Subgroups — C₂×#13125349 #16732523

Order 3328.

1942 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	C₂×#11719393	C₂	1	2	yes	maximal
2	#13125349	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718535	C₂	1	2	yes	maximal, complemented
4	C₂×#11718344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718725	C₂	1	2	yes	maximal, complemented
6	C₂×#11718516	C₂	1	2	yes	maximal, complemented
7	C₂×#11718541	C₂	1	2	yes	maximal
8	C₂×#11718521	C₂	1	2	yes	maximal
9	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#125	C₂²	1	2	yes	—
12	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
14	#11719393	—	2	8	no	—
15	#11719393	C₂²	1	16	yes	—
16	C₂×#88150	—	2	4	no	—
17	C₂×#88146	C₂²	1	4	yes	—
18	C₂×#88142	C₂²	1	4	yes	—
19	C₂×#88141	—	2	4	no	—
20	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88124	C₂²	1	2	yes	—
22	C₁₃×#499	C₂²	1	8	yes	complemented
23	C₂×C₁₃×#107	C₂²	1	2	yes	—
24	C₂×C₁₃×#106	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₂×#88058	C₂²	1	2	yes	—
27	C₂×#88057	C₂²	1	2	yes	—
28	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
30	#11718725	C₂²	1	16	yes	complemented
31	C₂×#88048	C₂²	1	4	yes	—
32	C₂×#88047	C₂²	1	4	yes	—
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	#11718541	C₂²	1	16	yes	—
35	#11718535	C₂²	1	16	yes	complemented
36	#11718521	C₂²	1	16	yes	—
37	#11718516	C₂²	1	16	yes	complemented
38	C₄×C₈×D₁₃	C₂²	1	8	yes	—
39	#11718344	C₂²	1	8	yes	complemented
40	C₂²×C₄×D₁₃	D₄	1	2	yes	—
41	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₆	—	2	4	no	—
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×#125	—	2	4	no	—
46	C₁₃×#125	C₂³	1	8	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×#6103	—	2	4	no	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
52	C₂×#6101	—	2	4	no	—
53	C₂×#6101	—	2	4	no	—
54	C₂×C₄×Dic₁₃	—	2	4	no	—
55	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
56	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₂²×#1055	C₂³	1	1	yes	characteristic
58	C₂²×#1055	C₂³	1	1	yes	characteristic
59	C₂×C₈×D₁₃	Q₈	1	8	yes	complemented
60	C₂×C₈×D₁₃	Q₈	1	8	yes	complemented
61	C₂×C₈×D₁₃	C₂³	1	4	yes	—
62	C₂×C₈×D₁₃	C₂³	1	4	yes	—
63	#88150	—	2	32	no	—
64	#88146	—	2	8	no	—
65	#88146	C₂³	1	16	yes	—
66	#88142	—	2	8	no	—
67	#88142	C₂³	1	16	yes	—
68	#88141	—	2	32	no	—
69	C₄²×D₁₃	D₄	1	4	yes	—
70	C₄²×D₁₃	C₂³	1	4	yes	—
71	#88124	—	2	4	no	—
72	#88124	C₂³	1	8	yes	—
73	C₁₃×#107	C₂³	1	8	yes	—
74	C₁₃×#106	C₂³	1	8	yes	—
75	C₄×C₈×C₁₃	C₂³	1	4	yes	—
76	#88058	C₂³	1	8	yes	—
77	#88057	C₂³	1	8	yes	—
78	C₈×Dic₁₃	C₂³	1	4	yes	—
79	C₈×Dic₁₃	C₂³	1	4	yes	—
80	#88048	C₂³	1	16	yes	—
81	#88047	C₂³	1	16	yes	—
82	C₄×#1055	C₂³	1	4	yes	—
83	C₂×#3160	—	13	1	no	maximal, Sylow
84	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₁₃	—	2	4	no	—
89	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₁₃	—	2	8	no	—
92	C₂×C₄×D₁₃	C₂×D₄	1	8	yes	—
93	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
95	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
96	C₂×C₄×D₁₃	C₂⁴	1	2	yes	—
97	C₂×Dic₂₆	—	2	16	no	—
98	C₂×Dic₂₆	—	4	16	no	—
99	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
102	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
103	C₁₃×C₄⋊C₄	—	2	16	no	—
104	C₁₃×C₄⋊C₄	—	2	4	no	—
105	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
106	C₄²×C₁₃	C₂×D₄	1	2	yes	—
107	C₄²×C₁₃	C₂⁴	1	2	yes	—
108	#6103	—	2	16	no	—
109	C₄⋊Dic₁₃	—	2	4	no	—
110	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
111	#6101	—	2	16	no	—
112	#6101	—	2	16	no	—
113	C₄×Dic₁₃	—	2	16	no	—
