Subgroups — C₂×#97×#261 #16796718

Order 3328.

1790 conjugacy classes of subgroups.

The subgroup lattice has 328 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×C₄²×#261	C₂	1	1	yes	maximal, characteristic
5	#97×#261	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₁₃×#97	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×#261	C₄	1	4	yes	—
8	C₂²×C₄×#261	C₂²	1	1	yes	characteristic
9	C₂²×C₄×#261	C₂²	1	1	yes	characteristic
10	C₂²×D₁₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
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	C₄²×#261	C₄	1	4	yes	—
19	C₄²×#261	C₂²	1	4	yes	—
20	#11719047	C₄	1	4	yes	complemented
21	#11719047	C₂²	1	4	yes	—
22	C₂×C₁₃×#97	C₄	1	1	yes	characteristic, complemented
23	C₂×#88055	C₄	1	2	yes	complemented
24	C₂×#88043	C₄	1	1	yes	characteristic, complemented
25	D₁₃×#97	C₄	1	4	yes	complemented
26	D₁₃×#97	C₂²	1	4	yes	—
27	C₂³×#261	C₂×C₄	1	2	yes	—
28	C₂³×#261	C₂×C₄	1	2	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	—	2	16	no	—
33	C₂×C₄×#261	C₂×C₄	1	16	yes	—
34	C₂×C₄×#261	C₂×C₄	1	2	yes	—
35	C₂×C₄×#261	C₂×C₄	1	2	yes	—
36	C₂×C₄×#261	C₂×C₄	1	8	yes	—
37	C₂×C₄×#261	C₂×C₄	1	8	yes	—
38	C₂×C₄×#261	C₂³	1	2	yes	—
39	C₂×C₄×#261	C₂³	1	2	yes	—
40	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
41	C₂×D₁₃⋊C₈	C₂×C₄	1	8	yes	—
42	C₂×D₁₃⋊C₈	C₂×C₄	1	8	yes	—
43	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
44	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
50	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
51	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
52	C₂×C₈×D₁₃	C₂³	1	4	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	#88110	C₂×C₄	1	16	yes	complemented
58	#88100	C₂×C₄	1	32	yes	complemented
59	C₈×#261	C₂×C₄	1	32	yes	complemented
60	C₁₃×#97	C₂×C₄	1	4	yes	complemented
61	#88055	C₂×C₄	1	8	yes	complemented
62	#88043	C₂×C₄	1	4	yes	complemented
63	C₂×C₄×#97	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×#261	—	2	8	no	—
66	C₂²×#261	C₄²	1	8	yes	—
67	C₂²×#261	C₄²	1	8	yes	—
68	C₂²×#261	M₁₆	1	16	yes	—
69	C₂²×#261	C₂²×C₄	1	4	yes	—
70	C₂²×#261	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₄²	1	2	yes	—
74	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	—	2	8	no	—
76	C₂×C₄×D₁₃	C₄²	1	4	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
81	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×#1057	C₄²	1	16	yes	complemented
85	C₂×#1057	C₂²×C₄	1	8	yes	—
86	C₄×#261	—	2	32	no	—
87	C₄×#261	C₄²	1	8	yes	—
88	C₄×#261	C₄²	1	8	yes	—
89	C₄×#261	C₂²×C₄	1	8	yes	—
90	C₄×#261	C₂²×C₄	1	8	yes	—
91	D₁₃⋊C₈	C₄²	1	16	yes	complemented
92	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
93	D₁₃⋊C₈	C₂²×C₄	1	8	yes	—
94	C₂×C₈×C₁₃	C₄²	1	8	yes	complemented
95	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₃	C₄²	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₁₃	C₄²	1	2	yes	—
99	C₄×Dic₁₃	C₄²	1	4	yes	—
100	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
102	C₂×#1055	C₄²	1	8	yes	complemented
103	C₂×#1055	C₂²×C₄	1	4	yes	—
104	C₈×D₁₃	C₄²	1	16	yes	complemented
105	C₈×D₁₃	C₂²×C₄	1	8	yes	—
106	C₈×D₁₃	C₂²×C₄	1	8	yes	—
107	C₂²×#97	—	13	2	no	—
108	C₂²×#97	—	13	1	no	—
109	C₂²×#97	—	13	1	no	—
110	C₂²×C₄×C₈	—	13	2	no	—
111	C₂×C₄³	—	13	1	no	—
112	C₄×#97	—	13	8	no	—
113	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
114	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
