Subgroups — C₂×#13113690 #15832455

Order 3328.

762 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#6105	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄×#1055	C₂	1	1	yes	maximal, characteristic
4	#13113690	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718071	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718048	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6105	C₄	1	1	yes	characteristic
11	C₂³×#1055	—	2	1	no	—
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88072	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊#1055	—	2	1	no	—
17	C₂×#88055	—	2	1	no	—
18	C₂×#88054	—	2	1	no	—
19	C₄×#6105	C₄	1	4	yes	—
20	C₄×#6105	C₂²	1	4	yes	—
21	D₄×#1055	—	2	2	no	—
22	D₄×#1055	C₂²	1	4	yes	—
23	C₂×#88050	C₄	1	1	yes	characteristic
24	C₂×#88048	C₄	1	1	yes	characteristic
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718492	—	2	2	no	—
29	#11718492	C₂²	1	4	yes	complemented
30	C₁₃×D₄⋊C₈	C₂²	1	4	yes	—
31	#11718071	C₂²	1	4	yes	complemented
32	#11718048	C₂²	1	4	yes	complemented
33	#11718047	C₂²	1	4	yes	complemented
34	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	—	2	1	no	—
36	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	—	2	1	no	—
39	C₄×D₄×C₁₃	—	2	1	no	—
40	C₄×D₄×C₁₃	—	2	1	no	—
41	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6105	—	2	4	no	—
47	C₂×#6105	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×#6101	—	2	1	no	—
50	C₂×C₄×Dic₁₃	—	2	1	no	—
51	C₂²×#1055	—	2	1	no	—
52	C₂²×#1055	—	2	4	no	—
53	C₂²×#1055	—	4	4	no	—
54	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
55	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
56	C₂²×#1055	D₄	1	1	yes	characteristic
57	C₂²×#1055	D₄	1	1	yes	characteristic
58	C₄×Dic₂₆	—	2	1	no	—
59	C₄×Dic₂₆	—	2	1	no	—
60	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₈	—	2	1	no	—
63	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
64	C₄×C₈×C₁₃	—	2	1	no	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	#88072	C₂×C₄	1	4	yes	—
67	C₂²⋊#1055	—	2	4	no	—
68	#88055	—	2	4	no	—
69	#88054	—	2	4	no	—
70	#88050	C₂×C₄	1	4	yes	—
71	#88048	C₂×C₄	1	4	yes	—
72	C₄⋊#1055	—	2	1	no	—
73	C₄⋊#1055	C₂³	1	2	yes	—
74	C₄×#1055	C₂×C₄	1	2	yes	—
75	C₄×#1055	D₄	1	2	yes	—
76	C₄×#1055	C₂³	1	2	yes	—
77	C₄×#1055	C₂³	1	2	yes	—
78	C₂×#1585	—	13	1	no	maximal, Sylow
79	C₂⁴×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	2	2	no	—
82	C₂×D₄×C₁₃	M₁₆	1	4	yes	—
83	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
92	C₂²×Dic₁₃	—	2	1	no	—
93	C₂²×Dic₁₃	—	2	1	no	—
94	C₂×Dic₂₆	—	2	2	no	—
95	C₂×Dic₂₆	—	2	2	no	—
96	C₂×Dic₂₆	—	2	1	no	—
97	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	—	2	4	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	2	2	no	—
105	C₁₃×C₂²⋊C₄	—	2	2	no	—
106	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	#41	1	1	yes	characteristic
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#6105	—	2	8	no	—
111	C₄⋊Dic₁₃	—	2	1	no	—
112	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
113	#6101	—	2	2	no	—
114	#6101	—	2	2	no	—
115	C₄×Dic₁₃	—	2	2	no	—
116	C₄×Dic₁₃	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	—	2	2	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	—	4	8	no	—
123	C₂×#1055	—	4	2	no	—
124	C₂×#1055	C₂²×C₄	1	2	yes	—
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	C₂×#1055	C₂×D₄	1	2	yes	—
127	C₂×#1055	C₂×D₄	1	2	yes	—
128	C₂×#446	—	13	1	no	—
129	C₂×C₄×SD₁₆	—	13	1	no	—
130	C₂×C₈×D₄	—	13	1	no	—
