Subgroups — C₂×#22562101 #31009265

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863518	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863503	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562101	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863219	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863212	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863206	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863173	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524050	—	2	1	no	—
10	C₂×Q₈×Dic₆₇	—	2	1	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	#16863518	—	2	2	no	—
13	#16863518	C₂²	1	4	yes	—
14	#16863503	—	2	2	no	—
15	#16863503	C₂²	1	4	yes	complemented
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	C₂×#14523975	C₂²	1	1	yes	characteristic
18	C₂×#14523970	—	2	1	no	—
19	C₂×#14523967	—	2	1	no	—
20	C₂×#14523953	C₂²	1	1	yes	characteristic
21	#16863219	C₂²	1	4	yes	complemented
22	#16863212	C₂²	1	4	yes	complemented
23	#16863206	C₂²	1	4	yes	—
24	#16863173	C₂²	1	4	yes	complemented
25	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	#510×C₆₇	C₄	1	4	yes	complemented
28	#510×C₆₇	C₂²	1	4	yes	—
29	C₂×Q₃₂×C₆₇	C₄	1	2	yes	complemented
30	C₂×SD₃₂×C₆₇	C₄	1	2	yes	complemented
31	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
32	#14524050	—	2	4	no	—
33	#14524050	—	4	2	no	—
34	C₂×#11761612	—	2	1	no	—
35	C₂×#11761612	—	4	1	no	—
36	C₂²×C₆₇⋊C₈	—	4	1	no	—
37	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
38	Q₈×Dic₆₇	—	2	4	no	—
39	Q₈×Dic₆₇	—	4	2	no	—
40	C₂×C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×C₄⋊Dic₆₇	—	4	1	no	—
42	C₂×C₄×Dic₆₇	—	4	1	no	—
43	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
44	#14523983	—	2	1	no	—
45	#14523983	C₂³	1	2	yes	—
46	C₂×C₆₇⋊C₁₆	—	2	1	no	—
47	#14523979	—	2	1	no	—
48	#14523979	C₂³	1	2	yes	—
49	#14523975	—	2	1	no	—
50	#14523975	C₂³	1	2	yes	—
51	#14523970	—	2	4	no	—
52	#14523967	—	2	4	no	—
53	#14523953	D₄	1	2	yes	—
54	#14523953	C₂³	1	2	yes	—
55	C₂×#41×C₆₇	—	2	1	no	—
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	#135×C₆₇	—	2	1	no	—
58	#135×C₆₇	—	2	1	no	—
59	#135×C₆₇	C₂×C₄	1	2	yes	—
60	#135×C₆₇	C₂³	1	2	yes	—
61	C₂×Q₁₆×C₆₇	—	2	1	no	—
62	C₂×Q₁₆×C₆₇	—	2	1	no	—
63	C₂×Q₁₆×C₆₇	—	2	1	no	—
64	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂×SD₁₆×C₆₇	—	2	1	no	—
69	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	Q₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
72	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
73	M₃₂×C₆₇	C₂×C₄	1	2	yes	—
74	M₃₂×C₆₇	C₂³	1	2	yes	—
75	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
76	C₂²×Dic₆₇	—	2	1	no	—
77	C₂²×Dic₆₇	—	4	1	no	—
78	#11761612	—	2	2	no	—
79	#11761612	—	4	4	no	—
80	#11761612	—	4	1	no	—
81	C₂×C₆₇⋊C₈	—	2	1	no	—
82	C₂×C₆₇⋊C₈	—	2	1	no	—
83	C₂×C₆₇⋊C₈	—	4	4	no	—
84	C₂×C₆₇⋊C₈	—	4	1	no	—
85	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	4	4	no	—
88	C₄⋊Dic₆₇	—	4	1	no	—
89	C₄×Dic₆₇	—	2	1	no	—
90	C₄×Dic₆₇	—	4	4	no	—
91	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
92	C₆₇⋊C₁₆	—	2	2	no	—
93	#41×C₆₇	—	2	2	no	—
94	#41×C₆₇	—	2	2	no	—
95	#41×C₆₇	—	4	1	no	—
96	#41×C₆₇	—	4	1	no	—
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	4	1	no	—
102	C₂×Q₈×C₆₇	—	4	1	no	—
103	C₂×Q₈×C₆₇	—	4	1	no	—
104	C₂×D₄×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	—	2	2	no	—
111	Q₁₆×C₆₇	—	2	1	no	—
112	Q₁₆×C₆₇	—	2	2	no	—
113	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
114	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	—
115	SD₁₆×C₆₇	—	2	2	no	—
116	SD₁₆×C₆₇	—	2	2	no	—
117	D₈×C₆₇	—	2	1	no	—
118	D₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
126	C₂³×C₆₇	—	4	1	no	—
127	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₆₇	—	2	2	no	—
129	Q₈×C₆₇	—	2	4	no	—
130	Q₈×C₆₇	—	4	2	no	—
131	Q₈×C₆₇	—	4	2	no	—
132	Q₈×C₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	4	1	no	—
134	D₄×C₆₇	—	2	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	4	2	no	—
152	C₂×Dic₆₇	—	4	2	no	—
153	C₂×Dic₆₇	—	8	2	no	—
154	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
158	C₆₇⋊C₈	—	2	2	no	—
159	C₆₇⋊C₈	—	4	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	4	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	#442	1	1	yes	characteristic
164	C₂²×C₆₇	#442	1	1	yes	characteristic
165	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	4	4	no	—
168	C₄×C₆₇	#442	1	1	yes	characteristic
169	C₄×C₆₇	#442	1	1	yes	characteristic
170	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	Dic₆₇	—	4	2	no	—
173	Dic₆₇	—	8	2	no	—
174	C₂×#2180	—	67	1	no	maximal, Sylow
