Subgroups — C₂×#13130492 #16796722

Order 3328.

1614 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719047	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88100	C₂	1	2	yes	maximal
3	C₂²×C₈×#261	C₂	1	2	yes	maximal
4	C₂×#11719057	C₂	1	1	yes	maximal, characteristic
5	#13130492	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×#261	C₂²	1	2	yes	—
8	C₂²×D₁₃⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×(C₂×#261)⋊C₄	C₄	1	4	yes	—
13	C₂×#88110	C₄	1	4	yes	complemented
14	C₂×#88100	C₄	1	16	yes	complemented
15	C₂×#88100	C₂²	1	8	yes	—
16	C₂×C₈×#261	C₄	1	16	yes	complemented
17	C₂×C₈×#261	C₂²	1	8	yes	—
18	#11719057	C₄	1	4	yes	—
19	#11719057	C₂²	1	4	yes	—
20	#11719047	C₄	1	4	yes	complemented
21	#11719047	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
23	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
24	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
25	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
26	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
27	C₄×C₈×D₁₃	C₂²	1	4	yes	—
28	C₂³×#261	C₂×C₄	1	4	yes	—
29	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
31	C₂²×#1057	C₂×C₄	1	4	yes	—
32	C₂×C₄×#261	C₂×C₄	1	4	yes	—
33	C₂×C₄×#261	C₂×C₄	1	8	yes	—
34	C₂×C₄×#261	C₂×C₄	1	8	yes	—
35	C₂×C₄×#261	C₂³	1	4	yes	—
36	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×D₁₃⋊C₈	C₂×C₄	1	8	yes	—
38	C₂×D₁₃⋊C₈	C₂×C₄	1	8	yes	—
39	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
50	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
51	C₂×C₈×D₁₃	C₂³	1	2	yes	—
52	C₂×C₈×D₁₃	C₂³	1	2	yes	—
53	C₄²×D₁₃	C₂×C₄	1	2	yes	—
54	C₄²×D₁₃	C₂×C₄	1	2	yes	—
55	C₄²×D₁₃	C₂×C₄	1	2	yes	—
56	C₄²×D₁₃	C₂³	1	2	yes	—
57	(C₂×#261)⋊C₄	C₂×C₄	1	16	yes	—
58	#88110	C₂×C₄	1	16	yes	complemented
59	#88100	C₂×C₄	1	32	yes	complemented
60	C₈×#261	C₂×C₄	1	32	yes	complemented
61	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
62	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
63	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
64	C₄×#1055	C₂×C₄	1	4	yes	complemented
65	C₂×#1726	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×#261	—	2	16	no	—
68	C₂²×#261	C₄²	1	16	yes	—
69	C₂²×#261	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₄²	1	2	yes	—
73	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	2	8	no	—
75	C₂×C₄×D₁₃	C₄²	1	4	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
83	C₂×#1057	C₄²	1	16	yes	—
84	C₂×#1057	C₂²×C₄	1	8	yes	—
85	C₄×#261	C₄²	1	16	yes	—
86	C₄×#261	C₂²×C₄	1	8	yes	—
87	C₄×#261	C₂²×C₄	1	8	yes	—
88	D₁₃⋊C₈	C₄²	1	16	yes	—
89	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
90	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
91	C₂×C₈×C₁₃	C₄²	1	4	yes	—
92	C₂×C₈×C₁₃	C₄²	1	4	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₄²	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₁₃	C₄²	1	2	yes	—
98	C₄×Dic₁₃	C₄²	1	4	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
101	C₂×#1055	C₄²	1	4	yes	—
102	C₂×#1055	C₄²	1	4	yes	—
103	C₂×#1055	C₂²×C₄	1	2	yes	—
104	C₂×#1055	C₂²×C₄	1	2	yes	—
105	C₈×D₁₃	C₄²	1	8	yes	—
106	C₈×D₁₃	C₄²	1	8	yes	—
107	C₈×D₁₃	C₂²×C₄	1	8	yes	—
108	C₈×D₁₃	C₂²×C₄	1	8	yes	—
109	C₂²×#97	—	13	2	no	—
110	C₂²×#97	—	13	1	no	—
