Subgroups — C₂×#13131056 #16806751

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88098	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719089	C₂	1	1	yes	maximal, characteristic
3	#13131056	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	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	C₂²×#6092	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	C₂×#88098	C₄	1	4	yes	complemented
16	C₂×#88098	C₄	1	2	yes	complemented
17	C₂×#88098	C₂²	1	1	yes	characteristic
18	C₂×#88098	C₂²	1	1	yes	characteristic
19	C₂×#88098	C₂²	1	4	yes	—
20	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
21	C₂×#88052	C₄	1	1	yes	characteristic, complemented
22	D₁₃×M₃₂	C₄	1	4	yes	complemented
23	D₁₃×M₃₂	C₂²	1	4	yes	—
24	C₂×#88038	C₂²	1	2	yes	—
25	C₂×C₁₆×D₁₃	C₄	1	2	yes	complemented
26	#11718218	C₂²	1	8	yes	complemented
27	#11718217	C₂²	1	8	yes	complemented
28	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊#261	—	2	1	no	—
30	C₂×#6122	—	2	1	no	—
31	#88237	—	2	2	no	—
32	#88237	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊#261	—	2	1	no	—
34	C₂×C₄×#261	C₂×C₄	1	2	yes	—
35	D₁₃⋊M₁₆	—	2	2	no	—
36	D₁₃⋊M₁₆	C₂×C₄	1	4	yes	—
37	C₂×#6118	—	2	1	no	—
38	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₁₃	C₂³	1	2	yes	—
49	C₂×#6092	C₂×C₄	1	2	yes	—
50	C₂×#6092	C₂×C₄	1	2	yes	—
51	C₂×#6092	C₂×C₄	1	8	yes	complemented
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	2	yes	—
56	#88098	C₂×C₄	1	2	yes	—
57	#88098	C₂×C₄	1	8	yes	complemented
58	#88098	C₂³	1	2	yes	—
59	#88098	C₂³	1	2	yes	—
60	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
61	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
62	#88052	C₂×C₄	1	4	yes	complemented
63	C₂×#6090	C₂×C₄	1	2	yes	—
64	#88038	C₂×C₄	1	4	yes	—
65	#88038	C₂³	1	4	yes	—
66	C₁₆×D₁₃	C₂×C₄	1	8	yes	complemented
67	C₂×#2110	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
69	C₂²×#261	—	2	2	no	—
70	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
71	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₄²	1	2	yes	—
73	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
76	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
77	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
80	C₂²⋊#261	—	2	4	no	—
81	#6122	—	2	4	no	—
82	C₂×#1057	—	2	2	no	—
83	C₄⋊#261	—	2	4	no	—
84	C₄×#261	—	2	2	no	—
85	C₄×#261	C₂²×C₄	1	4	yes	—
86	#6118	—	2	4	no	—
87	D₁₃⋊C₈	—	2	2	no	—
88	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₄²	1	2	yes	—
90	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×#1055	C₄²	1	2	yes	—
94	C₂×#1055	C₄²	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
96	C₂×#1055	C₂²×C₄	1	2	yes	—
97	C₈×D₁₃	C₄²	1	2	yes	—
98	C₈×D₁₃	C₄²	1	2	yes	—
99	C₈×D₁₃	C₄²	1	2	yes	—
100	C₈×D₁₃	C₂²×C₄	1	2	yes	—
101	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₈×D₁₃	C₂²×C₄	1	2	yes	—
103	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₈×D₁₃	C₂²×C₄	1	2	yes	—
106	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
107	#6092	C₂²×C₄	1	4	yes	—
108	#6092	C₂²×C₄	1	4	yes	—
109	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
110	#6090	C₂²×C₄	1	4	yes	—
111	C₂²×M₃₂	—	13	1	no	—
112	C₂²×M₃₂	—	13	1	no	—
113	C₂×#405	—	13	1	no	—
114	#2110	—	13	8	no	—