116	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
117	C₂²×D₁₃	C₂×M₁₆	1	8	yes	—
118	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×#261	—	2	8	no	—
120	C₂×#261	—	2	16	no	—
121	C₂×#261	#97	1	32	yes	direct product factor
122	C₂×#261	C₂×C₄²	1	8	yes	—
123	C₂×#261	C₂×C₄²	1	8	yes	—
124	C₂×#261	C₂×M₁₆	1	16	yes	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
132	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
134	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₄×D₁₃	—	2	8	no	—
136	C₄×D₁₃	—	2	8	no	—
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	C₂×C₄²	1	4	yes	—
141	C₄×D₁₃	C₂³×C₄	1	2	yes	—
142	C₄×D₁₃	C₂³×C₄	1	2	yes	—
143	#1057	C₂×C₄²	1	16	yes	complemented
144	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
145	#1055	C₂×C₄²	1	8	yes	complemented
146	C₂³×C₈	—	13	2	no	—
147	C₂³×C₈	—	13	2	no	—
148	C₂²×C₄²	—	13	1	no	—
149	C₂²×C₄²	—	13	1	no	—
150	C₂²×C₄²	—	13	1	no	—
151	C₂²×C₄²	—	13	4	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×#97	—	13	2	no	—
158	C₂×#97	—	13	1	no	—
159	C₂×#97	—	13	4	no	—
160	C₂×#97	—	13	4	no	—
161	C₂×#97	#261	1	1	yes	characteristic, direct product factor
162	C₂×C₄×C₈	—	13	16	no	—
163	C₂×C₄×C₈	—	13	8	no	—
164	C₄³	—	13	4	no	—
165	C₄³	—	13	4	no	—
166	C₂²×C₁₃	C₄³	1	2	yes	—
167	C₂²×C₁₃	C₄×M₁₆	1	4	yes	—
168	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
169	C₂×D₁₃	C₄³	1	2	yes	—
170	C₂×D₁₃	C₂×#97	1	8	yes	—
171	C₂×D₁₃	C₄×M₁₆	1	8	yes	—
172	C₂×D₁₃	C₄×M₁₆	1	4	yes	—
173	C₂×D₁₃	C₂²×C₄²	1	1	yes	characteristic
174	C₂×D₁₃	C₂²×C₄²	1	1	yes	characteristic
175	C₂×D₁₃	C₂²×M₁₆	1	4	yes	—
176	#261	—	2	8	no	—
177	#261	C₂×#97	1	16	yes	direct product factor
178	C₄×C₁₃	—	2	4	no	—
179	C₄×C₁₃	C₄³	1	2	yes	—
180	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
181	Dic₁₃	—	2	4	no	—
182	Dic₁₃	C₄³	1	2	yes	—
183	Dic₁₃	C₂²×C₄²	1	2	yes	—
184	C₂³×C₄	—	13	2	no	—
185	C₂³×C₄	—	13	1	no	—
186	C₂³×C₄	—	13	2	no	—
187	C₂³×C₄	—	13	2	no	—
188	C₂²×C₈	—	13	4	no	—
189	C₂²×C₈	—	13	8	no	—
190	C₂²×C₈	—	13	8	no	—
191	C₂²×C₈	—	13	4	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	—	13	8	no	—
195	C₂²×C₈	—	13	8	no	—
196	C₂²×C₈	—	13	4	no	—
197	C₂²×C₈	—	13	4	no	—
198	C₂²×C₈	C₂×#261	1	2	yes	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	13	8	no	—
204	C₂×C₄²	—	13	8	no	—
205	C₂×C₄²	—	13	2	no	—
206	C₂×C₄²	—	13	1	no	—
207	C₂×C₄²	—	13	2	no	—
208	C₂×C₄²	—	13	2	no	—
209	C₂×C₄²	—	13	2	no	—
210	C₂×C₄²	—	13	2	no	—
211	C₂×C₄²	—	13	16	no	—
212	C₂×C₄²	—	26	16	no	—
213	C₂×C₄²	C₂×#261	1	1	yes	characteristic
214	#97	—	13	32	no	—
215	#97	—	13	16	no	—
216	#97	—	13	8	no	—
217	#97	—	13	4	no	—
218	#97	C₂×#261	1	4	yes	direct product factor
219	C₄×C₈	—	13	32	no	—
220	C₂×C₁₃	C₄×#97	1	4	yes	complemented
221	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
222	C₂×C₁₃	C₂×C₄×M₁₆	1	2	yes	—
223	D₁₃	C₄×#97	1	4	yes	complemented
224	D₁₃	C₂²×#97	1	2	yes	—
225	D₁₃	C₂×C₄×M₁₆	1	2	yes	—
226	C₂⁴	—	13	1	no	—
227	C₂²×C₄	—	13	4	no	—
228	C₂²×C₄	—	13	8	no	—
229	C₂²×C₄	—	13	16	no	—
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	13	4	no	—
233	C₂²×C₄	—	13	4	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	4	no	—
238	C₂²×C₄	—	13	8	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	—	13	4	no	—
241	C₂²×C₄	—	13	4	no	—
242	C₂²×C₄	—	26	8	no	—
243	C₂²×C₄	—	26	8	no	—
244	C₂²×C₄	C₄×#261	1	2	yes	—