114	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
115	C₄×Dic₁₃	C₂×D₄	1	8	yes	—
116	C₄×Dic₁₃	C₂⁴	1	2	yes	—
117	C₂×#1055	C₂×Q₈	1	4	yes	—
118	C₂×#1055	C₂×Q₈	1	4	yes	—
119	C₂×#1055	C₂⁴	1	2	yes	—
120	C₂×#1055	C₂⁴	1	2	yes	—
121	C₈×D₁₃	C₂×Q₈	1	16	yes	complemented
122	C₈×D₁₃	C₂×Q₈	1	16	yes	complemented
123	C₂×#533	—	13	2	no	—
124	#3160	—	13	16	no	—
125	C₂×#500	—	13	2	no	—
126	C₂×#499	—	13	1	no	—
127	C₂×#499	—	13	2	no	—
128	C₂×#499	D₁₃	1	1	yes	characteristic, complemented
129	C₂×#498	—	13	2	no	—
130	C₂×#427	—	13	2	no	—
131	C₂×#427	—	13	2	no	—
132	C₂²×C₄×C₈	—	13	1	no	—
133	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²×D₁₃	—	2	4	no	—
135	C₂²×D₁₃	C₄⋊Q₈	1	4	yes	—
136	C₂²×D₁₃	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	8	no	—
139	C₂×C₄×C₁₃	—	4	8	no	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
145	C₂×Dic₁₃	—	2	8	no	—
146	C₂×Dic₁₃	—	4	8	no	—
147	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
148	C₂×Dic₁₃	#135	1	8	yes	—
149	C₂×Dic₁₃	C₂²×D₄	1	2	yes	—
150	C₂×Dic₁₃	C₂²×D₄	1	4	yes	—
151	C₄×D₁₃	—	2	16	no	—
152	C₄×D₁₃	C₄⋊Q₈	1	8	yes	—
153	C₄×D₁₃	C₂²×Q₈	1	4	yes	—
154	C₄×D₁₃	C₂²×Q₈	1	4	yes	—
155	Dic₂₆	—	4	32	no	—
156	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
157	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
158	#1055	C₂²×Q₈	1	4	yes	—
159	#1055	C₂²×Q₈	1	4	yes	—
160	C₂³×C₈	—	13	1	no	—
161	C₂³×C₈	—	13	1	no	—
162	#533	—	13	16	no	—
163	#533	—	26	8	no	—
164	C₂×C₄⋊Q₈	—	13	4	no	—
165	C₂×#125	—	13	2	no	—
166	C₂×#125	C₂×D₁₃	1	2	yes	—
167	C₂×C₂²⋊Q₈	—	26	4	no	—
168	C₂×C₄×Q₈	—	26	4	no	—
169	C₂×#117	—	13	4	no	—
170	C₂²×C₄²	—	13	1	no	—
171	#500	—	13	16	no	—
172	#499	—	13	16	no	—
173	#499	—	13	8	no	—
174	#499	C₂×D₁₃	1	8	yes	complemented
175	#498	—	13	16	no	—
176	#427	—	13	16	no	—
177	#427	—	13	16	no	—
178	C₂×#107	—	13	2	no	—
179	C₂×#107	—	13	4	no	—
180	C₂×#107	C₂×D₁₃	1	2	yes	—
181	C₂×#106	—	13	2	no	—
182	C₂×#106	—	13	4	no	—
183	C₂×#106	C₂×D₁₃	1	2	yes	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	13	1	no	—
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	—	13	8	no	—
188	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
189	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
190	C₂²×C₁₃	C₂×#135	1	4	yes	—
191	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
192	C₂×D₁₃	—	2	8	no	—
193	C₂×D₁₃	—	2	4	no	—
194	C₂×D₁₃	C₂×C₄⋊Q₈	1	4	yes	—
195	C₄×C₁₃	—	2	4	no	—
196	C₄×C₁₃	—	4	8	no	—
197	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
198	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
199	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
200	Dic₁₃	—	4	8	no	—
201	Dic₁₃	C₂×C₄⋊Q₈	1	4	yes	—
202	Dic₁₃	C₂×#135	1	8	yes	—
203	C₂²×Q₈	—	26	4	no	—
204	C₂³×C₄	—	13	1	no	—
205	C₂³×C₄	—	13	2	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	8	no	—
209	C₂²×C₈	—	13	8	no	—
210	C₂²×C₈	—	13	4	no	—
211	C₂²×C₈	—	13	4	no	—
212	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	13	16	no	—
215	C₄⋊Q₈	—	26	8	no	—
216	#125	—	2	4	no	—
217	#125	—	13	8	no	—
218	#125	—	26	4	no	—
219	#125	C₂²×D₁₃	1	8	yes	—
220	C₂²⋊Q₈	—	26	32	no	—
221	C₄×Q₈	—	26	32	no	—
222	#117	—	13	16	no	—
223	#117	—	26	8	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
225	C₂×C₄⋊C₄	—	13	4	no	—
226	C₂×C₄⋊C₄	—	26	4	no	—
227	C₂×C₄⋊C₄	—	26	4	no	—
228	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
229	C₂×C₂²⋊C₄	—	26	4	no	—
230	C₂×C₄²	—	13	2	no	—
231	C₂×C₄²	—	13	1	no	—
232	C₂×C₄²	—	13	4	no	—
233	C₂×C₄²	—	13	4	no	—
234	C₂×C₄²	—	26	4	no	—
235	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	#107	—	13	8	no	—
237	#107	—	13	16	no	—
