Subgroups — C₂×#13115579 #15866265

Order 3328.

1142 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718589	C₂	1	2	yes	maximal
4	#13115579	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718143	C₂	1	2	yes	maximal
6	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂²×#6101	—	2	2	no	—
13	C₂×C₁₃×#97	—	2	4	no	—
14	C₂×C₁₃×#97	C₂²	1	4	yes	—
15	C₂×#88071	—	2	2	no	—
16	C₂×#88058	—	2	4	no	—
17	C₂×#88058	C₂²	1	4	yes	—
18	C₂×#88057	C₄	1	8	yes	complemented
19	C₂×#88057	C₂²	1	4	yes	—
20	#11718589	—	2	4	no	—
21	#11718589	C₂²	1	8	yes	—
22	#11718143	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	2	no	—
27	C₂³×Dic₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	D₄	1	4	yes	—
32	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	—	2	2	no	—
38	C₂×C₄²×C₁₃	D₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	4	no	—
42	C₂×C₄⋊Dic₁₃	—	2	4	no	—
43	C₂×C₄⋊Dic₁₃	—	2	4	no	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
48	C₂×#6101	—	2	8	no	—
49	C₂×#6101	—	4	8	no	—
50	C₁₃×#97	—	2	8	no	—
51	#88071	—	2	8	no	—
52	#88058	—	2	8	no	—
53	#88057	C₂×C₄	1	16	yes	complemented
54	C₂×#1944	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	8	no	—
71	C₂²×Dic₁₃	—	2	4	no	—
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	—	4	4	no	—
74	C₂²×Dic₁₃	—	4	4	no	—
75	C₂²×Dic₁₃	—	4	4	no	—
76	C₂²×Dic₁₃	—	4	4	no	—
77	C₂²×Dic₁₃	—	4	4	no	—
78	C₂×C₈×C₁₃	—	2	8	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₄²×C₁₃	—	2	4	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	8	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	4	4	no	—
94	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
95	#6101	—	4	16	no	—
96	C₂²×#107	—	13	1	no	—
97	C₂²×#106	—	13	1	no	—
98	C₂²×#97	D₁₃	1	1	yes	characteristic
99	C₂×#389	—	13	2	no	—
100	#1944	—	13	8	no	—
101	C₂×#340	—	13	2	no	—
102	C₂³×C₁₃	C₄×D₄	1	2	yes	—
103	C₂³×C₁₃	#121	1	2	yes	—
104	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₂³×C₁₃	#137	1	4	yes	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
117	C₂×C₄×C₁₃	#121	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	#125	1	4	yes	—
120	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	16	no	—
131	C₂×Dic₁₃	—	4	4	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	8	8	no	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
137	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	13	2	no	—
140	C₂²×C₄⋊C₄	—	13	2	no	—
141	C₂²×C₄⋊C₄	—	26	2	no	—
142	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
143	C₂×#107	—	13	4	no	—
144	C₂×#107	—	26	4	no	—
145	C₂×#106	—	13	8	no	—
146	C₂×#106	—	13	4	no	—
147	C₂×#97	—	2	4	no	—
148	C₂×#97	C₂×D₁₃	1	4	yes	—
149	#389	—	13	8	no	—
150	#389	—	26	4	no	—
151	C₂×#95	—	26	2	no	—
152	#340	—	13	8	no	—
153	C₂²×C₁₃	#389	1	4	yes	—
154	C₂²×C₁₃	#440	1	8	yes	—
155	C₂²×C₁₃	#469	1	4	yes	—
156	C₂²×C₁₃	#470	1	4	yes	—
157	C₂²×C₁₃	#501	1	8	yes	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
162	C₂²×C₁₃	C₂×#137	1	2	yes	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	#389	1	4	yes	—
166	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
170	Dic₁₃	—	4	8	no	—
171	Dic₁₃	—	8	4	no	—
172	C₂³×C₄	—	26	2	no	—
173	C₂³×C₄	—	26	2	no	—
174	C₂³×C₄	D₅₂	1	1	yes	characteristic
175	C₂³×C₄	D₅₂	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	Dic₂₆	1	4	yes	—
180	C₂²×C₈	Dic₂₆	1	4	yes	—
181	C₂²×C₈	C₄×D₁₃	1	4	yes	—
182	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
183	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	—	13	8	no	—
186	C₂×C₄⋊C₄	—	13	4	no	—
187	C₂×C₄⋊C₄	—	13	4	no	—
188	C₂×C₄⋊C₄	—	26	2	no	—
189	C₂×C₄⋊C₄	—	26	4	no	—
190	C₂×C₄⋊C₄	—	26	4	no	—
191	C₂×C₄⋊C₄	—	26	4	no	—
192	C₂×C₄⋊C₄	—	26	2	no	—
193	C₂×C₄⋊C₄	—	26	8	no	—
194	C₂×C₄⋊C₄	—	52	8	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	D₅₂	1	2	yes	—
198	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
199	#107	—	26	8	no	—
200	#106	—	13	16	no	—
201	#97	—	2	8	no	—
