Subgroups — C₂×#13114289 #15845222

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718754	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718668	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
4	#13114289	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718071	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718138	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718078	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#476	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×#1055	—	2	1	no	—
12	C₂×#88123	C₂²	1	1	yes	characteristic
13	C₁₃×#476	—	2	2	no	—
14	C₁₃×#476	C₂²	1	4	yes	—
15	C₂×C₁₃×#106	—	2	1	no	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#102	—	2	1	no	—
18	C₂×#88072	—	2	1	no	—
19	C₂×C₂²⋊#1055	—	2	1	no	—
20	#11718754	—	2	2	no	—
21	#11718754	C₂²	1	4	yes	complemented
22	D₄×#1055	—	2	2	no	—
23	D₄×#1055	C₂²	1	4	yes	—
24	#11718668	—	2	2	no	—
25	#11718668	C₂²	1	4	yes	—
26	C₂×#88050	—	2	1	no	—
27	C₂×#88050	—	2	1	no	—
28	C₂×#88047	—	2	1	no	—
29	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	#11718138	C₂²	1	4	yes	complemented
33	#11718078	C₂²	1	4	yes	complemented
34	#11718071	C₂²	1	4	yes	complemented
35	C₂²×Q₈×C₁₃	—	2	1	no	—
36	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	—	2	1	no	—
38	C₂²×Dic₂₆	—	2	1	no	—
39	C₂²×C₈×C₁₃	—	2	1	no	—
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
44	C₄×D₄×C₁₃	—	2	1	no	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	D₄	1	2	yes	—
47	C₄×D₄×C₁₃	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
50	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊Dic₁₃	—	2	1	no	—
54	C₂×C₄⋊Dic₁₃	—	2	1	no	—
55	C₂²×#1055	—	2	1	no	—
56	C₂²×#1055	—	2	1	no	—
57	C₂²×#1055	—	2	4	no	—
58	C₂²×#1055	—	4	4	no	—
59	C₂²×#1055	D₄	1	1	yes	characteristic
60	C₂²×#1055	D₄	1	1	yes	characteristic
61	#88123	—	2	1	no	—
62	#88123	—	2	1	no	—
63	#88123	—	2	1	no	—
64	#88123	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	4	no	—
66	C₁₃×C₄⋊C₈	—	2	1	no	—
67	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₁₃×#102	—	2	4	no	—
69	#88072	—	2	4	no	—
70	C₂²⋊#1055	—	2	4	no	—
71	#88050	—	2	4	no	—
72	#88050	—	2	4	no	—
73	#88047	—	2	4	no	—
74	C₄⋊#1055	—	2	1	no	—
75	C₄⋊#1055	—	2	1	no	—
76	C₄⋊#1055	C₂³	1	2	yes	—
77	C₄⋊#1055	C₂³	1	2	yes	—
78	C₄×#1055	D₄	1	2	yes	—
79	C₄×#1055	C₂³	1	2	yes	—
80	C₂×#1683	—	13	1	no	maximal, Sylow
81	C₂⁴×C₁₃	—	2	1	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	4	1	no	—
84	C₂×Q₈×C₁₃	—	8	2	no	—
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	Q₁₆	1	4	yes	—
88	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃	—	2	1	no	—
90	C₂²×C₄×C₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₃	—	4	2	no	—
93	C₂²×C₄×C₁₃	—	4	2	no	—
94	C₂²×C₄×C₁₃	—	4	1	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂²×Dic₁₃	—	4	1	no	—
99	C₂×Dic₂₆	—	2	2	no	—
100	C₂×Dic₂₆	—	4	2	no	—
101	C₂×Dic₂₆	—	4	2	no	—
102	C₂×Dic₂₆	—	4	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
111	C₁₃×C₂²⋊C₄	—	2	2	no	—
112	C₁₃×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃	#41	1	1	yes	characteristic
116	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊Dic₁₃	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	—	4	1	no	—
120	C₄⋊Dic₁₃	—	4	1	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	2	no	—
125	C₂×#1055	—	4	8	no	—
126	C₂×#1055	—	4	2	no	—
127	C₂×#1055	—	4	2	no	—
128	C₂×#1055	C₂×D₄	1	2	yes	—
129	C₂×#1055	C₂×D₄	1	2	yes	—
130	C₂×#487	—	13	1	no	—
