Subgroups — C₂×#13129876 #16789052

Order 3328.

630 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	C₂²×#88100	C₂	1	1	yes	maximal, characteristic
3	#13129876	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×#261	C₄	1	1	yes	characteristic
6	C₂²×D₁₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₄	1	2	yes	complemented
9	C₂×#88100	—	2	2	no	—
10	C₂×#88100	—	2	2	no	—
11	C₂×#88100	C₄	1	2	yes	—
12	C₂×#88100	C₂²	1	2	yes	—
13	C₂×#88098	—	2	2	no	—
14	C₂×#88098	—	2	2	no	—
15	C₂×#88098	C₄	1	2	yes	complemented
16	C₂×#88098	C₂²	1	2	yes	—
17	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic, complemented
18	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
19	C₂×#88038	—	2	2	no	—
20	C₂×#88038	—	2	2	no	—
21	C₂×#88038	C₄	1	2	yes	complemented
22	C₂×#88038	C₂²	1	2	yes	—
23	C₂³×#261	—	2	1	no	—
24	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1057	—	2	1	no	—
26	C₂×C₄×#261	—	2	2	no	—
27	C₂×C₄×#261	—	2	1	no	—
28	C₂×C₄×#261	C₈	1	4	yes	—
29	C₂×C₄×#261	C₂×C₄	1	2	yes	—
30	C₂×D₁₃⋊C₈	—	2	2	no	—
31	C₂×D₁₃⋊C₈	—	2	1	no	—
32	C₂×D₁₃⋊C₈	C₈	1	4	yes	—
33	C₂×D₁₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	D₄	1	2	yes	—
41	C₂×C₈×D₁₃	D₄	1	2	yes	—
42	C₂×C₈×D₁₃	Q₈	1	2	yes	—
43	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
44	C₂×#6092	—	2	4	no	—
45	C₂×#6092	C₂×C₄	1	4	yes	complemented
46	#88100	—	2	4	no	—
47	#88100	—	2	2	no	—
48	#88100	—	2	2	no	—
49	#88098	—	2	4	no	—
50	#88098	—	2	2	no	—
51	#88098	—	2	2	no	—
52	C₂×C₁₃×C₁₆	—	2	2	no	—
53	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
54	C₂×#6090	—	2	2	no	—
55	C₂×#6090	C₂×C₄	1	2	yes	complemented
56	#88038	—	2	4	no	—
57	#88038	—	2	2	no	—
58	#88038	—	2	2	no	—
59	C₂×#1375	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×#261	—	2	4	no	—
62	C₂²×#261	—	2	2	no	—
63	C₂²×#261	—	4	2	no	—
64	C₂²×#261	—	4	2	no	—
65	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
66	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
73	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×#1057	—	2	4	no	—
76	C₂×#1057	—	2	2	no	—
77	C₄×#261	—	2	4	no	—
78	C₄×#261	—	2	2	no	—
79	C₄×#261	C₂×C₈	1	4	yes	—
80	D₁₃⋊C₈	—	2	4	no	—
81	D₁₃⋊C₈	—	2	2	no	—
82	D₁₃⋊C₈	C₂×C₈	1	4	yes	—
83	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×#1055	C₄²	1	1	yes	characteristic
88	C₂×#1055	C₂²⋊C₄	1	2	yes	—
89	C₂×#1055	C₄⋊C₄	1	2	yes	—
90	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
91	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
92	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
93	C₈×D₁₃	C₂²⋊C₄	1	2	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
95	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
96	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
97	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
98	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
99	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
100	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
101	#6092	—	2	4	no	—
102	C₁₃×C₁₆	—	2	2	no	—
103	#6090	—	2	2	no	—
104	C₂²×M₃₂	—	13	1	no	—
105	C₂²×M₃₂	—	13	1	no	—
106	C₂²×#97	—	13	1	no	—
107	#1375	—	13	4	no	—
108	C₂³×C₁₃	#97	1	1	yes	characteristic
109	C₂²×D₁₃	—	2	2	no	—
110	C₂²×D₁₃	—	2	2	no	—
111	C₂²×D₁₃	#97	1	1	yes	characteristic
