Subgroups — C₂×#22562425 #31010717

Order 17152.

604 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863650	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863443	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562425	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863218	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863210	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863177	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#365×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂²×#11761611	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₆₇	—	2	1	no	—
12	C₂×#14524018	—	2	1	no	—
13	C₂×#14523983	C₂²	1	1	yes	characteristic
14	#16863650	—	2	2	no	—
15	#16863650	C₂²	1	4	yes	—
16	#16863443	—	2	2	no	—
17	#16863443	C₂²	1	4	yes	complemented
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	C₂×#14523978	—	2	1	no	—
20	C₂×#14523963	—	2	1	no	—
21	#16863218	C₂²	1	4	yes	complemented
22	#16863210	C₂²	1	4	yes	complemented
23	#16863208	C₂²	1	4	yes	complemented
24	#16863177	C₂²	1	4	yes	complemented
25	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
28	#365×C₆₇	C₂²	1	4	yes	complemented
29	C₂²×D₂₆₈	—	2	1	no	—
30	C₂²×D₂₆₈	—	4	1	no	—
31	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
32	C₂×D₅₃₆	—	2	1	no	—
33	C₂×D₅₃₆	—	2	1	no	—
34	C₂×D₅₃₆	—	2	1	no	—
35	C₂×D₅₃₆	—	2	1	no	—
36	C₂×D₅₃₆	—	2	1	no	—
37	C₂×D₅₃₆	C₂³	1	2	yes	—
38	C₂×#11761612	—	2	1	no	—
39	C₂×#11761611	—	2	1	no	—
40	C₂×#11761611	—	2	1	no	—
41	C₂×#11761611	—	2	1	no	—
42	C₂×#11761611	—	2	1	no	—
43	C₂×#11761611	—	4	1	no	—
44	C₂×#11761611	D₄	1	2	yes	—
45	C₂×#11761611	C₂³	1	2	yes	—
46	M₁₆×D₆₇	—	2	4	no	—
47	M₁₆×D₆₇	—	4	2	no	—
48	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₈×D₆₇	—	4	1	no	—
50	#14524018	—	2	4	no	—
51	#14524018	—	4	2	no	—
52	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
53	#14523983	—	2	1	no	—
54	#14523983	C₂³	1	2	yes	—
55	C₂×C₆₇⋊C₁₆	—	2	1	no	—
56	#14523979	—	2	1	no	—
57	#14523979	C₂³	1	2	yes	—
58	#14523978	—	2	4	no	—
59	#14523963	—	2	4	no	—
60	C₂×M₁₆×C₆₇	—	2	1	no	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	M₃₂×C₆₇	—	2	1	no	—
64	M₃₂×C₆₇	C₂³	1	2	yes	—
65	C₂×C₁₆×C₆₇	—	2	1	no	—
66	#108×C₆₇	—	2	1	no	—
67	#108×C₆₇	C₂³	1	2	yes	—
68	#106×C₆₇	—	2	1	no	—
69	#106×C₆₇	C₂³	1	2	yes	—
70	C₂³×D₆₇	—	4	1	no	—
71	C₂³×D₆₇	SD₁₆	1	1	yes	characteristic
72	C₂×D₂₆₈	—	2	2	no	—
73	C₂×D₂₆₈	—	4	1	no	—
74	C₂×D₂₆₈	—	4	1	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	4	4	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂×D₂₆₈	—	8	4	no	—
80	C₂²×Dic₆₇	SD₁₆	1	1	yes	characteristic
81	C₂×C₄×D₆₇	—	2	1	no	—
82	C₂×C₄×D₆₇	—	2	1	no	—
83	C₂×C₄×D₆₇	—	2	1	no	—
84	C₂×C₄×D₆₇	—	2	1	no	—
85	C₂×C₄×D₆₇	SD₁₆	1	2	yes	—
86	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
87	D₅₃₆	—	2	2	no	—
