₆	—	13	8	no	—
280	C₁₆	—	13	2	no	—
281	C₁₆	—	13	2	no	—
282	C₁₆	C₂²×#261	1	2	yes	—
283	C₁₆	C₂²×#261	1	2	yes	—
284	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	13	1	no	—
286	C₂³	—	13	1	no	—
287	C₂³	—	13	2	no	—
288	C₂³	—	13	2	no	—
289	C₂³	—	52	2	no	—
290	C₂³	#88100	1	1	yes	characteristic
291	C₂×C₄	—	13	1	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	1	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	2	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	26	4	no	—
306	C₂×C₄	—	52	4	no	—
307	C₂×C₄	#88100	1	2	yes	—
308	C₂×C₄	#88100	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄×#261	1	2	yes	—
310	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
311	C₈	—	13	1	no	—
312	C₈	—	13	1	no	—
313	C₈	—	13	2	no	—
314	C₈	—	26	4	no	—
315	C₈	C₂×C₄×#261	1	1	yes	characteristic
316	C₈	C₂×C₄×#261	1	2	yes	—
317	C₈	C₂³×#261	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	13	1	no	—
320	C₂²	—	13	1	no	—
321	C₂²	—	13	2	no	—
322	C₂²	—	26	2	no	—
323	C₂²	—	26	2	no	—
324	C₂²	—	52	2	no	—
325	C₂²	—	52	2	no	—
326	C₂²	C₂×#88100	1	2	yes	—
327	C₂²	C₂×#88100	1	1	yes	characteristic
328	C₄	—	13	1	no	—
329	C₄	—	13	1	no	—
330	C₄	—	13	2	no	—
331	C₄	—	52	4	no	—
332	C₄	C₂×#88100	1	1	yes	characteristic
333	C₄	C₂×#88100	1	2	yes	—
334	C₄	C₂²×C₄×#261	1	1	yes	characteristic
335	C₂	—	2	2	no	—
336	C₂	—	26	1	no	—
337	C₂	—	26	1	no	—
338	C₂	—	26	2	no	—
339	C₂	#13131055	1	2	yes	direct product factor
340	C₂	C₂²×#88100	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₈	C₂³×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	C₂×#2110
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #13131055	C₂	direct (N × H)
4. C₂×#11718217	C₂	semidirect
5. C₂×#11718218	C₂	semidirect
10. C₂×#88106	C₄	semidirect
15. D₁₃⋊M₃₂	C₄	semidirect
18. C₂×D₁₃⋊C₁₆	C₄	semidirect
19. C₂²×C₁₃×C₁₆	C₄	semidirect
20. C₂²×#6090	C₄	semidirect
21. C₂×#88038	C₄	semidirect
22. C₂×#88038	C₄	semidirect
23. C₂×#88038	C₄	semidirect
27. #11718218	C₂²	semidirect
28. #11718217	C₂²	semidirect
50. #88106	C₂×C₄	semidirect
57. D₁₃⋊C₁₆	C₂×C₄	semidirect
60. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
63. C₂×#6090	C₂×C₄	semidirect
66. #88038	C₂×C₄	semidirect
67. #88038	C₂×C₄	semidirect
169. C₂²×C₁₆	#261	semidirect
230. C₂×C₁₆	C₂×#261	semidirect
235. C₂×C₁₃	#2110	semidirect
284. C₁₃	C₂×#2110	semidirect
339. C₂	#13131055	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×D₁₃⋊M₃₂	2
C₂×#11719089	2
#13131055	2
C₂×#11718217	2
C₂×#11718218	2
C₂²×#88038	2
C₂×#2110	13

Subgroups — C2×#13131055 #16806749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131055 #16806749