131	C₂×#476	D₁₃	1	1	yes	characteristic
132	C₂×#474	—	13	1	no	—
133	C₂×C₈×D₄	—	13	1	no	—
134	#1683	—	13	8	no	—
135	C₂×#335	—	13	1	no	—
136	C₂×#333	—	13	1	no	—
137	C₂×#332	—	13	1	no	—
138	C₂³×C₁₃	—	2	2	no	—
139	C₂³×C₁₃	—	2	4	no	—
140	C₂³×C₁₃	—	4	2	no	—
141	C₂³×C₁₃	—	4	2	no	—
142	C₂³×C₁₃	#121	1	1	yes	characteristic
143	Q₈×C₁₃	—	8	2	no	—
144	Q₈×C₁₃	—	8	2	no	—
145	D₄×C₁₃	—	2	2	no	—
146	D₄×C₁₃	—	2	4	no	—
147	D₄×C₁₃	C₂×Q₁₆	1	4	yes	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	4	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	8	2	no	—
156	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
160	C₂×C₄×C₁₃	#135	1	2	yes	—
161	C₂×C₄×C₁₃	#135	1	2	yes	—
162	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂×Dic₁₃	—	4	2	no	—
167	C₂×Dic₁₃	—	8	2	no	—
168	Dic₂₆	—	4	2	no	—
169	Dic₂₆	—	4	2	no	—
170	Dic₂₆	—	8	2	no	—
171	C₈×C₁₃	—	4	2	no	—
172	#1055	—	2	2	no	—
173	#1055	—	2	2	no	—
174	#1055	—	4	2	no	—
175	#1055	—	4	2	no	—
176	C₂³×C₈	—	26	1	no	—
177	C₂×C₄⋊Q₈	—	13	1	no	—
178	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
180	#487	—	13	4	no	—
181	#487	—	26	2	no	—
182	#476	—	2	2	no	—
183	#476	C₂×D₁₃	1	4	yes	—
184	#474	—	13	4	no	—
185	#474	—	26	2	no	—
186	C₈×D₄	—	13	4	no	—
187	C₈×D₄	—	26	2	no	—
188	C₂×#107	—	26	1	no	—
189	C₂×#106	—	2	1	no	—
190	C₂×C₄⋊C₈	—	13	1	no	—
191	C₂×C₄⋊C₈	—	13	1	no	—
192	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
193	C₂×#103	—	26	1	no	—
194	C₂×#103	—	26	1	no	—
195	C₂×#102	—	2	1	no	—
196	C₂×#102	—	26	1	no	—
197	C₂×C₂²⋊C₈	—	26	1	no	—
198	C₂×C₄×C₈	—	13	1	no	—
199	#335	—	13	4	no	—
200	#333	—	13	4	no	—
201	#332	—	13	4	no	—
202	C₂²×C₁₃	—	2	4	no	—
203	C₂²×C₁₃	—	4	8	no	—
204	C₂²×C₁₃	—	4	2	no	—
205	C₂²×C₁₃	—	4	2	no	—
206	C₂²×C₁₃	#463	1	2	yes	—
207	C₂²×C₁₃	#467	1	2	yes	—
208	C₂²×C₁₃	#471	1	2	yes	—
209	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	—	8	2	no	—
214	C₄×C₁₃	#463	1	2	yes	—
215	C₄×C₁₃	#467	1	2	yes	—
216	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
217	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
218	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
219	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	Dic₁₃	—	8	2	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×Q₈	—	26	1	no	—
223	C₂²×D₄	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂³×C₄	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	13	1	no	—
227	C₂²×C₈	—	13	1	no	—
228	C₂²×C₈	—	26	4	no	—
229	C₂²×C₈	—	26	1	no	—
230	C₂²×C₈	—	26	1	no	—
231	C₂²×C₈	—	52	4	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	13	2	no	—
236	C₄⋊Q₈	—	26	1	no	—
237	C₄⋊Q₈	—	26	1	no	—
238	C₄⋊Q₈	—	26	1	no	—
239	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
240	C₄×D₄	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	C₂²⋊D₁₃	1	2	yes	—
243	C₄×D₄	C₂²×D₁₃	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	26	1	no	—
247	C₂×C₄⋊C₄	—	26	1	no	—
248	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
251	#107	—	26	4	no	—
252	#106	—	2	4	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	13	2	no	—
255	C₄⋊C₈	—	13	2	no	—
256	C₄⋊C₈	—	26	1	no	—
257	C₄⋊C₈	—	26	1	no	—
258	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
259	#103	—	26	4	no	—
260	#103	—	26	4	no	—
261	#102	—	2	4	no	—
262	#102	—	26	4	no	—
263	C₂²⋊C₈	—	26	4	no	—
264	C₄×C₈	—	13	2	no	—
265	C₄×C₈	—	13	2	no	—
266	C₂×C₁₃	—	4	4	no	—
267	C₂×C₁₃	#1683	1	4	yes	complemented
268	C₂×C₁₃	C₂×#463	1	1	yes	characteristic
269	C₂×C₁₃	C₂×#467	1	1	yes	characteristic
