Subgroups — C₂×#13127704 #16743350

Order 3328.

1918 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719284	C₂	1	2	yes	maximal, complemented
2	#13127704	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718343	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718344	C₂	1	2	yes	maximal, complemented
5	C₂×#11718508	C₂	1	4	yes	maximal, complemented
6	C₂×#11718503	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×D₁₀₄	C₂	1	2	yes	maximal
9	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88200	—	2	2	no	—
12	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
13	#11719284	—	2	8	no	—
14	#11719284	C₂²	1	16	yes	complemented
15	C₂×#88150	—	2	4	no	—
16	C₂×C₄⋊C₄×D₁₃	—	2	2	no	—
17	C₂×C₄×D₅₂	C₂²	1	2	yes	—
18	C₂×C₄×D₅₂	C₂²	1	4	yes	—
19	C₂×#88124	C₂²	1	4	yes	—
20	C₂×#88123	C₂²	1	2	yes	—
21	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
22	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₂×#88061	C₂²	1	4	yes	—
27	C₂×#88061	C₂²	1	2	yes	—
28	C₂×#88058	C₂²	1	1	yes	characteristic
29	C₂×#88058	C₂²	1	2	yes	—
30	C₂×#88058	C₂²	1	2	yes	—
31	C₂×#88057	C₂²	1	4	yes	—
32	#11718508	C₂²	1	32	yes	complemented
33	#11718503	C₂²	1	16	yes	complemented
34	C₄×D₁₀₄	C₂²	1	8	yes	—
35	C₄×D₁₀₄	C₂²	1	16	yes	—
36	#11718344	C₂²	1	16	yes	complemented
37	#11718343	C₂²	1	8	yes	complemented
38	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×D₅₂	C₂³	1	2	yes	—
40	C₂²×C₄×D₁₃	—	2	2	no	—
41	C₂²×C₄×D₁₃	—	2	4	no	—
42	C₂²×Dic₂₆	—	2	2	no	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	D₄	1	4	yes	—
47	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
48	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	#88200	—	2	16	no	—
53	C₂×#6103	—	2	2	no	—
54	C₂×#6103	—	2	4	no	—
55	C₂×C₄⋊Dic₁₃	—	2	2	no	—
56	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
57	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
58	C₂×#6101	—	2	4	no	—
59	C₂×#6101	—	2	4	no	—
60	C₂×D₁₀₄	Q₈	1	8	yes	complemented
61	C₂×D₁₀₄	C₂³	1	4	yes	—
62	#88150	—	2	32	no	—
63	C₄⋊C₄×D₁₃	—	2	16	no	—
64	C₄×D₅₂	—	2	8	no	—
65	C₄×D₅₂	—	2	4	no	—
66	C₄×D₅₂	C₂³	1	8	yes	—
67	C₄×D₅₂	C₂³	1	16	yes	—
68	#88124	—	2	8	no	—
69	#88124	C₂³	1	16	yes	—
70	#88123	—	2	4	no	—
71	#88123	C₂³	1	8	yes	—
72	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
73	C₁₃×C₄⋊C₈	C₂³	1	8	yes	—
74	C₄×C₈×C₁₃	C₂³	1	4	yes	—
75	C₄×C₈×C₁₃	C₂³	1	8	yes	—
76	#88061	C₂³	1	16	yes	—
77	#88061	C₂³	1	8	yes	—
78	#88058	C₂³	1	8	yes	—
79	#88058	C₂³	1	8	yes	—
80	#88058	C₂³	1	4	yes	—
81	#88057	C₂³	1	16	yes	—
82	C₂×#3380	—	13	1	no	maximal, Sylow
83	C₂³×D₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	2	4	no	—
85	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₁₃	—	2	2	no	—
92	C₂²×Dic₁₃	—	2	4	no	—
93	C₂×D₅₂	—	2	2	no	—
94	C₂×D₅₂	—	2	4	no	—
95	C₂×D₅₂	C₂×Q₈	1	8	yes	—
96	C₂×D₅₂	C₂⁴	1	4	yes	—