88	D₅₃₆	—	2	2	no	—
89	D₅₃₆	—	4	1	no	—
90	D₅₃₆	—	4	1	no	—
91	#11761612	—	2	2	no	—
92	#11761612	—	4	1	no	—
93	#11761611	—	2	2	no	—
94	#11761611	—	2	2	no	—
95	#11761611	—	2	2	no	—
96	#11761611	—	2	2	no	—
97	#11761611	—	4	4	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	—	2	1	no	—
100	C₂×C₆₇⋊C₈	—	4	1	no	—
101	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₈×D₆₇	—	4	4	no	—
103	(C₂×D₆₇)⋊C₄	—	4	4	no	—
104	C₆₇⋊C₁₆	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	4	1	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	M₁₆×C₆₇	—	2	2	no	—
108	M₁₆×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	—	4	1	no	—
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
114	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₁₆×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂²×D₆₇	—	2	1	no	—
121	C₂²×D₆₇	—	2	1	no	—
122	C₂²×D₆₇	—	4	2	no	—
123	C₂²×D₆₇	—	4	2	no	—
124	C₂²×D₆₇	—	8	2	no	—
125	C₂²×D₆₇	—	8	2	no	—
126	C₂²×D₆₇	—	8	1	no	—
127	C₂²×D₆₇	—	8	1	no	—
128	C₂²×D₆₇	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	8	2	no	—
131	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₆₇	—	2	1	no	—
138	C₂×Dic₆₇	—	2	1	no	—
139	C₂×Dic₆₇	C₂×SD₁₆	1	2	yes	—
140	D₂₆₈	—	4	2	no	—
141	D₂₆₈	—	4	2	no	—
142	D₂₆₈	—	8	8	no	—
143	D₂₆₈	—	8	1	no	—
144	D₂₆₈	—	8	1	no	—
145	C₄×D₆₇	—	2	2	no	—
146	C₄×D₆₇	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	2	2	no	—
149	C₈×C₆₇	—	4	2	no	—
150	C₈×C₆₇	#123	1	1	yes	characteristic
151	C₈×C₆₇	#123	1	1	yes	characteristic
152	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₆₇⋊C₈	—	2	2	no	—
155	C₆₇⋊C₈	—	4	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	#481	1	1	yes	characteristic
158	C₂²×C₆₇	#481	1	1	yes	characteristic
159	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×D₆₇	—	2	2	no	—
161	C₂×D₆₇	—	4	2	no	—
162	C₂×D₆₇	—	4	2	no	—
163	C₂×D₆₇	—	8	1	no	—
164	C₂×D₆₇	—	8	2	no	—
165	C₂×D₆₇	—	8	2	no	—
166	C₂×D₆₇	—	8	2	no	—
167	C₂×D₆₇	—	16	2	no	—
168	C₂×D₆₇	—	16	2	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	#481	1	1	yes	characteristic
171	C₄×C₆₇	#481	1	1	yes	characteristic
172	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
173	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₆₇	—	2	2	no	—
175	C₂×#2238	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	#2238	1	2	yes	complemented
179	C₂×C₆₇	C₂×#481	1	1	yes	characteristic
180	D₆₇	—	4	2	no	—
181	D₆₇	—	16	2	no	—
182	C₂×#469	—	67	1	no	—
183	C₂×#430	—	67	1	no	—
184	#2238	—	67	8	no	—
185	C₂×#365	—	67	1	no	—
186	C₂×#365	D₆₇	1	1	yes	characteristic, complemented
187	C₂×#361	—	67	1	no	—
188	C₂×#361	—	67	1	no	—
189	C₂×#349	—	67	1	no	—
190	C₆₇	C₂×#2238	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₈	—	67	1	no	—
192	C₂²×M₁₆	—	67	1	no	—
193	C₂²×M₁₆	—	134	1	no	—
194	C₂×#121	—	134	1	no	—