115	C₂×#347	—	13	2	no	—
116	C₂×#347	—	13	2	no	—
117	C₂³×C₁₃	#97	1	1	yes	characteristic
118	C₂²×D₁₃	—	4	2	no	—
119	C₂²×D₁₃	#97	1	1	yes	characteristic
120	C₂²×D₁₃	#97	1	2	yes	—
121	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
123	C₂×#261	—	2	4	no	—
124	C₂×#261	—	4	4	no	—
125	C₂×C₄×C₁₃	#97	1	2	yes	—
126	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
129	C₂×Dic₁₃	#97	1	2	yes	—
130	C₂×Dic₁₃	#97	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
133	C₄×D₁₃	#97	1	2	yes	—
134	C₄×D₁₃	#97	1	2	yes	—
135	C₄×D₁₃	#97	1	2	yes	—
136	C₄×D₁₃	C₂×C₄²	1	2	yes	—
137	C₄×D₁₃	C₂×C₄²	1	1	yes	characteristic
138	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
139	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
140	C₄×D₁₃	C₂×M₁₆	1	1	yes	characteristic
141	C₄×D₁₃	C₂×M₁₆	1	2	yes	—
142	C₄×D₁₃	C₂³×C₄	1	1	yes	characteristic
143	#1057	—	2	4	no	—
144	C₈×C₁₃	C₂×C₄²	1	2	yes	—
145	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
146	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	#1055	C₂×C₄²	1	2	yes	—
148	#1055	C₂×C₄²	1	1	yes	characteristic
149	#1055	C₂³×C₄	1	1	yes	characteristic
150	C₂²×M₁₆	—	13	1	no	—
151	C₂³×C₈	—	13	1	no	—
152	C₂×#117	—	13	1	no	—
153	C₂×M₃₂	—	13	1	no	—
154	C₂×M₃₂	—	13	1	no	—
155	C₂×M₃₂	—	13	4	no	—
156	C₂×M₃₂	—	13	2	no	—
157	C₂×M₃₂	—	13	4	no	—
158	C₂×M₃₂	—	13	1	no	—
159	C₂×M₃₂	—	13	4	no	—
160	C₂×M₃₂	—	13	4	no	—
161	C₂×M₃₂	—	13	2	no	—
162	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
163	C₂²×C₁₆	—	13	2	no	—
164	C₂²×C₁₆	—	13	2	no	—
165	#405	—	13	4	no	—
166	#405	—	13	4	no	—
167	C₂×#97	—	13	2	no	—
168	C₂×C₄×C₈	—	13	2	no	—
169	#347	—	13	8	no	—
170	#347	—	13	8	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	C₂×#97	1	2	yes	—
173	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
174	C₂×D₁₃	—	2	2	no	—
175	C₂×D₁₃	—	2	2	no	—
176	C₂×D₁₃	—	4	2	no	—
177	C₂×D₁₃	—	4	2	no	—
178	C₂×D₁₃	C₂×#97	1	1	yes	characteristic
179	C₂×D₁₃	C₂×#97	1	2	yes	—
180	C₂×D₁₃	C₂²×M₁₆	1	1	yes	characteristic
181	#261	—	4	4	no	—
182	C₄×C₁₃	C₂×#97	1	2	yes	—
183	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
185	Dic₁₃	C₂×#97	1	1	yes	characteristic
186	Dic₁₃	C₂×#97	1	2	yes	—
187	Dic₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₂³×C₄	—	13	1	no	—
189	C₂×M₁₆	—	13	4	no	—
190	C₂×M₁₆	—	26	1	no	—
191	C₂×M₁₆	—	26	2	no	—
192	C₂×M₁₆	—	26	1	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	1	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	1	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	1	no	—
201	C₂²×C₈	—	13	2	no	—
202	C₂²×C₈	—	13	2	no	—
203	C₂²×C₈	C₂×#261	1	1	yes	characteristic
204	#117	—	13	4	no	—
205	#117	—	26	2	no	—
206	C₂×C₄⋊C₄	—	26	1	no	—
207	C₂×C₂²⋊C₄	—	26	1	no	—
208	C₂×C₄²	—	13	2	no	—
209	M₃₂	—	13	2	no	—
210	M₃₂	—	13	2	no	—
211	M₃₂	—	13	8	no	—
212	M₃₂	—	13	2	no	—
213	M₃₂	—	13	2	no	—
214	M₃₂	—	13	4	no	—
215	M₃₂	—	13	4	no	—
216	M₃₂	—	13	4	no	—
217	M₃₂	C₂×#261	1	4	yes	complemented
218	C₂×C₁₆	—	13	2	no	—
219	C₂×C₁₆	—	13	2	no	—
220	C₂×C₁₆	—	13	8	no	—
221	C₂×C₁₆	—	13	2	no	—
222	C₂×C₁₆	—	13	8	no	—
223	C₂×C₁₆	C₂×#261	1	2	yes	—
224	#97	—	13	4	no	—
225	#97	—	13	4	no	—
226	C₄×C₈	—	13	8	no	—
227	C₂×C₁₃	—	2	2	no	—
228	C₂×C₁₃	#2110	1	2	yes	complemented