245	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
246	C₂×C₈	—	13	8	no	—
247	C₂×C₈	—	13	8	no	—
248	C₂×C₈	—	13	8	no	—
249	C₂×C₈	—	13	16	no	—
250	C₂×C₈	—	13	16	no	—
251	C₂×C₈	—	13	8	no	—
252	C₂×C₈	—	13	4	no	—
253	C₂×C₈	—	13	8	no	—
254	C₂×C₈	—	13	8	no	—
255	C₂×C₈	—	13	16	no	—
256	C₂×C₈	C₄×#261	1	8	yes	complemented
257	C₂×C₈	C₂²×#261	1	4	yes	—
258	C₄²	—	13	8	no	—
259	C₄²	—	13	8	no	—
260	C₄²	—	13	8	no	—
261	C₄²	—	13	8	no	—
262	C₄²	—	13	4	no	—
263	C₄²	—	13	4	no	—
264	C₄²	—	13	2	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	26	32	no	—
267	C₄²	C₄×#261	1	2	yes	—
268	C₄²	C₂²×#261	1	2	yes	—
269	C₁₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	13	2	no	—
271	C₂³	—	13	2	no	—
272	C₂³	—	13	8	no	—
273	C₂³	—	13	1	no	—
274	C₂³	—	13	1	no	—
275	C₂³	C₂×C₄×#261	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	13	8	no	—
278	C₂×C₄	—	13	16	no	—
279	C₂×C₄	—	13	32	no	—
280	C₂×C₄	—	13	4	no	—
281	C₂×C₄	—	13	1	no	—
282	C₂×C₄	—	13	1	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	4	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	13	8	no	—
290	C₂×C₄	—	13	4	no	—
291	C₂×C₄	—	26	8	no	—
292	C₂×C₄	—	26	8	no	—
293	C₂×C₄	—	26	8	no	—
294	C₂×C₄	—	26	16	no	—
295	C₂×C₄	—	26	4	no	—
296	C₂×C₄	C₂×C₄×#261	1	4	yes	—
297	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄×#261	1	4	yes	—
299	C₂×C₄	C₂³×#261	1	1	yes	characteristic
300	C₈	—	13	16	no	—
301	C₈	—	13	8	no	—
302	C₈	C₂×C₄×#261	1	8	yes	complemented
303	C₂²	—	13	4	no	—
304	C₂²	—	13	4	no	—
305	C₂²	—	13	8	no	—
306	C₂²	—	13	8	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	13	1	no	—
309	C₂²	—	13	1	no	—
310	C₂²	C₄²×#261	1	2	yes	—
311	C₂²	M₁₆×#261	1	4	yes	—
312	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	13	16	no	—
315	C₄	—	13	2	no	—
316	C₄	—	13	2	no	—
317	C₄	—	26	8	no	—
318	C₄	—	26	4	no	—
319	C₄	C₄²×#261	1	2	yes	—
320	C₄	C₂²×C₄×#261	1	2	yes	—
321	C₂	—	13	2	no	—
322	C₂	—	13	2	no	—
323	C₂	—	13	4	no	—
324	C₂	#97×#261	1	4	yes	direct product factor
325	C₂	C₂×C₄²×#261	1	1	yes	characteristic
326	C₂	C₂×M₁₆×#261	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#97	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×#97	#261
hypocenter	C₁₃	C₂×C₄×#97
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
5. #97×#261	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×C₁₃×#97	C₄	semidirect
23. C₂×#88055	C₄	semidirect
24. C₂×#88043	C₄	semidirect
25. D₁₃×#97	C₄	semidirect
57. #88110	C₂×C₄	semidirect
58. #88100	C₂×C₄	semidirect
59. C₈×#261	C₂×C₄	semidirect
60. C₁₃×#97	C₂×C₄	semidirect
61. #88055	C₂×C₄	semidirect
62. #88043	C₂×C₄	semidirect
84. C₂×#1057	C₄²	semidirect
91. D₁₃⋊C₈	C₄²	semidirect
94. C₂×C₈×C₁₃	C₄²	semidirect
102. C₂×#1055	C₄²	semidirect
104. C₈×D₁₃	C₄²	semidirect
121. C₂×#261	#97	direct (N × H)
143. #1057	C₂×C₄²	semidirect
144. C₈×C₁₃	C₂×C₄²	semidirect
145. #1055	C₂×C₄²	semidirect
161. C₂×#97	#261	direct (N × H)
177. #261	C₂×#97	direct (N × H)
218. #97	C₂×#261	direct (N × H)
220. C₂×C₁₃	C₄×#97	semidirect
223. D₁₃	C₄×#97	semidirect
256. C₂×C₈	C₄×#261	semidirect
269. C₁₃	C₂×C₄×#97	semidirect
302. C₈	C₂×C₄×#261	semidirect
324. C₂	#97×#261	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×C₄²×#261	2
#97×#261	2
C₂×D₁₃×#97	2
C₂×C₄×#97	13

Subgroups — C2×#97×#261 #16796718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#97×#261 #16796718