111	C₂²×C₄×C₈	—	13	2	no	—
112	C₂²×C₄×C₈	—	13	1	no	—
113	C₂×#376	—	13	1	no	—
114	#1726	—	13	8	no	—
115	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
116	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
118	C₂²×D₁₃	#131	1	8	yes	—
119	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×#261	—	2	16	no	—
121	C₂×#261	—	2	32	no	—
122	C₂×#261	C₂×C₄²	1	16	yes	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
131	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
133	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₄×D₁₃	—	2	8	no	—
135	C₄×D₁₃	—	2	8	no	—
136	C₄×D₁₃	C₂×C₄²	1	4	yes	—
137	C₄×D₁₃	C₂×C₄²	1	2	yes	—
138	C₄×D₁₃	C₂×C₄²	1	2	yes	—
139	C₄×D₁₃	C₂×C₄²	1	4	yes	—
140	C₄×D₁₃	C₂³×C₄	1	2	yes	—
141	C₄×D₁₃	C₂³×C₄	1	2	yes	—
142	#1057	C₂×C₄²	1	16	yes	—
143	C₈×C₁₃	C₂×C₄²	1	4	yes	—
144	C₈×C₁₃	C₂×C₄²	1	4	yes	—
145	#1055	C₂×C₄²	1	4	yes	—
146	#1055	C₂×C₄²	1	4	yes	—
147	C₂³×C₈	—	13	2	no	—
148	C₂³×C₈	—	13	1	no	—
149	C₂³×C₈	—	13	1	no	—
150	C₂²×C₄²	—	13	2	no	—
151	C₂²×C₄²	—	13	1	no	—
152	C₂×#97	—	13	4	no	—
153	C₂×#97	—	13	4	no	—
154	C₂×#97	—	13	16	no	—
155	C₂×#97	—	13	8	no	—
156	C₂×#97	—	13	4	no	—
157	C₂×C₄×C₈	—	13	16	no	—
158	C₂×C₄×C₈	—	13	8	no	—
159	C₂×C₄×C₈	—	13	1	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	—	13	4	no	—
163	C₂×C₄×C₈	—	13	4	no	—
164	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
165	#376	—	13	4	no	—
166	#376	—	13	4	no	—
167	C₂×#95	—	13	4	no	—
168	C₂²×C₁₃	C₄³	1	2	yes	—
169	C₂²×C₁₃	#405	1	4	yes	—
170	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
171	C₂×D₁₃	C₄³	1	2	yes	—
172	C₂×D₁₃	#405	1	16	yes	—
173	C₂×D₁₃	#405	1	4	yes	—
174	C₂×D₁₃	C₂²×C₄²	1	2	yes	—
175	C₂×D₁₃	C₂×#131	1	4	yes	—
176	#261	—	2	16	no	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	C₄³	1	2	yes	—
179	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
180	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
181	Dic₁₃	—	2	4	no	—
182	Dic₁₃	C₄³	1	2	yes	—
183	Dic₁₃	C₂²×C₄²	1	2	yes	—
184	C₂³×C₄	—	13	1	no	—
185	C₂³×C₄	—	13	4	no	—
186	C₂³×C₄	—	13	2	no	—
187	C₂²×C₈	—	13	4	no	—
188	C₂²×C₈	—	13	8	no	—
189	C₂²×C₈	—	13	8	no	—
190	C₂²×C₈	—	13	4	no	—
191	C₂²×C₈	—	13	4	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	8	no	—
195	C₂²×C₈	—	13	8	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	C₂×#261	1	1	yes	characteristic
201	C₂²×C₈	C₂×#261	1	1	yes	characteristic
202	C₂×C₄²	—	13	4	no	—
203	C₂×C₄²	—	13	4	no	—
204	C₂×C₄²	—	13	8	no	—
205	C₂×C₄²	—	13	8	no	—
206	C₂×C₄²	—	13	2	no	—
207	C₂×C₄²	—	13	1	no	—
208	C₂×C₄²	—	13	2	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₄²	—	13	2	no	—
211	C₂×C₄²	—	13	2	no	—
212	C₂×C₄²	C₂×#261	1	1	yes	characteristic
213	#97	—	13	32	no	—
214	#97	—	13	16	no	—
215	C₄×C₈	—	13	32	no	—
216	C₄×C₈	—	13	4	no	—
217	C₄×C₈	—	13	4	no	—
218	C₄×C₈	—	13	4	no	—
219	C₄×C₈	C₂×#261	1	4	yes	complemented
220	#95	—	13	16	no	—
221	C₂×C₁₃	#1726	1	4	yes	complemented
222	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#405	1	2	yes	—
224	D₁₃	#1726	1	4	yes	complemented
225	D₁₃	C₂×#405	1	4	yes	—
226	C₂⁴	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	4	no	—
230	C₂²×C₄	—	13	4	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	8	no	—
235	C₂²×C₄	—	13	16	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	4	no	—