202	#95	—	26	8	no	—
203	C₂×C₁₃	#1944	1	8	yes	complemented
204	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#440	1	2	yes	—
206	C₂×C₁₃	C₂×#469	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#470	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#501	1	2	yes	—
209	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	26	8	no	—
215	C₂²×C₄	—	26	4	no	—
216	C₂²×C₄	—	26	4	no	—
217	C₂²×C₄	—	52	4	no	—
218	C₂²×C₄	—	52	4	no	—
219	C₂²×C₄	—	52	4	no	—
220	C₂²×C₄	—	52	4	no	—
221	C₂²×C₄	—	52	4	no	—
222	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
223	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
225	C₂²×C₄	C₂×D₅₂	1	2	yes	—
226	C₂²×C₄	C₂×D₅₂	1	2	yes	—
227	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
228	C₂²×C₄	#6127	1	2	yes	—
229	C₂²×C₄	#6127	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
231	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
232	C₂×C₈	—	2	8	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	#6101	1	16	yes	complemented
236	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
237	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
238	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
239	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
240	C₄⋊C₄	—	13	8	no	—
241	C₄⋊C₄	—	26	8	no	—
242	C₄⋊C₄	—	26	4	no	—
243	C₄⋊C₄	—	26	4	no	—
244	C₄⋊C₄	—	26	4	no	—
245	C₄⋊C₄	—	52	16	no	—
246	C₄⋊C₄	—	52	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₁₃	C₂×#1944	1	1	yes	characteristic, complemented, Sylow
251	C₂³	#88123	1	2	yes	—
252	C₂³	C₄×D₅₂	1	2	yes	—
253	C₂³	#88162	1	4	yes	—
254	C₂³	#88163	1	4	yes	—
255	C₂³	#88184	1	2	yes	—
256	C₂³	C₂²×D₅₂	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	26	8	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	52	16	no	—
267	C₂×C₄	—	52	4	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	104	8	no	—
271	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
272	C₂×C₄	#88123	1	2	yes	—
273	C₂×C₄	#88124	1	4	yes	—
274	C₂×C₄	C₄×D₅₂	1	2	yes	—
275	C₂×C₄	C₂×#6101	1	4	yes	—
276	C₂×C₄	#88178	1	4	yes	—
277	C₂×C₄	#88184	1	2	yes	—
278	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
279	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#6127	1	1	yes	characteristic
282	C₂×C₄	C₂×#6127	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	C₂×#6101	1	8	yes	complemented
286	C₂²	#11718363	1	8	yes	—
287	C₂²	#11718371	1	8	yes	—
288	C₂²	#11718589	1	4	yes	—
289	C₂²	#11718796	1	4	yes	—
290	C₂²	#11718798	1	4	yes	—
291	C₂²	C₂×#88123	1	1	yes	characteristic
292	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
293	C₂²	C₂×#88162	1	2	yes	—
294	C₂²	C₂×#88163	1	2	yes	—
295	C₂²	C₂×#88184	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	52	8	no	—
299	C₄	—	104	4	no	—
300	C₄	#11718589	1	4	yes	—
301	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
302	C₄	C₂×#88124	1	1	yes	characteristic
303	C₄	C₂²×#6101	1	1	yes	characteristic
304	C₄	C₂×#88178	1	1	yes	characteristic
305	C₂	#13115579	1	8	yes	direct product factor
306	C₂	C₂×#11718589	1	1	yes	characteristic
307	C₂	C₂×#11718371	1	2	yes	—
308	C₂	C₂×#11718796	1	1	yes	characteristic
309	C₂	C₂×#11718798	1	1	yes	characteristic
310	C₂	C₂×#11718363	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	C₂×#1944
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #13115579	C₂	direct (N × H)
18. C₂×#88057	C₄	semidirect
53. #88057	C₂×C₄	semidirect
81. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
136. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
203. C₂×C₁₃	#1944	semidirect
235. C₂×C₈	#6101	semidirect
250. C₁₃	C₂×#1944	semidirect
285. C₈	C₂×#6101	semidirect
305. C₂	#13115579	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×#88058	2
C₂²×#88057	2
C₂×#11718589	2
#13115579	2
C₂×#11718143	2
C₂²×C₁₃×#97	2
C₂×#1944	13

Subgroups — C2×#13115579 #15866265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115579 #15866265