112	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
113	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
114	C₂²×D₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×#261	—	2	4	no	—
116	C₂×#261	—	4	2	no	—
117	C₂×#261	—	4	2	no	—
118	C₂×#261	—	4	8	no	—
119	C₂×C₄×C₁₃	#95	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
121	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×Dic₁₃	#97	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
125	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×Dic₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₄×D₁₃	#95	1	2	yes	—
128	C₄×D₁₃	C₄×C₈	1	2	yes	—
129	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
130	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
131	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
132	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
133	C₄×D₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
137	#1057	—	2	4	no	—
138	C₈×C₁₃	#95	1	2	yes	—
139	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	#1055	#95	1	2	yes	—
142	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂³×C₈	—	13	1	no	—
145	C₂³×C₈	—	13	1	no	—
146	C₂²×C₄²	—	13	1	no	—
147	C₂×M₃₂	—	13	2	no	—
148	C₂×M₃₂	—	13	2	no	—
149	C₂×M₃₂	—	13	2	no	—
150	C₂×M₃₂	—	13	2	no	—
151	C₂×M₃₂	—	26	2	no	—
152	C₂×M₃₂	—	26	2	no	—
153	C₂×M₃₂	—	26	2	no	—
154	C₂×M₃₂	—	26	2	no	—
155	C₂²×C₁₆	—	13	2	no	—
156	C₂²×C₁₆	—	13	1	no	—
157	C₂²×C₁₆	#261	1	1	yes	characteristic, complemented
158	C₂×#97	—	13	2	no	—
159	C₂×#97	—	13	2	no	—
160	C₂×#97	—	26	2	no	—
161	C₂×#97	—	26	2	no	—
162	C₂²×C₁₃	#336	1	2	yes	—
163	C₂²×C₁₃	#347	1	2	yes	—
164	C₂²×C₁₃	#347	1	2	yes	—
165	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	4	no	—
171	C₂×D₁₃	—	2	4	no	—
172	C₂×D₁₃	#336	1	2	yes	—
173	C₂×D₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
174	C₂×D₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
175	#261	—	4	4	no	—
176	C₄×C₁₃	#336	1	2	yes	—
177	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
178	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
179	Dic₁₃	#336	1	2	yes	—
180	Dic₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	Dic₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
182	C₂³×C₄	—	13	1	no	—
183	C₂³×C₄	—	26	1	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	—	13	4	no	—
195	C₂²×C₈	—	26	2	no	—
196	C₂²×C₈	—	26	1	no	—
197	C₂²×C₈	—	26	1	no	—
198	C₂²×C₈	C₂×#261	1	1	yes	characteristic
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	4	no	—
201	C₂×C₄²	—	26	2	no	—
202	C₂×C₄²	—	26	1	no	—
203	M₃₂	—	26	2	no	—
204	M₃₂	—	26	2	no	—
205	M₃₂	—	26	4	no	—
206	M₃₂	—	26	4	no	—
207	M₃₂	—	26	2	no	—
208	M₃₂	—	26	2	no	—
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	—	13	4	no	—
211	C₂×C₁₆	—	13	2	no	—
212	C₂×C₁₆	—	26	4	no	—
213	C₂×C₁₆	—	26	2	no	—
214	C₂×C₁₆	C₂×#261	1	2	yes	complemented
215	#97	—	26	4	no	—
216	#97	—	26	2	no	—
217	#97	—	26	2	no	—
218	C₂×C₁₃	#1375	1	4	yes	complemented
219	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
222	D₁₃	—	2	1	no	—
223	D₁₃	—	2	2	no	—
224	D₁₃	—	2	1	no	—
225	C₂⁴	—	13	1	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	2	no	—
228	C₂²×C₄	—	13	2	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	13	2	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	26	4	no	—