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	8	2	no	—
178	C₂×C₆₇	#2180	1	2	yes	complemented
179	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
180	C₂×#510	D₆₇	1	1	yes	characteristic
181	C₂×#444	—	67	1	no	—
182	C₂×#441	—	67	1	no	—
183	#2180	—	67	8	no	—
184	C₂×#365	—	67	1	no	—
185	C₂×#362	—	67	1	no	—
186	C₂×#360	—	67	1	no	—
187	C₂×#347	—	67	1	no	—
188	C₆₇	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
190	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
191	C₂×#131	—	134	1	no	—
192	C₂×C₄×Q₈	—	134	1	no	—
193	#510	Dic₆₇	1	4	yes	complemented
194	#510	C₂×D₆₇	1	4	yes	—
195	C₂×Q₃₂	Dic₆₇	1	2	yes	complemented
196	C₂×SD₃₂	Dic₆₇	1	2	yes	complemented
197	C₂×M₃₂	—	67	1	no	—
198	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
199	#444	—	67	4	no	—
200	#444	—	134	2	no	—
201	#441	—	67	4	no	—
202	#441	—	134	2	no	—
203	C₂×#108	—	67	1	no	—
204	C₂×#106	—	67	1	no	—
205	C₂×#104	—	134	1	no	—
206	C₂×#103	—	134	1	no	—
207	C₂×#97	—	67	1	no	—
208	#365	—	67	4	no	—
209	#362	—	67	4	no	—
210	#360	—	67	4	no	—
211	#347	—	67	4	no	—
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	#135	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	C₂×Dic₆₇	1	2	yes	—
217	#135	C₂²×D₆₇	1	2	yes	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
222	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
223	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
224	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
225	C₂×SD₁₆	—	2	1	no	—
226	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
227	#131	—	268	2	no	—
228	#131	—	134	4	no	—
229	C₂×M₁₆	—	268	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	C₄×Q₈	—	268	2	no	—
235	C₄×Q₈	—	134	4	no	—
236	C₂×C₄⋊C₄	—	268	1	no	—
237	C₂×C₄⋊C₄	—	134	1	no	—
238	C₂×C₄²	—	268	1	no	—
239	C₂×C₄²	—	67	1	no	—
240	Q₃₂	C₂×Dic₆₇	1	8	yes	complemented
241	SD₃₂	C₂×Dic₆₇	1	8	yes	complemented
242	M₃₂	—	67	2	no	—
243	M₃₂	—	134	1	no	—
244	M₃₂	C₂×Dic₆₇	1	2	yes	—
245	M₃₂	C₂²×D₆₇	1	2	yes	—
246	C₂×C₁₆	—	134	1	no	—
247	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
248	#108	—	67	2	no	—
249	#108	—	134	1	no	—
250	#106	—	67	2	no	—
251	#106	—	134	1	no	—
252	#104	—	134	4	no	—
253	#103	—	134	4	no	—
254	#97	—	67	2	no	—
255	#97	—	67	2	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	268	1	no	—
272	C₂²×C₄	—	134	1	no	—
273	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	C₂²×Dic₆₇	1	4	yes	—
279	Q₁₆	C₂²×Dic₆₇	1	2	yes	—
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	C₂²×Dic₆₇	1	2	yes	—
284	M₁₆	—	268	1	no	—
285	M₁₆	—	268	4	no	—
286	M₁₆	—	134	2	no	—
287	C₂×C₈	—	268	1	no	—
288	C₂×C₈	—	268	4	no	—
289	C₂×C₈	—	67	2	no	—
290	C₂×C₈	—	134	1	no	—
291	C₂×C₈	—	134	1	no	—
292	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
294	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
295	C₂×C₈	#11761627	1	1	yes	characteristic
296	C₂×C₈	#11761627	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
298	C₄⋊C₄	—	268	4	no	—
299	C₄⋊C₄	—	268	1	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₄²	—	268	4	no	—
302	C₄²	—	67	2	no	—
303	C₄²	—	134	1	no	—
304	C₁₆	—	134	2	no	—
305	C₁₆	C₂²×Dic₆₇	1	4	yes	—
306	C₂³	—	4	1	no	—
307	C₂³	#14524067	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	—	2	4	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	268	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	536	2	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
330	C₂×C₄	#14524067	1	1	yes	characteristic
331	C₂×C₄	#14524067	1	1	yes	characteristic
332	C₂×C₄	#14524067	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
334	C₈	—	268	2	no	—
335	C₈	—	134	2	no	—
336	C₈	D₄×Dic₆₇	1	1	yes	characteristic
337	C₈	D₄×Dic₆₇	1	1	yes	characteristic
338	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
339	C₈	C₂×#11761627	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	#16863508	1	1	yes	characteristic
344	C₂²	#16863508	1	1	yes	characteristic
345	C₂²	C₂×#14524067	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	268	2	no	—
349	C₄	—	536	2	no	—
350	C₄	#16863508	1	1	yes	characteristic
351	C₄	#16863508	1	1	yes	characteristic
352	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
353	C₄	C₂×#14524067	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	8	2	no	—
357	C₂	#22562101	1	2	yes	direct product factor
358	C₂	C₂×#16863508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863508
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#510×C₆₇	C₂
socle	C₂²×C₆₇	#442
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#510	D₆₇
hypocenter	C₆₇	C₂×#2180
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#510