229	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
230	D₁₃	—	2	1	no	—
231	D₁₃	—	2	1	no	—
232	D₁₃	—	2	2	no	—
233	C₂⁴	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	2	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	2	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	C₄×#261	1	1	yes	characteristic
245	M₁₆	—	26	4	no	—
246	M₁₆	—	26	4	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	4	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	C₄×#261	1	2	yes	—
265	C₂×C₈	C₄×#261	1	1	yes	characteristic
266	C₂×C₈	C₂²×#261	1	2	yes	—
267	C₂×C₈	C₂²×#261	1	1	yes	characteristic
268	C₄⋊C₄	—	26	4	no	—
269	C₂²⋊C₄	—	26	4	no	—
270	C₄²	—	13	4	no	—
271	C₄²	—	26	2	no	—
272	C₁₆	—	13	4	no	—
273	C₁₆	—	13	4	no	—
274	C₁₆	—	13	4	no	—
275	C₁₆	C₂²×#261	1	4	yes	—
276	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	13	1	no	—
278	C₂³	—	13	1	no	—
279	C₂³	—	13	2	no	—
280	C₂³	—	13	2	no	—
281	C₂³	—	52	2	no	—
282	C₂³	#88100	1	1	yes	characteristic
283	C₂×C₄	—	13	1	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	1	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	13	2	no	—
291	C₂×C₄	—	13	2	no	—
292	C₂×C₄	—	13	2	no	—
293	C₂×C₄	—	13	2	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	1	no	—
297	C₂×C₄	—	26	4	no	—
298	C₂×C₄	—	52	4	no	—
299	C₂×C₄	#88100	1	1	yes	characteristic
300	C₂×C₄	#88100	1	2	yes	—
301	C₂×C₄	C₂×C₄×#261	1	2	yes	—
302	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
303	C₈	—	13	1	no	—
304	C₈	—	13	2	no	—
305	C₈	—	13	1	no	—
306	C₈	—	26	4	no	—
307	C₈	C₂×C₄×#261	1	2	yes	—
308	C₈	C₂×C₄×#261	1	1	yes	characteristic
309	C₈	C₂³×#261	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	13	1	no	—
312	C₂²	—	13	1	no	—
313	C₂²	—	13	2	no	—
314	C₂²	—	26	2	no	—
315	C₂²	—	26	2	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	52	2	no	—
318	C₂²	C₂×#88100	1	2	yes	—
319	C₂²	C₂×#88100	1	1	yes	characteristic
320	C₄	—	13	1	no	—
321	C₄	—	13	1	no	—
322	C₄	—	13	2	no	—
323	C₄	—	52	4	no	—
324	C₄	C₂×#88100	1	2	yes	—
325	C₄	C₂×#88100	1	1	yes	characteristic
326	C₄	C₂²×C₄×#261	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	26	1	no	—
329	C₂	—	26	1	no	—
330	C₂	—	26	2	no	—
331	C₂	#13131056	1	2	yes	direct product factor
332	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	C₂×#2110
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. #13131056	C₂	direct (N × H)
4. C₂×#11718217	C₂	semidirect
5. C₂×#11718218	C₂	semidirect
10. C₂²×#6092	C₄	semidirect
15. C₂×#88098	C₄	semidirect
16. C₂×#88098	C₄	semidirect
20. C₂×C₁₃×M₃₂	C₄	semidirect
21. C₂×#88052	C₄	semidirect
22. D₁₃×M₃₂	C₄	semidirect
25. C₂×C₁₆×D₁₃	C₄	semidirect
26. #11718218	C₂²	semidirect
27. #11718217	C₂²	semidirect
51. C₂×#6092	C₂×C₄	semidirect
57. #88098	C₂×C₄	semidirect
60. C₁₃×M₃₂	C₂×C₄	semidirect
62. #88052	C₂×C₄	semidirect
66. C₁₆×D₁₃	C₂×C₄	semidirect
162. C₂×M₃₂	#261	semidirect
217. M₃₂	C₂×#261	semidirect
228. C₂×C₁₃	#2110	semidirect
276. C₁₃	C₂×#2110	semidirect
331. C₂	#13131056	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×#88098	2
C₂×#11719089	2
#13131056	2
C₂×#11718217	2
C₂×#11718218	2
C₂×D₁₃×M₃₂	2
C₂×#2110	13

Subgroups — C2×#13131056 #16806751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131056 #16806751