Subgroups — C₂×#13131051 #16806745

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 362 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	C₂²×#88100	C₂	1	1	yes	maximal, characteristic
3	#13131051	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	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×#261	C₄	1	1	yes	characteristic
8	C₂²×D₁₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₄	1	2	yes	complemented
11	C₂×#88100	C₄	1	4	yes	—
12	C₂×#88100	C₄	1	1	yes	characteristic
13	C₂×#88100	C₄	1	1	yes	characteristic
14	C₂×#88100	C₂²	1	1	yes	characteristic
15	C₂×#88100	C₂²	1	1	yes	characteristic
16	C₂×#88100	C₂²	1	4	yes	—
17	D₁₃⋊M₃₂	C₄	1	4	yes	complemented
18	D₁₃⋊M₃₂	C₂²	1	4	yes	—
19	C₂×#88098	C₂²	1	2	yes	—
20	C₂×D₁₃⋊C₁₆	C₄	1	2	yes	complemented
21	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
22	C₂×#88052	C₄	1	1	yes	characteristic, complemented
23	D₁₃×M₃₂	C₄	1	4	yes	complemented
24	D₁₃×M₃₂	C₂²	1	4	yes	—
25	C₂×#88038	C₂²	1	2	yes	—
26	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
27	#11718218	C₂²	1	8	yes	complemented
28	#11718217	C₂²	1	8	yes	complemented
29	C₂³×#261	—	2	1	no	—
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1057	—	2	1	no	—
32	C₂×C₄×#261	—	2	2	no	—
33	C₂×C₄×#261	—	2	1	no	—
34	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×#261	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×#261	C₂×C₄	1	4	yes	—
37	C₂×D₁₃⋊C₈	—	2	2	no	—
38	C₂×D₁₃⋊C₈	—	2	1	no	—
39	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×D₁₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	2	yes	—
52	#88106	C₂×C₄	1	8	yes	complemented
53	C₂×#6092	C₂×C₄	1	4	yes	complemented
54	#88100	C₂×C₄	1	2	yes	—
55	#88100	C₂×C₄	1	2	yes	—
56	#88100	C₂×C₄	1	4	yes	—
57	#88100	C₂×C₄	1	4	yes	—
58	#88100	C₂³	1	2	yes	—
59	#88100	C₂³	1	2	yes	—
60	#88098	C₂×C₄	1	4	yes	complemented
61	#88098	C₂³	1	4	yes	—
62	D₁₃⋊C₁₆	C₂×C₄	1	8	yes	complemented
63	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
64	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
65	#88052	C₂×C₄	1	4	yes	complemented
66	C₂×#6090	C₂×C₄	1	2	yes	complemented
67	#88038	C₂×C₄	1	4	yes	complemented
68	#88038	C₂³	1	4	yes	—
69	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
70	C₂²×#347	—	13	1	no	maximal, Sylow
71	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
72	C₂²×#261	—	2	1	no	—
73	C₂²×#261	—	2	1	no	—
74	C₂²×#261	—	2	4	no	—
75	C₂²×#261	—	4	4	no	—
76	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
77	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₄²	1	2	yes	—
79	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
82	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
83	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×#1057	—	2	4	no	—
87	C₂×#1057	—	2	1	no	—
88	C₂×#1057	—	2	1	no	—
89	C₄×#261	—	2	1	no	—
90	C₄×#261	—	2	1	no	—
91	C₄×#261	—	2	4	no	—
92	C₄×#261	C₂²×C₄	1	2	yes	—
93	C₄×#261	C₂²×C₄	1	2	yes	—