238	C₂²×C₄	—	13	4	no	—
239	C₂²×C₄	—	26	8	no	—
240	C₂²×C₄	—	26	16	no	—
241	C₂²×C₄	C₄×#261	1	2	yes	—
242	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
243	C₂×C₈	—	13	8	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	16	no	—
247	C₂×C₈	—	13	16	no	—
248	C₂×C₈	—	13	4	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	8	no	—
253	C₂×C₈	—	13	8	no	—
254	C₂×C₈	—	13	8	no	—
255	C₂×C₈	—	13	8	no	—
256	C₂×C₈	C₄×#261	1	4	yes	—
257	C₂×C₈	C₄×#261	1	4	yes	—
258	C₂×C₈	C₂²×#261	1	2	yes	—
259	C₂×C₈	C₂²×#261	1	2	yes	—
260	C₄²	—	13	8	no	—
261	C₄²	—	13	8	no	—
262	C₄²	—	13	16	no	—
263	C₄²	—	13	4	no	—
264	C₄²	—	13	4	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	13	2	no	—
267	C₄²	C₄×#261	1	2	yes	—
268	C₄²	C₂²×#261	1	2	yes	—
269	C₁₃	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	13	4	no	—
271	C₂³	—	13	8	no	—
272	C₂³	—	13	1	no	—
273	C₂³	—	13	1	no	—
274	C₂³	C₂×C₄×#261	1	1	yes	characteristic
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	13	4	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	4	no	—
280	C₂×C₄	—	13	4	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	16	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	26	8	no	—
288	C₂×C₄	—	26	8	no	—
289	C₂×C₄	—	26	16	no	—
290	C₂×C₄	—	26	32	no	—
291	C₂×C₄	—	26	4	no	—
292	C₂×C₄	C₂×C₄×#261	1	4	yes	—
293	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄×#261	1	2	yes	—
295	C₂×C₄	C₂×C₄×#261	1	2	yes	—
296	C₂×C₄	C₂³×#261	1	1	yes	characteristic
297	C₈	—	13	16	no	—
298	C₈	—	13	4	no	—
299	C₈	—	13	4	no	—
300	C₈	C₂×C₄×#261	1	4	yes	—
301	C₈	C₂×C₄×#261	1	4	yes	—
302	C₂²	—	13	4	no	—
303	C₂²	—	13	4	no	—
304	C₂²	—	13	16	no	—
305	C₂²	—	13	2	no	—
306	C₂²	—	13	2	no	—
307	C₂²	C₄²×#261	1	2	yes	—
308	C₂²	#11719089	1	4	yes	—
309	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	13	2	no	—
312	C₄	—	13	2	no	—
313	C₄	—	26	16	no	—
314	C₄	—	26	4	no	—
315	C₄	C₄²×#261	1	2	yes	—
316	C₄	C₂²×C₄×#261	1	1	yes	characteristic
317	C₄	C₂²×C₄×#261	1	1	yes	characteristic
318	C₂	—	13	4	no	—
319	C₂	—	13	4	no	—
320	C₂	#13130492	1	4	yes	direct product factor
321	C₂	C₂×C₄²×#261	1	1	yes	characteristic
322	C₂	C₂×#11719089	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1726
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #13130492	C₂	direct (N × H)
13. C₂×#88110	C₄	semidirect
14. C₂×#88100	C₄	semidirect
16. C₂×C₈×#261	C₄	semidirect
20. #11719047	C₄	semidirect
22. C₂×C₄×C₈×C₁₃	C₄	semidirect
23. C₂×C₈×Dic₁₃	C₄	semidirect
24. C₂×C₈×Dic₁₃	C₄	semidirect
25. C₂×C₄×#1055	C₄	semidirect
26. C₄×C₈×D₁₃	C₄	semidirect
58. #88110	C₂×C₄	semidirect
59. #88100	C₂×C₄	semidirect
60. C₈×#261	C₂×C₄	semidirect
61. C₄×C₈×C₁₃	C₂×C₄	semidirect
62. C₈×Dic₁₃	C₂×C₄	semidirect
63. C₈×Dic₁₃	C₂×C₄	semidirect
64. C₄×#1055	C₂×C₄	semidirect
164. C₂×C₄×C₈	#261	semidirect
219. C₄×C₈	C₂×#261	semidirect
221. C₂×C₁₃	#1726	semidirect
224. D₁₃	#1726	semidirect
269. C₁₃	C₂×#1726	semidirect
320. C₂	#13130492	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×#11719047	2
C₂²×#88100	2
C₂²×C₈×#261	2
C₂×#11719057	2
#13130492	2
C₂×C₄×C₈×D₁₃	2
C₂×#1726	13

Subgroups — C2×#13130492 #16796722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130492 #16796722