Subgroups — C₂×#13114733 #15860974

Order 3328.

1318 conjugacy classes of subgroups.

The subgroup lattice has 300 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×C₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
4	#13114733	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×C₄×Dic₁₃	—	2	2	no	—
13	C₂×C₁₃×#97	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	4	yes	—
15	C₂×#88071	—	2	2	no	—
16	C₂×#88058	C₄	1	8	yes	complemented
17	C₂×#88058	C₂²	1	4	yes	—
18	C₂×#88057	C₄	1	8	yes	complemented
19	C₂×#88057	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₁₃	—	2	4	no	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	8	yes	—
22	#11718143	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂³×Dic₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
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	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
46	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
47	C₂×C₄×Dic₁₃	—	2	8	no	—
48	C₂×C₄×Dic₁₃	—	4	8	no	—
49	C₁₃×#97	C₂×C₄	1	16	yes	complemented
50	#88071	—	2	8	no	—
51	#88058	C₂×C₄	1	16	yes	complemented
52	#88057	C₂×C₄	1	16	yes	complemented
53	C₂×#1776	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	8	no	—
68	C₂²×Dic₁₃	—	2	8	no	—
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	—	4	4	no	—
72	C₂²×Dic₁₃	—	4	4	no	—
73	C₂²×Dic₁₃	—	4	4	no	—
74	C₂²×Dic₁₃	—	4	4	no	—
75	C₂×C₈×C₁₃	—	2	8	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
85	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
86	C₄⋊Dic₁₃	—	2	8	no	—
87	C₄⋊Dic₁₃	—	2	8	no	—
88	C₄⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
92	C₄×Dic₁₃	—	4	16	no	—
93	C₂²×#107	—	13	1	no	—
94	C₂²×#106	—	13	1	no	—
95	C₂²×#97	D₁₃	1	1	yes	characteristic
96	C₂×C₄×C₄⋊C₄	—	13	2	no	—
97	#1776	—	13	8	no	—
98	C₂×#340	—	13	2	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂³×C₁₃	C₄×D₄	1	2	yes	—
101	C₂³×C₁₃	C₄×D₄	1	2	yes	—
102	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
103	C₂³×C₁₃	#137	1	4	yes	—
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	16	no	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	4	yes	—
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	8	no	—
123	C₂×Dic₁₃	—	2	8	no	—
124	C₂×Dic₁₃	—	4	4	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	16	no	—
129	C₂×Dic₁₃	—	4	16	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
132	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	—	13	2	no	—
135	C₂²×C₄⋊C₄	—	13	2	no	—
136	C₂²×C₄²	—	26	2	no	—
137	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
138	C₂×#107	—	13	8	no	—
139	C₂×#107	—	13	4	no	—
140	C₂×#106	—	13	8	no	—
141	C₂×#106	—	13	4	no	—
142	C₂×#97	Dic₁₃	1	8	yes	complemented
143	C₂×#97	C₂×D₁₃	1	4	yes	—
144	C₄×C₄⋊C₄	—	13	8	no	—
145	C₄×C₄⋊C₄	—	26	4	no	—
146	C₂×#95	—	26	2	no	—
147	#340	—	13	8	no	—
148	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
149	C₂²×C₁₃	#428	1	8	yes	—
150	C₂²×C₁₃	#440	1	8	yes	—
151	C₂²×C₁₃	#441	1	4	yes	—
152	C₂²×C₁₃	#442	1	4	yes	—
153	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
157	C₂²×C₁₃	C₂×#137	1	2	yes	—
158	C₄×C₁₃	—	2	8	no	—
159	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
160	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
161	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	Dic₁₃	—	4	8	no	—
165	Dic₁₃	—	4	8	no	—
166	C₂³×C₄	—	26	2	no	—
167	C₂³×C₄	—	26	2	no	—
168	C₂³×C₄	D₅₂	1	2	yes	—
169	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₄×D₁₃	1	4	yes	—
173	C₂²×C₈	C₄×D₁₃	1	4	yes	—
174	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
175	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
177	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	—	13	8	no	—
179	C₂×C₄⋊C₄	—	13	8	no	—
180	C₂×C₄⋊C₄	—	13	4	no	—
181	C₂×C₄⋊C₄	—	13	4	no	—
182	C₂×C₄⋊C₄	—	26	2	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	4	no	—
185	C₂×C₄⋊C₄	—	26	4	no	—
186	C₂×C₄²	—	26	8	no	—
187	C₂×C₄²	—	52	8	no	—
188	C₂×C₄²	Dic₂₆	1	4	yes	—
189	C₂×C₄²	D₅₂	1	2	yes	—
190	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
191	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
192	#107	—	13	16	no	—