131	#1585	—	13	8	no	—
132	C₂×#335	—	13	1	no	—
133	C₂×Q₈⋊C₈	—	13	1	no	—
134	C₂×D₄⋊C₈	D₁₃	1	1	yes	characteristic
135	C₂×#322	—	13	1	no	—
136	C₂³×C₁₃	—	2	2	no	—
137	C₂³×C₁₃	—	2	4	no	—
138	C₂³×C₁₃	—	4	2	no	—
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
141	D₄×C₁₃	—	2	2	no	—
142	D₄×C₁₃	—	2	4	no	—
143	D₄×C₁₃	C₂×M₁₆	1	4	yes	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	4	no	—
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
154	C₂×C₄×C₁₃	#131	1	2	yes	—
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₁₃	#137	1	2	yes	—
157	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
160	C₂×Dic₁₃	—	2	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	C₂×Dic₁₃	—	4	2	no	—
163	C₂×Dic₁₃	—	4	2	no	—
164	Dic₂₆	—	2	2	no	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	4	2	no	—
167	C₈×C₁₃	—	2	4	no	—
168	C₈×C₁₃	—	4	2	no	—
169	#1055	—	2	2	no	—
170	#1055	—	2	2	no	—
171	#1055	—	2	2	no	—
172	#1055	—	2	2	no	—
173	#1055	—	4	2	no	—
174	C₂²×SD₁₆	—	13	1	no	—
175	C₂³×C₈	—	26	1	no	—
176	C₂×C₄×Q₈	—	13	1	no	—
177	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
178	#446	—	13	4	no	—
179	#446	—	26	2	no	—
180	C₄×SD₁₆	—	13	4	no	—
181	C₄×SD₁₆	—	13	4	no	—
182	C₈×D₄	—	13	4	no	—
183	C₈×D₄	—	26	2	no	—
184	C₂×#106	—	13	1	no	—
185	C₂×C₄⋊C₈	—	13	1	no	—
186	C₂×C₄⋊C₈	—	26	1	no	—
187	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
188	C₂×#103	—	13	1	no	—
189	C₂×#102	—	13	1	no	—
190	C₂×C₂²⋊C₈	—	26	1	no	—
191	C₂×#97	—	26	1	no	—
192	C₂×C₄×C₈	—	13	1	no	—
193	C₂×C₄×C₈	—	13	1	no	—
194	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
195	#335	—	13	4	no	—
196	Q₈⋊C₈	—	13	4	no	—
197	D₄⋊C₈	C₂×D₁₃	1	4	yes	—
198	#322	—	13	4	no	—
199	C₂²×C₁₃	—	2	4	no	—
200	C₂²×C₁₃	—	4	8	no	—
201	C₂²×C₁₃	—	4	2	no	—
202	C₂²×C₁₃	—	4	2	no	—
203	C₂²×C₁₃	#435	1	2	yes	—
204	C₂²×C₁₃	#440	1	2	yes	—
205	C₂²×C₁₃	#443	1	2	yes	—
206	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	4	2	no	—
210	C₄×C₁₃	#435	1	2	yes	—
211	C₄×C₁₃	#440	1	2	yes	—
212	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
213	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
214	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
216	Dic₁₃	—	4	2	no	—
217	Dic₁₃	—	4	2	no	—
218	C₂²×Q₈	—	13	1	no	—
219	C₂²×D₄	C₄×D₁₃	1	1	yes	characteristic
220	C₂³×C₄	—	2	1	no	—
221	C₂×SD₁₆	—	13	4	no	—
222	C₂×SD₁₆	—	26	4	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	13	1	no	—
227	C₂²×C₈	—	13	1	no	—
228	C₂²×C₈	—	13	1	no	—
229	C₂²×C₈	—	26	4	no	—
230	C₂²×C₈	—	26	1	no	—
231	C₂²×C₈	—	52	4	no	—
232	C₄×Q₈	—	13	2	no	—
233	C₄×Q₈	—	13	2	no	—
234	C₄×Q₈	—	26	1	no	—
235	C₄×Q₈	—	26	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	C₄×D₁₃	1	2	yes	—
239	C₄×D₄	C₂²×D₁₃	1	2	yes	—
240	C₂×C₄⋊C₄	—	13	1	no	—
241	C₂×C₄⋊C₄	—	26	1	no	—
242	C₂×C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	26	1	no	—
245	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
246	#106	—	13	4	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	13	2	no	—
249	C₄⋊C₈	—	26	4	no	—
250	C₄⋊C₈	—	26	1	no	—
251	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
252	#103	—	13	4	no	—
253	#102	—	13	4	no	—
254	C₂²⋊C₈	—	26	4	no	—
255	#97	—	26	4	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	—	13	2	no	—
261	C₄×C₈	C₂²×D₁₃	1	2	yes	—
262	C₂×C₁₃	—	4	4	no	—
263	C₂×C₁₃	#1585	1	4	yes	complemented
264	C₂×C₁₃	C₂×#435	1	1	yes	characteristic
265	C₂×C₁₃	C₂×#440	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