97	C₂×C₄×D₁₃	—	2	8	no	—
98	C₂×C₄×D₁₃	—	2	16	no	—
99	C₂×C₄×D₁₃	—	4	8	no	—
100	C₂×C₄×D₁₃	—	8	8	no	—
101	C₂×Dic₂₆	—	2	8	no	—
102	C₂×Dic₂₆	—	8	4	no	—
103	C₂×C₈×C₁₃	—	2	4	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
106	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
107	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
108	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
109	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
110	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
111	C₄²×C₁₃	C₂×D₄	1	2	yes	—
112	C₄²×C₁₃	C₂×D₄	1	4	yes	—
113	C₄²×C₁₃	C₂⁴	1	2	yes	—
114	C₄²×C₁₃	C₂⁴	1	4	yes	—
115	#6103	—	2	8	no	—
116	#6103	—	2	16	no	—
117	C₄⋊Dic₁₃	—	2	4	no	—
118	C₄⋊Dic₁₃	—	2	8	no	—
119	C₄⋊Dic₁₃	—	2	2	no	—
120	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
121	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
122	#6101	—	2	16	no	—
123	#6101	—	2	16	no	—
124	D₁₀₄	C₂×Q₈	1	16	yes	complemented
125	C₂×#554	—	13	2	no	—
126	#3380	—	13	16	no	—
127	C₂×#500	—	13	1	no	—
128	C₂×#499	—	13	2	no	—
129	C₂×#478	—	13	4	no	—
130	C₂×#476	—	13	2	no	—
131	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic, complemented
132	C₂×C₄×D₈	—	13	2	no	—
133	C₂×C₄×D₈	—	13	1	no	—
134	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²×D₁₃	—	2	8	no	—
136	C₂²×D₁₃	—	2	4	no	—
137	C₂²×D₁₃	—	4	4	no	—
138	C₂²×D₁₃	—	4	4	no	—
139	Q₈×C₁₃	—	2	8	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	#135	1	4	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
145	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
146	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
150	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
151	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	4	no	—
153	C₂×Dic₁₃	—	2	8	no	—
154	C₂×Dic₁₃	—	4	4	no	—
155	C₂×Dic₁₃	—	4	8	no	—
156	D₅₂	—	2	8	no	—
157	D₅₂	—	2	4	no	—
158	D₅₂	C₂²×Q₈	1	8	yes	—
159	C₄×D₁₃	—	4	16	no	—
160	C₄×D₁₃	—	8	16	no	—
161	Dic₂₆	—	8	8	no	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
165	C₂²×D₈	—	13	1	no	—
166	#554	—	13	16	no	—
167	#554	—	26	8	no	—
168	C₂×C₄⋊Q₈	—	13	2	no	—
169	C₂×#125	—	13	4	no	—
170	C₂×C₂²⋊Q₈	—	26	2	no	—
171	C₂×C₂²⋊Q₈	—	26	4	no	—
172	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×C₄×D₄	—	13	4	no	—
174	C₂×C₄×D₄	—	13	2	no	—
175	C₂²×C₄⋊C₄	—	26	2	no	—
176	#500	—	13	8	no	—
177	#499	—	13	16	no	—
178	#478	—	13	32	no	—
179	#476	—	13	16	no	—
180	C₈×Q₈	C₂×D₁₃	1	8	yes	complemented
181	C₄×D₈	—	13	16	no	—
182	C₄×D₈	—	13	8	no	—
183	C₂×#107	—	13	2	no	—
184	C₂×#107	—	13	1	no	—
185	C₂×#107	—	13	2	no	—
186	C₂×#106	—	13	4	no	—
187	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
188	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
189	C₂×#102	—	13	2	no	—
190	C₂×#102	—	13	4	no	—
191	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