94	D₁₃⋊C₈	—	2	1	no	—
95	D₁₃⋊C₈	—	2	1	no	—
96	D₁₃⋊C₈	—	2	4	no	—
97	D₁₃⋊C₈	C₂²×C₄	1	2	yes	—
98	D₁₃⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₄²	1	2	yes	—
100	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
103	C₂×#1055	C₄²	1	2	yes	—
104	C₂×#1055	C₄²	1	1	yes	characteristic
105	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
106	C₂×#1055	C₂²×C₄	1	2	yes	—
107	C₈×D₁₃	C₄²	1	2	yes	—
108	C₈×D₁₃	C₄²	1	2	yes	—
109	C₈×D₁₃	C₄²	1	2	yes	—
110	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₁₃	C₂²×C₄	1	2	yes	—
112	C₈×D₁₃	C₂²×C₄	1	2	yes	—
113	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₈×D₁₃	C₂²×C₄	1	2	yes	—
115	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
117	#6092	C₂²×C₄	1	8	yes	complemented
118	C₁₃×C₁₆	C₂²×C₄	1	4	yes	complemented
119	#6090	C₂²×C₄	1	4	yes	complemented
120	C₂²×M₃₂	—	13	1	no	—
121	C₂²×M₃₂	—	13	1	no	—
122	C₂²×#97	—	13	1	no	—
123	C₂×#347	—	13	2	no	—
124	C₂×#347	—	13	2	no	—
125	C₂×#347	—	13	8	no	—
126	C₂³×C₁₃	#97	1	1	yes	characteristic
127	C₂²×D₁₃	—	2	4	no	—
128	C₂²×D₁₃	#97	1	2	yes	—
129	C₂²×D₁₃	#97	1	1	yes	characteristic
130	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
132	C₂×#261	—	2	2	no	—
133	C₂×#261	—	2	2	no	—
134	C₂×#261	—	4	2	no	—
135	C₂×#261	—	4	2	no	—
136	C₂×#261	—	4	8	no	—
137	C₂×C₄×C₁₃	#97	1	2	yes	—
138	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
141	C₂×Dic₁₃	#97	1	2	yes	—
142	C₂×Dic₁₃	#97	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
144	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
145	C₄×D₁₃	#97	1	2	yes	—
146	C₄×D₁₃	#97	1	2	yes	—
147	C₄×D₁₃	#97	1	2	yes	—
148	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
149	C₄×D₁₃	C₂×C₄²	1	2	yes	—
150	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
151	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
152	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
153	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
154	C₄×D₁₃	C₂³×C₄	1	1	yes	characteristic
155	#1057	—	2	2	no	—
156	#1057	—	2	2	no	—
157	C₈×C₁₃	C₂×C₄²	1	2	yes	—
158	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
159	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	#1055	C₂×C₄²	1	2	yes	—
161	#1055	C₂×C₄²	1	1	yes	characteristic
162	#1055	C₂³×C₄	1	1	yes	characteristic
163	C₂³×C₈	—	13	1	no	—
164	C₂³×C₈	—	13	1	no	—
165	C₂²×C₄²	—	13	1	no	—
166	C₂×M₃₂	—	13	2	no	—
167	C₂×M₃₂	—	13	4	no	—
168	C₂×M₃₂	—	13	4	no	—
169	C₂×M₃₂	—	13	2	no	—
170	C₂×M₃₂	—	13	1	no	—
171	C₂×M₃₂	—	13	4	no	—
172	C₂×M₃₂	—	13	4	no	—
173	C₂×M₃₂	—	13	2	no	—
174	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
175	C₂²×C₁₆	—	13	2	no	—
176	C₂²×C₁₆	—	13	2	no	—
177	C₂×#97	—	13	1	no	—
178	C₂×#97	—	13	1	no	—
179	C₂×#97	—	13	4	no	—
180	C₂×#97	—	13	4	no	—
181	C₂×#97	—	13	1	no	—
182	C₂×#97	—	13	1	no	—
183	#347	—	13	8	no	—
184	#347	—	13	8	no	—
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	C₂×#97	1	2	yes	—