238	#107	C₂²×D₁₃	1	8	yes	—
239	#106	—	13	8	no	—
240	#106	—	13	16	no	—
241	#106	C₂²×D₁₃	1	8	yes	—
242	C₄×C₈	—	13	4	no	—
243	C₄×C₈	—	13	4	no	—
244	C₄×C₈	—	13	4	no	—
245	C₄×C₈	C₂²×D₁₃	1	4	yes	—
246	C₂×C₁₃	#3160	1	4	yes	complemented
247	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₁₃	C₂²×#135	1	2	yes	—
249	D₁₃	—	2	4	no	—
250	C₂⁴	—	13	1	no	—
251	C₂×Q₈	—	26	16	no	—
252	C₂×Q₈	—	52	16	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	4	no	—
258	C₂²×C₄	—	13	4	no	—
259	C₂²×C₄	—	13	2	no	—
260	C₂²×C₄	—	13	8	no	—
261	C₂²×C₄	—	26	8	no	—
262	C₂²×C₄	—	26	4	no	—
263	C₂²×C₄	D₄×D₁₃	1	2	yes	—
264	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	13	4	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	16	no	—
270	C₂×C₈	—	13	16	no	—
271	C₂×C₈	Q₈×D₁₃	1	4	yes	—
272	C₂×C₈	Q₈×D₁₃	1	4	yes	—
273	C₂×C₈	C₂³×D₁₃	1	2	yes	—
274	C₂×C₈	C₂³×D₁₃	1	2	yes	—
275	C₄⋊C₄	—	2	16	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	13	8	no	—
278	C₄⋊C₄	—	26	16	no	—
279	C₄⋊C₄	—	26	4	no	—
280	C₄⋊C₄	—	26	16	no	—
281	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
282	C₂²⋊C₄	—	26	16	no	—
283	C₄²	—	13	8	no	—
284	C₄²	—	13	2	no	—
285	C₄²	—	13	2	no	—
286	C₄²	—	26	16	no	—
287	C₄²	D₄×D₁₃	1	2	yes	—
288	C₄²	C₂³×D₁₃	1	2	yes	—
289	C₁₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
290	C₂³	—	13	4	no	—
291	C₂³	—	13	2	no	—
292	C₂³	—	26	4	no	—
293	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
294	Q₈	—	52	32	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	13	4	no	—
299	C₂×C₄	—	13	8	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	13	8	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	26	16	no	—
306	C₂×C₄	—	26	8	no	—
307	C₂×C₄	—	52	8	no	—
308	C₂×C₄	C₂×D₄×D₁₃	1	4	yes	—
309	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
312	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
313	C₈	—	13	4	no	—
314	C₈	—	13	4	no	—
315	C₈	C₂×Q₈×D₁₃	1	4	yes	—
316	C₈	C₂×Q₈×D₁₃	1	4	yes	—
317	C₂²	—	13	4	no	—
318	C₂²	—	26	4	no	—
319	C₂²	—	26	8	no	—
320	C₂²	D₁₃×C₄⋊Q₈	1	2	yes	—
321	C₂²	D₁₃×#135	1	4	yes	—
322	C₂²	C₂²×D₄×D₁₃	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	13	8	no	—
326	C₄	—	13	4	no	—
327	C₄	—	52	8	no	—
328	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
329	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
331	C₂	—	26	4	no	—
332	C₂	#13125349	1	4	yes	direct product factor
333	C₂	C₂×D₁₃×C₄⋊Q₈	1	1	yes	characteristic
334	C₂	C₂×D₁₃×#135	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#499	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂³×D₁₃	C₂×D₄
hypercenter	C₂×#499	D₁₃
hypocenter	C₁₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #13125349	C₂	direct (N × H)
3. C₂×#11718535	C₂	semidirect
4. C₂×#11718344	C₂	semidirect
5. C₂×#11718725	C₂	semidirect
6. C₂×#11718516	C₂	semidirect
10. C₂×C₁₃×#499	C₂	semidirect
22. C₁₃×#499	C₂²	semidirect
30. #11718725	C₂²	semidirect
35. #11718535	C₂²	semidirect
37. #11718516	C₂²	semidirect
39. #11718344	C₂²	semidirect
59. C₂×C₈×D₁₃	Q₈	semidirect
60. C₂×C₈×D₁₃	Q₈	semidirect
121. C₈×D₁₃	C₂×Q₈	semidirect
122. C₈×D₁₃	C₂×Q₈	semidirect
128. C₂×#499	D₁₃	semidirect
174. #499	C₂×D₁₃	semidirect
246. C₂×C₁₃	#3160	semidirect
289. C₁₃	C₂×#3160	semidirect
332. C₂	#13125349	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×#11719393	2
#13125349	2
C₂×#11718535	2
C₂×#11718344	2
C₂×#11718725	2
C₂×#11718516	2
C₂×#11718541	2
C₂×#11718521	2
C₂×C₄×C₈×D₁₃	2
C₂×C₁₃×#499	2
C₂×#3160	13

Subgroups — C2×#13125349 #16732523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13125349 #16732523