194	C₂²×C₁₃	C₂×#135	1	2	yes	—
195	C₂²×C₁₃	#576	1	2	yes	—
196	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
197	C₂×D₁₃	—	2	8	no	—
198	C₂×D₁₃	—	4	16	no	—
199	C₂×D₁₃	—	4	4	no	—
200	C₂×D₁₃	—	4	4	no	—
201	C₄×C₁₃	—	2	4	no	—
202	C₄×C₁₃	—	2	2	no	—
203	C₄×C₁₃	—	2	4	no	—
204	C₄×C₁₃	D₄×Q₈	1	2	yes	—
205	C₄×C₁₃	C₂×#135	1	4	yes	—
206	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
207	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
208	Dic₁₃	—	4	4	no	—
209	Dic₁₃	—	4	8	no	—
210	C₂²×Q₈	—	26	2	no	—
211	C₂²×Q₈	D₅₂	1	1	yes	characteristic
212	C₂²×D₄	—	13	2	no	—
213	C₂³×C₄	—	26	2	no	—
214	C₂³×C₄	—	26	4	no	—
215	C₂×D₈	—	13	8	no	—
216	C₂×D₈	—	13	4	no	—
217	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
218	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
220	C₄⋊Q₈	—	13	8	no	—
221	C₄⋊Q₈	—	26	4	no	—
222	#125	—	13	16	no	—
223	#125	—	26	8	no	—
224	C₂²⋊Q₈	—	26	16	no	—
225	C₂²⋊Q₈	—	26	32	no	—
226	C₄×Q₈	D₅₂	1	4	yes	—
227	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
228	C₄×D₄	—	13	8	no	—
229	C₄×D₄	—	13	16	no	—
230	C₄×D₄	—	26	8	no	—
231	C₄×D₄	—	26	4	no	—
232	C₂×C₄⋊C₄	—	13	2	no	—
233	C₂×C₄⋊C₄	—	13	4	no	—
234	C₂×C₄⋊C₄	—	26	4	no	—
235	C₂×C₄⋊C₄	—	26	2	no	—
236	C₂×C₄⋊C₄	—	26	4	no	—
237	C₂×C₄⋊C₄	—	26	16	no	—
238	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
239	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
240	C₂×C₂²⋊C₄	—	26	2	no	—
241	C₂×C₂²⋊C₄	—	26	4	no	—
242	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
243	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
244	#107	—	13	4	no	—
245	#107	—	13	8	no	—
246	#107	—	13	8	no	—
247	#106	—	13	16	no	—
248	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
249	C₄⋊C₈	C₂²×D₁₃	1	8	yes	—
250	#102	—	13	16	no	—
251	#102	—	13	8	no	—
252	C₄×C₈	C₂²×D₁₃	1	8	yes	—
253	C₄×C₈	C₂²×D₁₃	1	4	yes	—
254	C₂×C₁₃	#3380	1	4	yes	complemented
255	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
256	C₂×C₁₃	C₂²×#135	1	1	yes	characteristic
257	C₂×C₁₃	C₂×#576	1	1	yes	characteristic
258	D₁₃	—	4	8	no	—
259	C₂⁴	—	26	2	no	—
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	26	8	no	—
262	C₂×Q₈	—	104	4	no	—
263	C₂×Q₈	C₂×D₅₂	1	4	yes	—
264	C₂×D₄	—	13	8	no	—
265	C₂×D₄	—	13	4	no	—
266	C₂×D₄	—	26	2	no	—
267	C₂×D₄	—	26	4	no	—
268	C₂²×C₄	—	26	2	no	—
269	C₂²×C₄	—	26	4	no	—
270	C₂²×C₄	—	26	8	no	—
271	C₂²×C₄	—	26	16	no	—
272	C₂²×C₄	—	52	8	no	—
273	C₂²×C₄	—	104	8	no	—
274	C₂²×C₄	C₂×D₅₂	1	2	yes	—
275	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₅₂	1	2	yes	—
277	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
278	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
279	D₈	—	13	16	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	Q₈×D₁₃	1	4	yes	—
283	C₂×C₈	C₂³×D₁₃	1	4	yes	—
284	C₂×C₈	C₂³×D₁₃	1	2	yes	—
285	C₂×C₈	C₂³×D₁₃	1	2	yes	—