187	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
188	C₂×D₁₃	—	2	2	no	—
189	C₂×D₁₃	—	2	4	no	—
190	C₂×D₁₃	—	2	4	no	—
191	C₂×D₁₃	#347	1	4	yes	complemented
192	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
193	C₂×D₁₃	C₂×#97	1	2	yes	—
194	C₂×D₁₃	C₂²×M₁₆	1	1	yes	characteristic
195	#261	—	4	2	no	—
196	#261	—	4	2	no	—
197	C₄×C₁₃	C₂×#97	1	2	yes	—
198	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
199	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
200	Dic₁₃	C₂×#97	1	2	yes	—
201	Dic₁₃	C₂×#97	1	1	yes	characteristic
202	Dic₁₃	C₂²×M₁₆	1	1	yes	characteristic
203	C₂³×C₄	—	13	1	no	—
204	C₂³×C₄	—	26	1	no	—
205	C₂²×C₈	—	13	1	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	13	2	no	—
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	2	no	—
212	C₂²×C₈	—	13	2	no	—
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	4	no	—
215	C₂²×C₈	—	13	1	no	—
216	C₂²×C₈	—	13	1	no	—
217	C₂²×C₈	—	26	1	no	—
218	C₂²×C₈	—	26	2	no	—
219	C₂²×C₈	—	26	1	no	—
220	C₂²×C₈	C₂×#261	1	1	yes	characteristic
221	C₂×C₄²	—	13	4	no	—
222	C₂×C₄²	—	13	1	no	—
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	26	1	no	—
225	C₂×C₄²	—	26	2	no	—
226	M₃₂	—	13	4	no	—
227	M₃₂	—	13	4	no	—
228	M₃₂	—	13	8	no	—
229	M₃₂	—	13	4	no	—
230	M₃₂	—	13	4	no	—
231	M₃₂	—	13	4	no	—
232	M₃₂	C₂×#261	1	4	yes	complemented
233	C₂×C₁₆	—	13	8	no	—
234	C₂×C₁₆	—	13	4	no	—
235	C₂×C₁₆	—	13	2	no	—
236	C₂×C₁₆	—	13	8	no	—
237	C₂×C₁₆	C₂×#261	1	2	yes	complemented
238	#97	—	13	2	no	—
239	#97	—	13	2	no	—
240	#97	—	13	4	no	—
241	#97	—	13	4	no	—
242	#97	—	13	2	no	—
243	#97	—	13	2	no	—
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	C₂×#347	1	2	yes	complemented
246	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
247	D₁₃	—	2	1	no	—
248	D₁₃	—	2	1	no	—
249	D₁₃	C₂×#347	1	4	yes	complemented
250	C₂⁴	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	2	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	1	no	—
260	C₂²×C₄	—	26	4	no	—
261	C₂²×C₄	—	26	1	no	—
262	C₂²×C₄	—	26	1	no	—
263	C₂²×C₄	—	52	4	no	—
264	C₂²×C₄	C₄×#261	1	1	yes	characteristic
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	1	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	—	13	1	no	—
273	C₂×C₈	—	13	2	no	—
274	C₂×C₈	—	13	2	no	—
275	C₂×C₈	—	13	2	no	—
276	C₂×C₈	—	13	1	no	—
277	C₂×C₈	—	13	2	no	—
278	C₂×C₈	—	13	1	no	—
279	C₂×C₈	—	13	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	26	4	no	—
282	C₂×C₈	—	26	1	no	—
283	C₂×C₈	—	26	1	no	—
284	C₂×C₈	—	26	4	no	—
285	C₂×C₈	—	26	1	no	—
286	C₂×C₈	—	26	1	no	—
287	C₂×C₈	C₄×#261	1	2	yes	—
288	C₂×C₈	C₄×#261	1	1	yes	characteristic
289	C₂×C₈	C₂²×#261	1	2	yes	—
290	C₂×C₈	C₂²×#261	1	1	yes	characteristic
291	C₄²	—	13	2	no	—
292	C₄²	—	13	2	no	—
293	C₄²	—	26	4	no	—
294	C₄²	—	26	1	no	—
295	C₄²	—	26	1	no	—
296	C₁₆	—	13	8	no	—
297	C₁₆	—	13	4	no	—
298	C₁₆	C₂²×#261	1	4	yes	complemented
299	C₁₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	13	1	no	—
301	C₂³	—	13	2	no	—
