Subgroups — C₂×#13112943 #15777835

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	#13112943	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
7	C₂×#88180	—	2	1	no	—
8	C₂×C₁₃×#106	—	2	1	no	—
9	C₂×C₁₃×#106	—	2	2	no	—
10	C₂×C₁₃×#106	C₄	1	4	yes	complemented
11	C₂×C₁₃×#106	C₂²	1	2	yes	—
12	#11718767	—	2	1	no	—
13	#11718767	—	2	1	no	—
14	#11718767	C₄	1	2	yes	complemented
15	#11718767	C₂²	1	2	yes	—
16	C₂×#88058	C₄	1	2	yes	complemented
17	C₂×#88057	—	2	1	no	—
18	C₂×#88052	—	2	1	no	—
19	C₂×#88052	—	2	2	no	—
20	C₂×#88052	C₄	1	4	yes	complemented
21	C₂×#88052	C₂²	1	2	yes	—
22	C₂²×#6090	—	2	1	no	—
23	C₂³×C₄×C₁₃	—	4	1	no	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
40	#88180	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	4	1	no	—
43	#88180	—	4	1	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	4	1	no	—
46	C₁₃×#106	—	2	4	no	—
47	C₁₃×#106	—	2	2	no	—
48	C₁₃×#106	C₂×C₄	1	4	yes	complemented
49	#88058	—	2	2	no	—
50	#88058	C₂×C₄	1	4	yes	complemented
51	#88057	—	2	2	no	—
52	#88057	—	2	2	no	—
53	#88052	—	2	4	no	—
54	#88052	—	2	2	no	—
55	#88052	C₂×C₄	1	4	yes	complemented
56	C₂×#6090	—	2	4	no	—
57	C₂×#6090	—	2	2	no	—
58	C₂×#1386	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	8	4	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
69	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	C₄²	1	2	yes	—
75	C₂×C₈×C₁₃	C₄²	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₄⋊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×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	4	4	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₂²⋊Dic₁₃	—	4	2	no	—
96	C₂²⋊Dic₁₃	—	4	2	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	4	2	no	—
99	C₄×Dic₁₃	—	4	2	no	—
100	C₄×Dic₁₃	—	4	2	no	—
101	#6090	—	2	4	no	—
102	C₂²×M₃₂	—	13	1	no	—
103	C₂×#427	—	13	1	no	—
104	C₂²×#106	D₁₃	1	1	yes	characteristic
105	#1386	—	13	4	no	—
106	C₂³×C₁₃	—	2	4	no	—
107	C₂³×C₁₃	#102	1	2	yes	—
108	C₂³×C₁₃	#102	1	1	yes	characteristic
109	C₂³×C₁₃	#102	1	1	yes	characteristic
110	C₂³×C₁₃	#107	1	2	yes	—
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	—	8	8	no	—
115	C₂×C₄×C₁₃	#95	1	2	yes	—
116	C₂×C₄×C₁₃	#95	1	2	yes	—
117	C₂×C₄×C₁₃	#102	1	2	yes	—
118	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#103	1	2	yes	—
121	C₂×C₄×C₁₃	#103	1	2	yes	—
122	C₂×C₄×C₁₃	#103	1	2	yes	—
123	C₂×C₄×C₁₃	#103	1	2	yes	—
124	C₂×C₄×C₁₃	#106	1	2	yes	—
125	C₂×C₄×C₁₃	#106	1	2	yes	—
126	C₂×C₄×C₁₃	#107	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	8	4	no	—
135	C₈×C₁₃	#95	1	2	yes	—
136	C₈×C₁₃	#95	1	2	yes	—
137	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
138	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
142	C₂×#117	—	26	1	no	—
143	C₂²×C₄⋊C₄	—	2	1	no	—
144	C₂×M₃₂	—	13	4	no	—
145	C₂×M₃₂	—	13	2	no	—
146	C₂×M₃₂	—	26	1	no	—
147	C₂×M₃₂	—	26	2	no	—
148	C₂²×C₁₆	—	26	1	no	—
149	#427	—	13	2	no	—
150	#427	—	13	2	no	—
151	#427	—	26	1	no	—
152	#427	—	26	1	no	—
153	C₂×#107	—	13	2	no	—
154	C₂×#106	—	2	1	no	—
155	C₂×#106	—	2	2	no	—
156	C₂×#106	—	26	1	no	—
157	C₂×#106	Dic₁₃	1	4	yes	complemented
158	C₂×#106	C₂×D₁₃	1	2	yes	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	4	no	—
163	C₂²×C₁₃	#340	1	2	yes	—
164	C₂²×C₁₃	#340	1	2	yes	—
165	C₂²×C₁₃	#360	1	4	yes	—
166	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
169	C₄×C₁₃	—	8	4	no	—
170	C₄×C₁₃	#340	1	2	yes	—
171	C₄×C₁₃	#340	1	2	yes	—
172	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
176	Dic₁₃	—	8	4	no	—
177	C₂³×C₄	—	4	1	no	—
178	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₂²×C₈	Dic₂₆	1	2	yes	—
180	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₁₃	1	2	yes	—
183	C₂²×C₈	D₅₂	1	1	yes	characteristic
184	C₂²×C₈	D₅₂	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
186	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	#117	—	26	2	no	—
190	#117	—	26	2	no	—
191	#117	—	52	1	no	—
192	#117	—	52	1	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₂²⋊C₄	—	52	1	no	—
199	C₂×C₄²	—	52	1	no	—
200	M₃₂	—	13	4	no	—
201	M₃₂	—	26	4	no	—
202	M₃₂	—	26	2	no	—
203	C₂×C₁₆	—	26	4	no	—
204	C₂×C₁₆	—	26	2	no	—
205	#107	—	13	4	no	—
206	#107	—	26	2	no	—
207	#106	—	2	4	no	—
208	#106	—	2	2	no	—