270	C₂×C₁₃	C₂×#471	1	1	yes	characteristic
271	C₂⁴	—	2	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	8	2	no	—
275	C₂×Q₈	—	26	2	no	—
276	C₂×Q₈	—	52	1	no	—
277	C₂×Q₈	—	52	2	no	—
278	C₂×Q₈	—	52	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	#6107	1	4	yes	—
282	C₂×D₄	C₂×C₂²⋊D₁₃	1	2	yes	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	52	1	no	—
290	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
291	C₂²×C₄	#6129	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	—	13	2	no	—
296	C₂×C₈	—	13	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	—	52	8	no	—
302	C₂×C₈	—	52	2	no	—
303	C₂×C₈	—	52	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₄⋊C₄	—	52	1	no	—
313	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	D₄×D₁₃	1	1	yes	characteristic
317	C₄²	#6129	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
319	C₄²	C₂³×D₁₃	1	1	yes	characteristic
320	C₁₃	C₂×#1683	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	2	4	no	—
322	C₂³	—	2	2	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	#88193	1	1	yes	characteristic
326	Q₈	—	8	2	no	—
327	Q₈	—	8	2	no	—
328	Q₈	—	52	2	no	—
329	Q₈	—	52	2	no	—
330	Q₈	—	104	2	no	—
331	D₄	—	2	4	no	—
332	D₄	—	2	2	no	—
333	D₄	C₂×#6107	1	4	yes	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	104	2	no	—
344	C₂×C₄	#88168	1	2	yes	—
345	C₂×C₄	#88170	1	2	yes	—
346	C₂×C₄	#88193	1	1	yes	characteristic
347	C₂×C₄	#88193	1	1	yes	characteristic
348	C₂×C₄	#88193	1	1	yes	characteristic
349	C₂×C₄	C₂×#6107	1	2	yes	—
350	C₂×C₄	#88204	1	2	yes	—
351	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
352	C₂×C₄	C₂×#6129	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
354	C₈	—	4	2	no	—
355	C₈	—	26	2	no	—
356	C₈	—	26	2	no	—
357	C₈	—	52	2	no	—
358	C₈	—	52	2	no	—
359	C₂²	—	2	4	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	8	no	—
362	C₂²	—	4	2	no	—
363	C₂²	#11718712	1	2	yes	—
364	C₂²	#11718826	1	2	yes	—
365	C₂²	#11718861	1	2	yes	—
366	C₂²	C₂×#88193	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	104	2	no	—
372	C₄	#11718712	1	2	yes	—
373	C₄	#11718826	1	2	yes	—
374	C₄	C₂×#88168	1	1	yes	characteristic
375	C₄	C₂×#88170	1	1	yes	characteristic
376	C₄	C₂²×#6107	1	1	yes	characteristic
377	C₄	C₂×#88204	1	1	yes	characteristic
378	C₂	—	4	4	no	—
379	C₂	#13114289	1	4	yes	direct product factor
380	C₂	C₂×#11718826	1	1	yes	characteristic
381	C₂	C₂×#11718712	1	1	yes	characteristic
382	C₂	C₂×#11718861	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88193
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#476	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂²×D₄×C₁₃	D₄
hypercenter	C₂×#476	D₁₃
hypocenter	C₁₃	C₂×#1683
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#476

Complements

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

N	H ≅ G/N	type
1. C₂×#11718754	C₂	semidirect
4. #13114289	C₂	direct (N × H)
5. C₂×#11718071	C₂	semidirect
6. C₂×#11718138	C₂	semidirect
7. C₂×#11718078	C₂	semidirect
21. #11718754	C₂²	semidirect
32. #11718138	C₂²	semidirect
33. #11718078	C₂²	semidirect
34. #11718071	C₂²	semidirect
267. C₂×C₁₃	#1683	semidirect
320. C₁₃	C₂×#1683	semidirect
379. C₂	#13114289	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×#11718754	2
C₂×#11718668	2
C₂×D₄×#1055	2
#13114289	2
C₂×#11718071	2
C₂×#11718138	2
C₂×#11718078	2
C₂×C₁₃×#476	2
C₂×#1683	13

Subgroups — C2×#13114289 #15845222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114289 #15845222