Subgroups — C₂×#13125422 #16732663

Order 3328.

2006 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719175	C₂	1	2	yes	maximal
2	#13125422	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718342	C₂	1	2	yes	maximal, complemented
6	C₂×#11718777	C₂	1	4	yes	maximal
7	C₂×C₄×Dic₅₂	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88180	—	2	2	no	—
12	C₂×#88178	—	2	2	no	—
13	C₂×#88178	C₂²	1	4	yes	—
14	C₂²×Dic₅₂	C₂²	1	4	yes	—
15	#11719175	—	2	8	no	—
16	#11719175	C₂²	1	16	yes	—
17	C₂×#88124	—	2	2	no	—
18	C₂×#88123	C₂²	1	2	yes	—
19	C₂×#88123	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₆	—	2	2	no	—
21	C₂×C₄×Dic₂₆	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	#11718777	C₂²	1	32	yes	—
27	C₂×#88058	C₂²	1	4	yes	—
28	C₂×#88056	C₂²	1	8	yes	—
29	#11718360	C₂²	1	8	yes	complemented
30	C₄×Dic₅₂	C₂²	1	32	yes	complemented
31	#11718343	C₂²	1	8	yes	complemented
32	#11718342	C₂²	1	16	yes	complemented
33	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₂₆	—	2	2	no	—
37	C₂²×Dic₂₆	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	—	2	8	no	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	8	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
48	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
49	#88180	—	2	16	no	—
50	C₂×C₄⋊Dic₁₃	—	2	2	no	—
51	C₂×C₄⋊Dic₁₃	—	2	2	no	—
52	C₂×C₄⋊Dic₁₃	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
54	#88178	—	2	8	no	—
55	#88178	—	2	16	no	—
56	#88178	C₂³	1	16	yes	—
57	C₂×#6101	—	2	4	no	—
58	C₂×#6101	—	2	4	no	—
59	C₂×C₄×Dic₁₃	—	2	4	no	—
60	C₂×Dic₅₂	—	2	16	no	—
61	C₂×Dic₅₂	C₂³	1	16	yes	—
62	#88124	—	2	16	no	—
63	#88123	—	2	4	no	—
64	#88123	—	2	4	no	—
65	#88123	C₂³	1	8	yes	—
66	#88123	C₂³	1	8	yes	—
67	C₄×Dic₂₆	—	2	16	no	—
68	C₄×Dic₂₆	—	2	8	no	—
69	C₄×Dic₂₆	C₂³	1	16	yes	—
70	C₄×C₈×C₁₃	C₂³	1	4	yes	—
71	C₄×C₈×C₁₃	C₂³	1	4	yes	—
72	C₄×C₈×C₁₃	C₂³	1	8	yes	—
73	#88058	C₂³	1	16	yes	—
74	#88056	C₂³	1	32	yes	—
75	C₂×#3170	—	13	1	no	maximal, Sylow
76	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	Q₁₆	1	8	yes	complemented
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₁₃	—	2	4	no	—
91	C₂²×Dic₁₃	—	2	4	no	—
92	C₂×Dic₂₆	—	2	16	no	—
93	C₂×Dic₂₆	—	2	4	no	—
94	C₂×Dic₂₆	—	2	8	no	—
95	C₂×Dic₂₆	—	4	8	no	—
96	C₂×Dic₂₆	C₂⁴	1	8	yes	—
97	C₂×C₈×C₁₃	—	2	4	no	—
98	C₂×C₈×C₁₃	—	2	16	no	—
99	C₂×C₈×C₁₃	#41	1	8	yes	—
100	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
101	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
102	C₄²×C₁₃	C₂×D₄	1	4	yes	—
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃	C₂×D₄	1	2	yes	—
105	C₄²×C₁₃	C₂×D₄	1	4	yes	—
106	C₄²×C₁₃	C₂⁴	1	2	yes	—
107	C₄²×C₁₃	C₂⁴	1	2	yes	—
108	C₂²⋊Dic₁₃	—	2	16	no	—
109	C₄⋊Dic₁₃	—	2	4	no	—
110	C₄⋊Dic₁₃	—	2	8	no	—
111	C₄⋊Dic₁₃	—	2	8	no	—
112	C₄⋊Dic₁₃	—	2	8	no	—
113	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
114	#6101	—	2	16	no	—
115	#6101	—	2	16	no	—
116	C₄×Dic₁₃	—	2	16	no	—
117	Dic₅₂	—	2	32	no	—
118	C₂×#533	—	13	2	no	—
119	#3170	—	13	16	no	—
120	C₂×#500	—	13	1	no	—
121	C₂×#494	—	13	1	no	—
122	C₂×#487	—	13	2	no	—
123	C₂×#467	—	13	4	no	—
124	C₂×C₄×Q₁₆	—	13	4	no	—
125	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
126	C₂³×C₁₃	—	2	2	no	—
127	C₂³×C₁₃	—	2	2	no	—
128	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