209	#106	—	26	2	no	—
210	#106	—	26	2	no	—
211	#106	C₂×Dic₁₃	1	4	yes	complemented
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	#1386	1	4	yes	complemented
216	C₂×C₁₃	C₂×#340	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#360	1	2	yes	—
218	C₂⁴	#6104	1	1	yes	characteristic
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	8	4	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	#6101	1	2	yes	—
224	C₂²×C₄	#6103	1	1	yes	characteristic
225	C₂²×C₄	#6103	1	1	yes	characteristic
226	C₂²×C₄	#6104	1	1	yes	characteristic
227	C₂²×C₄	#6105	1	2	yes	—
228	C₂²×C₄	#6106	1	1	yes	characteristic
229	C₂²×C₄	#6106	1	1	yes	characteristic
230	C₂²×C₄	#6107	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
234	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
235	C₂×C₈	#6101	1	2	yes	—
236	C₂×C₈	#6101	1	2	yes	—
237	C₂×C₈	#6101	1	2	yes	—
238	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
239	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
240	C₂×C₈	#6103	1	1	yes	characteristic
241	C₂×C₈	#6103	1	1	yes	characteristic
242	C₂×C₈	#6103	1	2	yes	—
243	C₂×C₈	#6103	1	2	yes	—
244	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
245	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
249	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	26	4	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₂²⋊C₄	—	52	2	no	—
257	C₂²⋊C₄	—	52	2	no	—
258	C₄²	—	52	2	no	—
259	C₄²	—	52	2	no	—
260	C₁₆	—	26	4	no	—
261	C₁₃	C₂×#1386	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	2	4	no	—
263	C₂³	#88047	1	2	yes	—
264	C₂³	#88049	1	1	yes	characteristic
265	C₂³	#88049	1	1	yes	characteristic
266	C₂³	#88072	1	2	yes	—
267	C₂³	C₂×#6104	1	1	yes	characteristic
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	8	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	—	104	4	no	—
273	C₂×C₄	#88047	1	2	yes	—
274	C₂×C₄	#88048	1	2	yes	—
275	C₂×C₄	#88048	1	2	yes	—
276	C₂×C₄	#88049	1	1	yes	characteristic
277	C₂×C₄	#88049	1	1	yes	characteristic
278	C₂×C₄	#88050	1	2	yes	—
279	C₂×C₄	#88050	1	2	yes	—
280	C₂×C₄	#88071	1	2	yes	—
281	C₂×C₄	#88071	1	2	yes	—
282	C₂×C₄	#88072	1	2	yes	—
283	C₂×C₄	#88075	1	2	yes	—
284	C₂×C₄	#88075	1	2	yes	—
285	C₂×C₄	C₂×#6101	1	1	yes	characteristic
286	C₂×C₄	C₂×#6103	1	1	yes	characteristic
287	C₂×C₄	C₂×#6105	1	1	yes	characteristic
288	C₂×C₄	C₂×#6106	1	1	yes	characteristic
289	C₂×C₄	C₂×#6107	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
291	C₈	#88071	1	2	yes	—
292	C₈	#88071	1	2	yes	—
293	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
294	C₈	C₂×#6101	1	1	yes	characteristic
295	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
296	C₈	C₂×#6103	1	1	yes	characteristic
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#11718080	1	4	yes	—
302	C₂²	#11718121	1	2	yes	—
303	C₂²	#11718121	1	2	yes	—
304	C₂²	C₂×#88047	1	1	yes	characteristic
305	C₂²	C₂×#88049	1	1	yes	characteristic
306	C₂²	C₂×#88072	1	1	yes	characteristic
307	C₄	—	8	4	no	—
308	C₄	—	104	4	no	—
309	C₄	#11718121	1	2	yes	—
310	C₄	#11718121	1	2	yes	—
311	C₄	C₂×#88048	1	1	yes	characteristic
312	C₄	C₂×#88050	1	1	yes	characteristic
313	C₄	C₂×#88071	1	1	yes	characteristic
314	C₄	C₂×#88075	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	#13112943	1	4	yes	direct product factor
319	C₂	C₂×#11718121	1	1	yes	characteristic
320	C₂	C₂×#11718080	1	2	yes	—

	subgroup	G/H
center	C₂³	#88049
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle	C₂³×C₁₃	#102
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂×#1386
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. #13112943	C₂	direct (N × H)
10. C₂×C₁₃×#106	C₄	semidirect
14. #11718767	C₄	semidirect
16. C₂×#88058	C₄	semidirect
20. C₂×#88052	C₄	semidirect
48. C₁₃×#106	C₂×C₄	semidirect
50. #88058	C₂×C₄	semidirect
55. #88052	C₂×C₄	semidirect
157. C₂×#106	Dic₁₃	semidirect
211. #106	C₂×Dic₁₃	semidirect
215. C₂×C₁₃	#1386	semidirect
261. C₁₃	C₂×#1386	semidirect
318. C₂	#13112943	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×#88052	2
C₂×#11718767	2
#13112943	2
C₂²×C₁₃×#106	2
C₂×#1386	13

Subgroups — C2×#13112943 #15777835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112943 #15777835