195	C₂×M₃₂	—	67	1	no	—
196	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
197	#469	—	67	4	no	—
198	#469	—	134	2	no	—
199	#430	—	67	4	no	—
200	#430	—	134	2	no	—
201	C₂×#108	—	67	1	no	—
202	C₂×#108	—	134	1	no	—
203	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
204	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
205	C₂×#102	—	134	1	no	—
206	#365	—	67	4	no	—
207	#365	C₂×D₆₇	1	4	yes	complemented
208	#361	—	67	4	no	—
209	#361	—	67	4	no	—
210	#349	—	67	4	no	—
211	C₂²×D₄	—	268	1	no	—
212	C₂²×D₄	—	134	1	no	—
213	C₂³×C₄	—	67	1	no	—
214	C₂×D₈	—	67	2	no	—
215	C₂×D₈	—	134	1	no	—
216	C₂×D₈	—	134	1	no	—
217	C₂×D₈	—	134	1	no	—
218	C₂×D₈	—	134	1	no	—
219	C₂×D₈	—	134	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	268	1	no	—
222	C₂×M₁₆	—	268	2	no	—
223	C₂×M₁₆	—	67	2	no	—
224	C₂×M₁₆	—	67	2	no	—
225	C₂×M₁₆	—	134	4	no	—
226	C₂×M₁₆	—	134	1	no	—
227	C₂×M₁₆	—	134	1	no	—
228	C₂×M₁₆	—	134	1	no	—
229	C₂×M₁₆	—	134	1	no	—
230	C₂×M₁₆	—	134	1	no	—
231	C₂²×C₈	—	268	1	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	#121	—	268	2	no	—
235	#121	—	134	4	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	268	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	67	2	no	—
240	M₃₂	—	134	1	no	—
241	M₃₂	C₂²×D₆₇	1	2	yes	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	134	1	no	—
244	#108	—	2	1	no	—
245	#108	—	67	2	no	—
246	#108	—	134	4	no	—
247	#108	—	134	1	no	—
248	#108	C₂²×D₆₇	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	C₂²×D₆₇	1	2	yes	—
251	#102	—	134	4	no	—
252	C₂⁴	—	268	1	no	—
253	C₂⁴	—	67	1	no	—
254	C₂×D₄	—	268	4	no	—
255	C₂×D₄	—	268	1	no	—
256	C₂×D₄	—	268	1	no	—
257	C₂×D₄	—	268	1	no	—
258	C₂×D₄	—	268	1	no	—
259	C₂×D₄	—	268	1	no	—
260	C₂×D₄	—	536	4	no	—
261	C₂×D₄	—	134	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	67	1	no	—
264	C₂²×C₄	—	67	2	no	—
265	C₂²×C₄	—	67	2	no	—
266	C₂²×C₄	—	134	1	no	—
267	C₂²×C₄	—	134	1	no	—
268	C₂²×C₄	—	134	1	no	—
269	C₂²×C₄	—	134	1	no	—
270	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
271	D₈	—	268	1	no	—
272	D₈	—	268	1	no	—
273	D₈	—	134	2	no	—
274	D₈	—	134	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	268	4	no	—
278	M₁₆	—	268	1	no	—
279	M₁₆	—	134	2	no	—
280	M₁₆	—	134	2	no	—
281	M₁₆	—	134	2	no	—
282	M₁₆	—	134	2	no	—
283	M₁₆	—	134	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	268	4	no	—
286	C₂×C₈	—	268	1	no	—
287	C₂×C₈	—	67	2	no	—
288	C₂×C₈	—	134	1	no	—
289	C₂×C₈	—	134	1	no	—
290	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	#11761628	1	1	yes	characteristic
292	C₂×C₈	#11761628	1	1	yes	characteristic
293	C₂×C₈	#11761629	1	1	yes	characteristic
294	C₂×C₈	#11761629	1	1	yes	characteristic
295	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	268	4	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	134	2	no	—