267	C₂⁴	—	2	1	no	—
268	C₂×Q₈	—	13	2	no	—
269	C₂×Q₈	—	26	1	no	—
270	C₂×Q₈	—	26	2	no	—
271	C₂×Q₈	—	26	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	#6094	1	4	yes	—
275	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	26	1	no	—
282	C₂²×C₄	—	26	1	no	—
283	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
285	C₂²×C₄	#6129	1	1	yes	characteristic
286	SD₁₆	—	26	8	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	13	2	no	—
292	C₂×C₈	—	13	2	no	—
293	C₂×C₈	—	13	2	no	—
294	C₂×C₈	—	13	2	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	52	8	no	—
301	C₂×C₈	—	52	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	13	2	no	—
304	C₄⋊C₄	—	26	1	no	—
305	C₄⋊C₄	—	26	2	no	—
306	C₄⋊C₄	—	26	2	no	—
307	C₄⋊C₄	C₂×C₄×D₁₃	1	2	yes	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₄²	—	26	2	no	—
311	C₄²	—	26	2	no	—
312	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₄²	D₄×D₁₃	1	1	yes	characteristic
314	C₄²	#6129	1	1	yes	characteristic
315	C₄²	C₂³×D₁₃	1	1	yes	characteristic
316	C₁₃	C₂×#1585	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	2	2	no	—
318	C₂³	—	2	4	no	—
319	C₂³	—	4	2	no	—
320	C₂³	—	4	2	no	—
321	C₂³	#88135	1	1	yes	characteristic
322	Q₈	—	26	2	no	—
323	Q₈	—	26	2	no	—
324	Q₈	—	52	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	4	no	—
327	D₄	C₂×#6094	1	4	yes	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	52	2	no	—
338	C₂×C₄	#88135	1	1	yes	characteristic
339	C₂×C₄	#88135	1	1	yes	characteristic
340	C₂×C₄	#88135	1	1	yes	characteristic
341	C₂×C₄	C₂×#6094	1	2	yes	—
342	C₂×C₄	#88161	1	2	yes	—
343	C₂×C₄	#88165	1	2	yes	—
344	C₂×C₄	#88169	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
347	C₂×C₄	C₂×#6129	1	1	yes	characteristic
348	C₈	—	2	4	no	—
349	C₈	—	4	2	no	—
350	C₈	—	26	2	no	—
351	C₈	—	26	2	no	—
352	C₈	—	26	2	no	—
353	C₈	—	26	2	no	—
354	C₈	—	52	2	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	8	no	—
358	C₂²	—	4	2	no	—
359	C₂²	#11718391	1	2	yes	—
360	C₂²	#11718418	1	2	yes	—
361	C₂²	#11718485	1	2	yes	—
362	C₂²	C₂×#88135	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	2	no	—
366	C₄	—	52	2	no	—
367	C₄	—	52	2	no	—
368	C₄	#11718391	1	2	yes	—
369	C₄	#11718418	1	2	yes	—
370	C₄	C₂²×#6094	1	1	yes	characteristic
371	C₄	C₂×#88161	1	1	yes	characteristic
372	C₄	C₂×#88165	1	1	yes	characteristic
373	C₄	C₂×#88169	1	1	yes	characteristic
374	C₂	—	4	4	no	—
375	C₂	#13113690	1	4	yes	direct product factor
376	C₂	C₂×#11718391	1	1	yes	characteristic
377	C₂	C₂×#11718418	1	1	yes	characteristic
378	C₂	C₂×#11718485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×D₄⋊C₈	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×D₄⋊C₈	D₁₃
hypocenter	C₁₃	C₂×#1585
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×D₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11718492	C₂	semidirect
4. #13113690	C₂	direct (N × H)
5. C₂×#11718071	C₂	semidirect
6. C₂×#11718048	C₂	semidirect
7. C₂×#11718047	C₂	semidirect
29. #11718492	C₂²	semidirect
31. #11718071	C₂²	semidirect
32. #11718048	C₂²	semidirect
33. #11718047	C₂²	semidirect
263. C₂×C₁₃	#1585	semidirect
316. C₁₃	C₂×#1585	semidirect
375. C₂	#13113690	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×#11718492	2
C₂×C₄×#6105	2
C₂×D₄×#1055	2
#13113690	2
C₂×#11718071	2
C₂×#11718048	2
C₂×#11718047	2
C₂×C₁₃×D₄⋊C₈	2
C₂×#1585	13

Subgroups — C2×#13113690 #15832455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113690 #15832455