Complements

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

N	H ≅ G/N	type
2. C₂×#16863503	C₂	semidirect
3. #22562101	C₂	direct (N × H)
4. C₂×#16863219	C₂	semidirect
5. C₂×#16863212	C₂	semidirect
7. C₂×#16863173	C₂	semidirect
15. #16863503	C₂²	semidirect
21. #16863219	C₂²	semidirect
22. #16863212	C₂²	semidirect
24. #16863173	C₂²	semidirect
27. #510×C₆₇	C₄	semidirect
29. C₂×Q₃₂×C₆₇	C₄	semidirect
30. C₂×SD₃₂×C₆₇	C₄	semidirect
71. Q₃₂×C₆₇	C₂×C₄	semidirect
72. SD₃₂×C₆₇	C₂×C₄	semidirect
178. C₂×C₆₇	#2180	semidirect
188. C₆₇	C₂×#2180	semidirect
193. #510	Dic₆₇	semidirect
195. C₂×Q₃₂	Dic₆₇	semidirect
196. C₂×SD₃₂	Dic₆₇	semidirect
240. Q₃₂	C₂×Dic₆₇	semidirect
241. SD₃₂	C₂×Dic₆₇	semidirect
357. C₂	#22562101	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×#16863518	2
C₂×#16863503	2
#22562101	2
C₂×#16863219	2
C₂×#16863212	2
C₂×#16863206	2
C₂×#16863173	2
C₂×#510×C₆₇	2
C₂×#2180	67

Subgroups — C2×#22562101 #31009265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562101 #31009265