302	C₂³	—	13	1	no	—
303	C₂³	—	13	2	no	—
304	C₂³	—	26	4	no	—
305	C₂³	#88100	1	1	yes	characteristic
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	13	1	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	13	1	no	—
310	C₂×C₄	—	13	2	no	—
311	C₂×C₄	—	13	1	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	2	no	—
315	C₂×C₄	—	13	2	no	—
316	C₂×C₄	—	13	2	no	—
317	C₂×C₄	—	13	1	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	13	1	no	—
320	C₂×C₄	—	26	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	52	8	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	#88100	1	1	yes	characteristic
326	C₂×C₄	#88100	1	2	yes	—
327	C₂×C₄	C₂×C₄×#261	1	2	yes	—
328	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
329	C₈	—	13	1	no	—
330	C₈	—	13	2	no	—
331	C₈	—	13	1	no	—
332	C₈	—	26	2	no	—
333	C₈	—	26	2	no	—
334	C₈	C₂×C₄×#261	1	2	yes	—
335	C₈	C₂×C₄×#261	1	1	yes	characteristic
336	C₈	C₂³×#261	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	13	4	no	—
339	C₂²	—	13	2	no	—
340	C₂²	—	13	1	no	—
341	C₂²	—	13	1	no	—
342	C₂²	—	26	4	no	—
343	C₂²	—	26	4	no	—
344	C₂²	—	26	2	no	—
345	C₂²	C₂×#88100	1	2	yes	—
346	C₂²	C₂×#88100	1	1	yes	characteristic
347	C₄	—	13	1	no	—
348	C₄	—	13	2	no	—
349	C₄	—	13	1	no	—
350	C₄	—	52	2	no	—
351	C₄	—	52	2	no	—
352	C₄	C₂×#88100	1	2	yes	—
353	C₄	C₂×#88100	1	1	yes	characteristic
354	C₄	C₂²×C₄×#261	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	13	4	no	—
357	C₂	—	26	1	no	—
358	C₂	—	26	1	no	—
359	C₂	#13131051	1	2	yes	direct product factor
360	C₂	C₂²×#88100	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	C₂²×#347
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13131051	C₂	direct (N × H)
4. C₂×#11718217	C₂	semidirect
5. C₂×#11718218	C₂	semidirect
10. C₂×#88106	C₄	semidirect
17. D₁₃⋊M₃₂	C₄	semidirect
20. C₂×D₁₃⋊C₁₆	C₄	semidirect
21. C₂×C₁₃×M₃₂	C₄	semidirect
22. C₂×#88052	C₄	semidirect
23. D₁₃×M₃₂	C₄	semidirect
26. C₂×C₁₆×D₁₃	C₄	semidirect
27. #11718218	C₂²	semidirect
28. #11718217	C₂²	semidirect
52. #88106	C₂×C₄	semidirect
53. C₂×#6092	C₂×C₄	semidirect
60. #88098	C₂×C₄	semidirect
62. D₁₃⋊C₁₆	C₂×C₄	semidirect
63. C₁₃×M₃₂	C₂×C₄	semidirect
64. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
65. #88052	C₂×C₄	semidirect
66. C₂×#6090	C₂×C₄	semidirect
67. #88038	C₂×C₄	semidirect
69. C₁₆×D₁₃	C₂×C₄	semidirect
117. #6092	C₂²×C₄	semidirect
118. C₁₃×C₁₆	C₂²×C₄	semidirect
119. #6090	C₂²×C₄	semidirect
174. C₂×M₃₂	#261	semidirect
191. C₂×D₁₃	#347	semidirect
232. M₃₂	C₂×#261	semidirect
237. C₂×C₁₆	C₂×#261	semidirect
245. C₂×C₁₃	C₂×#347	semidirect
249. D₁₃	C₂×#347	semidirect
298. C₁₆	C₂²×#261	semidirect
299. C₁₃	C₂²×#347	semidirect
359. C₂	#13131051	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
C₂²×#88100	2
#13131051	2
C₂×#11718217	2
C₂×#11718218	2
C₂×D₁₃×M₃₂	2
C₂²×#347	13

Subgroups — C2×#13131051 #16806745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131051 #16806745