129	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
130	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	8	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
137	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
138	C₂×C₄×C₁₃	C₂×Q₁₆	1	16	yes	complemented
139	C₂×C₄×C₁₃	#135	1	4	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
147	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
148	C₂×Dic₁₃	—	2	8	no	—
149	C₂×Dic₁₃	—	2	8	no	—
150	C₂×Dic₁₃	—	4	8	no	—
151	C₂×Dic₁₃	—	4	8	no	—
152	Dic₂₆	—	2	16	no	—
153	Dic₂₆	—	4	8	no	—
154	Dic₂₆	—	4	16	no	—
155	C₈×C₁₃	—	2	4	no	—
156	C₈×C₁₃	C₂×#41	1	8	yes	—
157	C₂²×Q₁₆	—	13	4	no	—
158	C₂³×C₈	C₂×D₁₃	1	2	yes	—
159	#533	—	13	16	no	—
160	#533	—	26	8	no	—
161	C₂×C₄⋊Q₈	—	13	2	no	—
162	C₂×C₄⋊Q₈	—	13	2	no	—
163	C₂×#125	—	26	2	no	—
164	C₂×C₂²⋊Q₈	—	13	4	no	—
165	C₂×C₂²⋊Q₈	—	26	2	no	—
166	C₂×C₄×Q₈	—	13	4	no	—
167	C₂×C₄×Q₈	—	26	2	no	—
168	C₂×#117	—	26	2	no	—
169	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
170	#500	—	13	8	no	—
171	#494	—	13	8	no	—
172	#487	—	13	16	no	—
173	#467	—	13	32	no	—
174	C₄×Q₁₆	—	13	32	no	—
175	C₂×#107	—	13	4	no	—
176	C₂×#103	—	13	8	no	—
177	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₁₃	1	8	yes	—
180	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
181	C₂²×C₁₃	—	2	8	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	2	2	no	—
184	C₂²×C₁₃	#532	1	2	yes	—
185	C₂²×C₁₃	C₂²×Q₁₆	1	4	yes	—
186	C₂²×C₁₃	C₂²×Q₁₆	1	2	yes	—
187	C₂²×C₁₃	C₂×#135	1	2	yes	—
188	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	#532	1	2	yes	—
192	C₄×C₁₃	C₂²×Q₁₆	1	4	yes	—
193	C₄×C₁₃	C₂²×Q₁₆	1	4	yes	—
194	C₄×C₁₃	C₂×#135	1	4	yes	—
195	C₄×C₁₃	C₂²×#41	1	2	yes	—
196	Dic₁₃	—	4	8	no	—
197	Dic₁₃	—	4	8	no	—
198	C₂²×Q₈	—	13	4	no	—
199	C₂²×Q₈	—	26	2	no	—
200	C₂³×C₄	D₅₂	1	1	yes	characteristic
201	C₂³×C₄	D₅₂	1	1	yes	characteristic
202	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×Q₁₆	—	13	16	no	—
204	C₂×Q₁₆	—	26	16	no	—
205	C₂²×C₈	—	2	8	no	—
206	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₁₃	1	8	yes	—
208	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
209	C₄⋊Q₈	—	13	8	no	—
210	C₄⋊Q₈	—	13	8	no	—
211	C₄⋊Q₈	—	26	4	no	—
212	C₄⋊Q₈	—	26	4	no	—
213	#125	—	26	16	no	—
214	C₂²⋊Q₈	—	13	16	no	—
215	C₂²⋊Q₈	—	26	8	no	—
216	C₂²⋊Q₈	—	26	16	no	—
217	C₄×Q₈	—	13	16	no	—
218	C₄×Q₈	—	26	8	no	—
219	C₄×Q₈	—	26	16	no	—
220	#117	—	26	16	no	—
221	C₂×C₄⋊C₄	—	13	4	no	—
222	C₂×C₄⋊C₄	—	26	2	no	—
223	C₂×C₄⋊C₄	—	26	2	no	—
224	C₂×C₄⋊C₄	—	26	4	no	—
225	C₂×C₄⋊C₄	—	26	4	no	—
226	C₂×C₄⋊C₄	—	26	2	no	—
227	C₂×C₂²⋊C₄	—	26	4	no	—
228	C₂×C₄²	—	26	4	no	—
229	C₂×C₄²	D₅₂	1	1	yes	characteristic
230	C₂×C₄²	D₅₂	1	4	yes	—
231	C₂×C₄²	D₅₂	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	#107	—	13	16	no	—
236	#103	—	13	32	no	—
237	C₄×C₈	C₂²×D₁₃	1	8	yes	—
238	C₄×C₈	C₂²×D₁₃	1	4	yes	—
239	C₄×C₈	C₂²×D₁₃	1	4	yes	—
240	C₂×C₁₃	—	2	4	no	—
241	C₂×C₁₃	#3170	1	4	yes	complemented
242	C₂×C₁₃	C₂×#532	1	1	yes	characteristic
243	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
244	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
245	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
246	C₂×Q₈	—	13	8	no	—
247	C₂×Q₈	—	26	8	no	—
248	C₂×Q₈	—	26	4	no	—
249	C₂×Q₈	—	26	16	no	—
250	C₂×Q₈	—	52	8	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	26	4	no	—