301	C₂³	—	268	2	no	—
302	C₂³	—	268	2	no	—
303	C₂³	—	536	2	no	—
304	C₂³	—	536	2	no	—
305	C₂³	—	536	1	no	—
306	C₂³	—	536	1	no	—
307	C₂³	—	67	2	no	—
308	C₂³	—	134	1	no	—
309	C₂³	—	134	1	no	—
310	C₂³	SD₁₆×D₆₇	1	1	yes	characteristic
311	D₄	—	268	2	no	—
312	D₄	—	268	2	no	—
313	D₄	—	536	8	no	—
314	D₄	—	536	1	no	—
315	D₄	—	536	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	67	2	no	—
319	C₂×C₄	—	134	1	no	—
320	C₂×C₄	—	134	1	no	—
321	C₂×C₄	—	134	2	no	—
322	C₂×C₄	—	134	2	no	—
323	C₂×C₄	—	134	2	no	—
324	C₂×C₄	—	134	2	no	—
325	C₂×C₄	#14524028	1	1	yes	characteristic
326	C₂×C₄	#14524028	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₆₇	1	1	yes	characteristic
328	C₂×C₄	#14524070	1	1	yes	characteristic
329	C₂×C₄	#14524070	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	268	2	no	—
333	C₈	—	134	2	no	—
334	C₈	#14524028	1	1	yes	characteristic
335	C₈	#14524028	1	1	yes	characteristic
336	C₈	C₂×#11761628	1	1	yes	characteristic
337	C₈	C₂×#11761629	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	268	2	no	—
340	C₂²	—	268	2	no	—
341	C₂²	—	536	1	no	—
342	C₂²	—	536	2	no	—
343	C₂²	—	536	2	no	—
344	C₂²	—	536	2	no	—
345	C₂²	—	1072	2	no	—
346	C₂²	—	1072	2	no	—
347	C₂²	—	134	2	no	—
348	C₂²	#16863716	1	1	yes	characteristic
349	C₂²	#16863716	1	1	yes	characteristic
350	C₂²	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
351	C₄	—	8	2	no	—
352	C₄	—	134	2	no	—
353	C₄	#16863716	1	1	yes	characteristic
354	C₄	#16863716	1	1	yes	characteristic
355	C₄	C₂×#14524028	1	1	yes	characteristic
356	C₄	C₂×#14524070	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	268	2	no	—
360	C₂	—	1072	2	no	—
361	C₂	#22562425	1	2	yes	direct product factor
362	C₂	C₂×#16863716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863716
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#365×C₆₇	C₂
socle	C₂²×C₆₇	#481
prime radical	C₂×#16863650	C₂
hypercenter	C₂×#365	D₆₇
hypocenter	C₆₇	C₂×#2238
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. C₂×#16863443	C₂	semidirect
3. #22562425	C₂	direct (N × H)
4. C₂×#16863218	C₂	semidirect
5. C₂×#16863210	C₂	semidirect
6. C₂×#16863208	C₂	semidirect
7. C₂×#16863177	C₂	semidirect
8. C₂×#365×C₆₇	C₂	semidirect
17. #16863443	C₂²	semidirect
21. #16863218	C₂²	semidirect
22. #16863210	C₂²	semidirect
23. #16863208	C₂²	semidirect
24. #16863177	C₂²	semidirect
28. #365×C₆₇	C₂²	semidirect
178. C₂×C₆₇	#2238	semidirect
186. C₂×#365	D₆₇	semidirect
190. C₆₇	C₂×#2238	semidirect
207. #365	C₂×D₆₇	semidirect
361. C₂	#22562425	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×#16863650	2
C₂×#16863443	2
#22562425	2
C₂×#16863218	2
C₂×#16863210	2
C₂×#16863208	2
C₂×#16863177	2
C₂×#365×C₆₇	2
C₂×#2238	67

Subgroups — C2×#22562425 #31010717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562425 #31010717