286	C₄⋊C₄	—	13	4	no	—
287	C₄⋊C₄	—	13	8	no	—
288	C₄⋊C₄	—	26	2	no	—
289	C₄⋊C₄	—	26	4	no	—
290	C₄⋊C₄	—	26	16	no	—
291	C₄⋊C₄	—	26	8	no	—
292	C₄⋊C₄	—	26	16	no	—
293	C₄⋊C₄	C₂×D₅₂	1	4	yes	—
294	C₄⋊C₄	C₂×D₅₂	1	8	yes	—
295	C₂²⋊C₄	—	26	8	no	—
296	C₂²⋊C₄	—	26	16	no	—
297	C₄²	C₂×D₅₂	1	4	yes	—
298	C₄²	C₂×D₅₂	1	2	yes	—
299	C₄²	C₂³×D₁₃	1	4	yes	—
300	C₄²	C₂³×D₁₃	1	2	yes	—
301	C₁₃	C₂×#3380	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	26	8	no	—
303	C₂³	—	26	4	no	—
304	C₂³	—	52	4	no	—
305	C₂³	—	52	4	no	—
306	C₂³	C₂²×D₅₂	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	Q₈	—	104	8	no	—
309	D₄	—	13	8	no	—
310	D₄	—	26	8	no	—
311	D₄	—	26	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	26	4	no	—
316	C₂×C₄	—	26	8	no	—
317	C₂×C₄	—	52	4	no	—
318	C₂×C₄	—	52	8	no	—
319	C₂×C₄	—	52	16	no	—
320	C₂×C₄	—	104	16	no	—
321	C₂×C₄	#88158	1	4	yes	—
322	C₂×C₄	C₂²×D₅₂	1	4	yes	—
323	C₂×C₄	C₂²×D₅₂	1	2	yes	—
324	C₂×C₄	C₂²×D₅₂	1	4	yes	—
325	C₂×C₄	C₂²×D₅₂	1	2	yes	—
326	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
327	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
328	C₂×C₄	#88257	1	2	yes	—
329	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	C₂×Q₈×D₁₃	1	4	yes	—
333	C₂²	—	26	8	no	—
334	C₂²	—	52	16	no	—
335	C₂²	—	52	4	no	—
336	C₂²	—	52	4	no	—
337	C₂²	Q₈×D₅₂	1	2	yes	—
338	C₂²	C₂×#88158	1	2	yes	—
339	C₂²	#11719457	1	2	yes	—
340	C₂²	C₂³×D₅₂	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	2	2	no	—
344	C₄	—	52	4	no	—
345	C₄	—	52	8	no	—
346	C₄	Q₈×D₅₂	1	2	yes	—
347	C₄	C₂×#88158	1	4	yes	—
348	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
349	C₄	C₂×#88257	1	1	yes	characteristic
350	C₂	—	52	8	no	—
351	C₂	#13127704	1	4	yes	direct product factor
352	C₂	C₂×Q₈×D₅₂	1	1	yes	characteristic
353	C₂	C₂²×#88158	1	1	yes	characteristic
354	C₂	C₂×#11719457	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	C₂×#3380
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11719284	C₂	semidirect
2. #13127704	C₂	direct (N × H)
3. C₂×#11718343	C₂	semidirect
4. C₂×#11718344	C₂	semidirect
5. C₂×#11718508	C₂	semidirect
6. C₂×#11718503	C₂	semidirect
9. C₂×C₈×Q₈×C₁₃	C₂	semidirect
14. #11719284	C₂²	semidirect
21. C₈×Q₈×C₁₃	C₂²	semidirect
32. #11718508	C₂²	semidirect
33. #11718503	C₂²	semidirect
36. #11718344	C₂²	semidirect
37. #11718343	C₂²	semidirect
60. C₂×D₁₀₄	Q₈	semidirect
124. D₁₀₄	C₂×Q₈	semidirect
131. C₂×C₈×Q₈	D₁₃	semidirect
180. C₈×Q₈	C₂×D₁₃	semidirect
254. C₂×C₁₃	#3380	semidirect
301. C₁₃	C₂×#3380	semidirect
351. C₂	#13127704	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×#11719284	2
#13127704	2
C₂×#11718343	2
C₂×#11718344	2
C₂×#11718508	2
C₂×#11718503	2
C₂×C₄×D₁₀₄	2
C₂×C₈×Q₈×C₁₃	2
C₂×#3380	13

Subgroups — C2×#13127704 #16743350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13127704 #16743350