236	C₂²×C₄	—	26	2	no	—
237	C₂²×C₄	—	52	2	no	—
238	C₂²×C₄	—	52	2	no	—
239	C₂²×C₄	C₄×#261	1	1	yes	characteristic
240	C₂×C₈	—	13	2	no	—
241	C₂×C₈	—	13	2	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	13	2	no	—
245	C₂×C₈	—	13	2	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	1	no	—
251	C₂×C₈	—	13	1	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	—	13	4	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	4	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	26	2	no	—
259	C₂×C₈	C₄×#261	1	1	yes	characteristic
260	C₂×C₈	C₄⋊#261	1	2	yes	—
261	C₂×C₈	C₂²⋊#261	1	2	yes	—
262	C₂×C₈	C₂²×#261	1	1	yes	characteristic
263	C₄²	—	13	4	no	—
264	C₄²	—	26	2	no	—
265	C₄²	—	26	4	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	26	4	no	—
268	C₁₆	—	26	2	no	—
269	C₁₃	C₂×#1375	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	13	2	no	—
271	C₂³	—	13	2	no	—
272	C₂³	—	13	1	no	—
273	C₂³	—	13	1	no	—
274	C₂³	—	26	2	no	—
275	C₂³	—	26	2	no	—
276	C₂³	#88100	1	1	yes	characteristic
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	2	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	1	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	13	1	no	—
291	C₂×C₄	—	26	4	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	8	no	—
295	C₂×C₄	C₈×#261	1	2	yes	—
296	C₂×C₄	#88100	1	1	yes	characteristic
297	C₂×C₄	(C₂×#261)⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
299	C₈	—	13	1	no	—
300	C₈	—	13	2	no	—
301	C₈	—	13	1	no	—
302	C₈	—	26	4	no	—
303	C₈	(C₂×#261)⋊C₄	1	2	yes	—
304	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
305	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
306	C₂²	—	13	2	no	—
307	C₂²	—	13	1	no	—
308	C₂²	—	13	1	no	—
309	C₂²	—	26	1	no	—
310	C₂²	—	26	1	no	—
311	C₂²	—	26	1	no	—
312	C₂²	—	26	1	no	—
313	C₂²	—	26	4	no	—
314	C₂²	—	26	4	no	—
315	C₂²	#11718217	1	2	yes	—
316	C₂²	#11718218	1	2	yes	—
317	C₂²	(C₂×#261)⋊C₈	1	2	yes	—
318	C₂²	C₂×#88100	1	1	yes	characteristic
319	C₄	—	13	2	no	—
320	C₄	—	13	1	no	—
321	C₄	—	13	1	no	—
322	C₄	—	52	4	no	—
323	C₄	(C₂×#261)⋊C₈	1	2	yes	—
324	C₄	C₂×C₈×#261	1	1	yes	characteristic
325	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
326	C₂	—	26	1	no	—
327	C₂	—	26	2	no	—
328	C₂	—	26	1	no	—
329	C₂	#13129876	1	4	yes	direct product factor
330	C₂	C₂×(C₂×#261)⋊C₈	1	1	yes	characteristic
331	C₂	C₂×#11718218	1	1	yes	characteristic
332	C₂	C₂×#11718217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂³×C₁₃	#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	C₂×#1375
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #13129876	C₂	direct (N × H)
8. C₂²×#6092	C₄	semidirect
15. C₂×#88098	C₄	semidirect
17. C₂²×C₁₃×C₁₆	C₄	semidirect
18. C₂²×#6090	C₄	semidirect
21. C₂×#88038	C₄	semidirect
45. C₂×#6092	C₂×C₄	semidirect
53. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
55. C₂×#6090	C₂×C₄	semidirect
157. C₂²×C₁₆	#261	semidirect
214. C₂×C₁₆	C₂×#261	semidirect
218. C₂×C₁₃	#1375	semidirect
269. C₁₃	C₂×#1375	semidirect
329. C₂	#13129876	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
C₂²×#88100	2
#13129876	2
C₂²×#88038	2
C₂×#1375	13

Subgroups — C2×#13129876 #16789052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129876 #16789052