254	C₂²×C₄	—	26	4	no	—
255	C₂²×C₄	Dic₅₂	1	8	yes	complemented
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₅₂	1	2	yes	—
258	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₅₂	1	4	yes	—
260	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₅₂	1	4	yes	—
263	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
265	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
266	Q₁₆	—	26	32	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	16	no	—
269	C₂×C₈	#6127	1	8	yes	—
270	C₂×C₈	C₂³×D₁₃	1	4	yes	—
271	C₂×C₈	C₂³×D₁₃	1	4	yes	—
272	C₄⋊C₄	—	13	8	no	—
273	C₄⋊C₄	—	26	4	no	—
274	C₄⋊C₄	—	26	8	no	—
275	C₄⋊C₄	—	26	8	no	—
276	C₄⋊C₄	—	26	16	no	—
277	C₄⋊C₄	—	26	16	no	—
278	C₄⋊C₄	—	26	8	no	—
279	C₂²⋊C₄	—	26	16	no	—
280	C₄²	—	26	16	no	—
281	C₄²	C₂×D₅₂	1	4	yes	—
282	C₄²	C₂×D₅₂	1	2	yes	—
283	C₄²	C₂×D₅₂	1	4	yes	—
284	C₄²	C₂×D₅₂	1	2	yes	—
285	C₄²	C₂³×D₁₃	1	2	yes	—
286	C₄²	C₂³×D₁₃	1	2	yes	—
287	C₁₃	C₂×#3170	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	C₂×Dic₅₂	1	4	yes	—
291	C₂³	C₂²×D₅₂	1	2	yes	—
292	C₂³	C₂²×D₅₂	1	1	yes	characteristic
293	Q₈	—	26	16	no	—
294	Q₈	—	52	8	no	—
295	Q₈	—	52	16	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	26	8	no	—
302	C₂×C₄	—	26	8	no	—
303	C₂×C₄	—	52	8	no	—
304	C₂×C₄	—	52	8	no	—
305	C₂×C₄	#88158	1	4	yes	—
306	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
307	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
308	C₂×C₄	C₂×Dic₅₂	1	16	yes	complemented
309	C₂×C₄	C₂²×D₅₂	1	2	yes	—
310	C₂×C₄	C₂²×D₅₂	1	2	yes	—
311	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₅₂	1	2	yes	—
313	C₂×C₄	C₂²×D₅₂	1	2	yes	—
314	C₂×C₄	C₂²×D₅₂	1	2	yes	—
315	C₂×C₄	C₂×#6127	1	4	yes	—
316	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
317	C₈	—	2	4	no	—
318	C₈	C₂×#6127	1	8	yes	—
319	C₂²	—	2	8	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#11719182	1	2	yes	—
323	C₂²	C₂×#88158	1	2	yes	—
324	C₂²	C₂²×Dic₅₂	1	4	yes	—
325	C₂²	C₂²×Dic₅₂	1	2	yes	—
326	C₂²	C₂³×D₅₂	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	52	8	no	—
330	C₄	—	52	8	no	—
331	C₄	#11719182	1	2	yes	—
332	C₄	C₂×#88158	1	4	yes	—
333	C₄	C₂²×Dic₅₂	1	4	yes	—
334	C₄	C₂²×Dic₅₂	1	4	yes	—
335	C₄	C₂²×#6127	1	2	yes	—
336	C₂	—	2	4	no	—
337	C₂	#13125422	1	4	yes	direct product factor
338	C₂	C₂×#11719182	1	1	yes	characteristic
339	C₂	C₂³×Dic₅₂	1	1	yes	characteristic
340	C₂	C₂²×#88158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#3170
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13125422	C₂	direct (N × H)
3. C₂×#11718343	C₂	semidirect
4. C₂×#11718360	C₂	semidirect
5. C₂×#11718342	C₂	semidirect
7. C₂×C₄×Dic₅₂	C₂	semidirect
29. #11718360	C₂²	semidirect
30. C₄×Dic₅₂	C₂²	semidirect
31. #11718343	C₂²	semidirect
32. #11718342	C₂²	semidirect
79. C₂²×C₄×C₁₃	Q₁₆	semidirect
138. C₂×C₄×C₁₃	C₂×Q₁₆	semidirect
241. C₂×C₁₃	#3170	semidirect
255. C₂²×C₄	Dic₅₂	semidirect
287. C₁₃	C₂×#3170	semidirect
308. C₂×C₄	C₂×Dic₅₂	semidirect
337. C₂	#13125422	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×#11719175	2
#13125422	2
C₂×#11718343	2
C₂×#11718360	2
C₂×#11718342	2
C₂×#11718777	2
C₂×C₄×Dic₅₂	2
C₂²×C₄×C₈×C₁₃	2
C₂×#3170	13

Subgroups — C2×#13125422 #16732663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13125422 #16732663