193	#106	—	13	16	no	—
194	#97	C₂×Dic₁₃	1	16	yes	complemented
195	#95	—	26	8	no	—
196	C₂×C₁₃	#1776	1	8	yes	complemented
197	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#428	1	2	yes	—
199	C₂×C₁₃	C₂×#440	1	2	yes	—
200	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
202	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	26	8	no	—
206	C₂²×C₄	—	26	8	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	—	26	4	no	—
209	C₂²×C₄	—	52	4	no	—
210	C₂²×C₄	—	52	4	no	—
211	C₂²×C₄	—	52	4	no	—
212	C₂²×C₄	—	52	4	no	—
213	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
214	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
217	C₂²×C₄	C₂×D₅₂	1	4	yes	—
218	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
219	C₂²×C₄	#6127	1	2	yes	—
220	C₂²×C₄	#6127	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
227	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
229	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
230	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
231	C₄⋊C₄	—	13	8	no	—
232	C₄⋊C₄	—	13	8	no	—
233	C₄⋊C₄	—	26	8	no	—
234	C₄⋊C₄	—	26	8	no	—
235	C₄⋊C₄	—	26	4	no	—
236	C₄⋊C₄	—	26	4	no	—
237	C₄²	—	52	16	no	—
238	C₄²	C₄⋊Dic₁₃	1	8	yes	—
239	C₄²	C₂×Dic₂₆	1	4	yes	—
240	C₄²	C₂²×Dic₁₃	1	4	yes	—
241	C₁₃	C₂×#1776	1	1	yes	characteristic, complemented, Sylow
242	C₂³	C₄×D₅₂	1	2	yes	—
243	C₂³	C₄×D₅₂	1	2	yes	—
244	C₂³	#88162	1	4	yes	—
245	C₂³	#88163	1	4	yes	—
246	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
247	C₂³	C₂²×D₅₂	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	16	no	—
251	C₂×C₄	—	26	8	no	—
252	C₂×C₄	—	26	8	no	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	—	52	4	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	—	52	16	no	—
258	C₂×C₄	—	52	16	no	—
259	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
260	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
261	C₂×C₄	C₄×D₅₂	1	2	yes	—
262	C₂×C₄	C₄×D₅₂	1	2	yes	—
263	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
265	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
266	C₂×C₄	#88180	1	4	yes	—
267	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#6127	1	1	yes	characteristic
271	C₂×C₄	C₂×#6127	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	C₂×C₄×Dic₁₃	1	8	yes	complemented
275	C₂²	#11718371	1	8	yes	—
276	C₂²	#11718372	1	4	yes	—
277	C₂²	#11718377	1	4	yes	—
278	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
279	C₂²	#11718782	1	8	yes	—
280	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
281	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
282	C₂²	C₂×#88162	1	2	yes	—
283	C₂²	C₂×#88163	1	2	yes	—
284	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
285	C₄	—	2	8	no	—
286	C₄	—	52	8	no	—
287	C₄	—	52	8	no	—
288	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
289	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
290	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
291	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
292	C₄	C₂×#88180	1	1	yes	characteristic
293	C₂	#13114733	1	8	yes	direct product factor
294	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
295	C₂	C₂×#11718782	1	2	yes	—
296	C₂	C₂×#11718371	1	2	yes	—
297	C₂	C₂×#11718377	1	1	yes	characteristic
298	C₂	C₂×#11718372	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
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×#1776
LCS	G ▷ 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. #13114733	C₂	direct (N × H)
13. C₂×C₁₃×#97	C₄	semidirect
16. C₂×#88058	C₄	semidirect
18. C₂×#88057	C₄	semidirect
49. C₁₃×#97	C₂×C₄	semidirect
51. #88058	C₂×C₄	semidirect
52. #88057	C₂×C₄	semidirect
78. C₂×C₈×C₁₃	C₄²	semidirect
131. C₈×C₁₃	C₂×C₄²	semidirect
142. C₂×#97	Dic₁₃	semidirect
194. #97	C₂×Dic₁₃	semidirect
196. C₂×C₁₃	#1776	semidirect
226. C₂×C₈	C₄×Dic₁₃	semidirect
241. C₁₃	C₂×#1776	semidirect
274. C₈	C₂×C₄×Dic₁₃	semidirect
293. C₂	#13114733	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×C₄×C₄⋊Dic₁₃	2
#13114733	2
C₂×#11718143	2
C₂²×C₁₃×#97	2
C₂×#1776	13

Subgroups — C2×#13114733 #15860974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114733 #15860974