Subgroups — C₂×#22590229 #32150883

Order 18176.

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×#17986440	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986122	C₂	1	1	yes	maximal, characteristic
3	#22590229	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985821	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17985815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985808	C₂	1	1	yes	maximal, characteristic
7	C₂×#17985777	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#441×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556150	C₂²	1	1	yes	characteristic
10	C₂×#14556134	—	2	1	no	—
11	#17986440	C₄	1	4	yes	complemented
12	#17986440	C₂²	1	4	yes	—
13	C₂×#14556074	C₄	1	2	yes	complemented
14	C₂×#14556071	C₄	1	2	yes	complemented
15	C₂×#14556068	C₂²	1	1	yes	characteristic
16	C₂×#14556068	C₂²	1	1	yes	characteristic
17	#17986122	—	2	2	no	—
18	#17986122	C₂²	1	4	yes	—
19	C₂×#14556064	C₂²	1	1	yes	characteristic
20	C₂×#14556056	—	2	1	no	—
21	#17985821	C₂²	1	4	yes	complemented
22	#17985815	C₂²	1	4	yes	complemented
23	#17985808	C₂²	1	4	yes	—
24	#17985777	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
27	#441×C₇₁	—	2	2	no	—
28	#441×C₇₁	C₂²	1	4	yes	complemented
29	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×#103×C₇₁	—	2	1	no	—
31	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×#11765865	—	2	1	no	—
33	#14556150	—	2	1	no	—
34	#14556150	—	2	1	no	—
35	#14556150	C₂×C₄	1	2	yes	—
36	#14556150	C₂³	1	2	yes	—
37	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
38	C₂×#11765854	—	2	1	no	—
39	C₂×D₅₆₈	C₂×C₄	1	1	yes	characteristic
40	#14556134	—	2	4	no	—
41	#14556134	—	4	2	no	—
42	C₂×#11765849	—	2	1	no	—
43	C₂×#11765848	—	4	1	no	—
44	C₂×C₈×D₇₁	—	4	1	no	—
45	#14556074	C₂×C₄	1	8	yes	complemented
46	#14556071	C₂×C₄	1	8	yes	complemented
47	#14556068	—	2	1	no	—
48	#14556068	C₂×C₄	1	2	yes	—
49	#14556068	C₂³	1	2	yes	—
50	#14556068	C₂³	1	2	yes	—
51	C₂×C₇₁⋊C₁₆	—	2	1	no	—
52	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
53	#14556064	—	2	1	no	—
54	#14556064	C₂³	1	2	yes	—
55	#14556056	—	2	4	no	—
56	C₂²×Q₈×C₇₁	—	2	1	no	—
57	C₂×Q₁₆×C₇₁	—	2	1	no	—
58	C₂×Q₁₆×C₇₁	—	2	1	no	—
59	C₂×Q₁₆×C₇₁	—	2	1	no	—
60	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
61	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
62	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
63	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₇₁	—	2	4	no	—
66	C₄×Q₈×C₇₁	—	4	2	no	—
67	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
69	C₂×C₄²×C₇₁	—	4	1	no	—
70	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
71	#106×C₇₁	—	2	1	no	—
72	#106×C₇₁	C₂³	1	2	yes	—
73	#103×C₇₁	—	2	4	no	—
74	#97×C₇₁	D₄	1	2	yes	—
75	#97×C₇₁	C₂³	1	2	yes	—
76	#11765865	—	2	2	no	—
77	#11765865	—	2	2	no	—
78	#11765865	—	4	1	no	—
79	#11765865	—	4	1	no	—
80	C₂×Dic₁₄₂	—	2	1	no	—
81	C₂×D₂₈₄	—	2	1	no	—
82	C₂×C₂²⋊D₇₁	—	4	1	no	—
83	C₂×C₄×D₇₁	—	4	1	no	—
84	Dic₂₈₄	—	2	1	no	—
85	Dic₂₈₄	C₂²×C₄	1	2	yes	—
86	#11765854	—	2	2	no	—
87	#11765854	—	2	2	no	—
88	D₅₆₈	—	2	1	no	—
89	D₅₆₈	C₂²×C₄	1	2	yes	—
90	#11765849	—	2	2	no	—
91	#11765849	—	4	1	no	—
92	#11765848	—	4	4	no	—
93	C₂×C₇₁⋊C₈	—	4	1	no	—
94	C₈×D₇₁	—	4	4	no	—
95	C₇₁⋊C₁₆	—	2	2	no	—
96	C₇₁⋊C₁₆	C₂²×C₄	1	4	yes	—
97	C₂×Q₈×C₇₁	—	2	2	no	—
98	C₂×Q₈×C₇₁	—	2	2	no	—
99	C₂×Q₈×C₇₁	—	2	2	no	—
100	C₂×Q₈×C₇₁	—	4	1	no	—
101	C₂×Q₈×C₇₁	—	4	1	no	—
102	C₂×Q₈×C₇₁	—	4	1	no	—
103	C₂²×C₄×C₇₁	—	2	1	no	—
104	C₂²×C₄×C₇₁	—	4	1	no	—
105	C₂²×C₄×C₇₁	—	4	1	no	—
106	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
107	Q₁₆×C₇₁	—	2	2	no	—
108	Q₁₆×C₇₁	—	2	2	no	—
109	Q₁₆×C₇₁	—	2	2	no	—
110	Q₁₆×C₇₁	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₇₁	—	2	1	no	—
112	C₂×C₈×C₇₁	—	2	1	no	—
113	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
119	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₇₁	—	2	2	no	—
121	C₄⋊C₄×C₇₁	—	4	4	no	—
122	C₄⋊C₄×C₇₁	—	4	1	no	—
123	C₄²×C₇₁	—	2	1	no	—
124	C₄²×C₇₁	—	4	4	no	—
125	C₄²×C₇₁	C₂×D₄	1	2	yes	—
126	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₇₁	—	4	1	no	—
128	Q₈×C₇₁	—	2	4	no	—
129	Q₈×C₇₁	—	4	2	no	—
130	Q₈×C₇₁	—	4	2	no	—
131	Q₈×C₇₁	—	4	2	no	—
132	C₂×C₄×C₇₁	—	2	2	no	—
133	C₂×C₄×C₇₁	—	4	2	no	—
134	C₂×C₄×C₇₁	—	4	2	no	—
135	C₂×C₄×C₇₁	—	4	2	no	—
136	C₂×C₄×C₇₁	—	4	2	no	—
137	C₂×C₄×C₇₁	—	4	2	no	—
138	C₂×C₄×C₇₁	—	8	2	no	—
139	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₇₁	—	4	2	no	—
146	C₂²⋊D₇₁	—	4	2	no	—
147	C₂×Dic₇₁	—	4	1	no	—
148	D₂₈₄	—	2	2	no	—
149	D₂₈₄	—	4	1	no	—
150	C₄×D₇₁	—	4	2	no	—
151	C₄×D₇₁	—	4	2	no	—
152	Dic₁₄₂	—	2	2	no	—
153	Dic₁₄₂	—	4	1	no	—
154	C₈×C₇₁	—	2	2	no	—
155	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
156	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
157	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
159	C₇₁⋊C₈	—	4	2	no	—
160	C₂²×C₇₁	—	2	2	no	—
161	C₂²×C₇₁	#442	1	1	yes	characteristic
162	C₂²×C₇₁	#442	1	1	yes	characteristic
163	C₂²×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×D₇₁	—	4	2	no	—
165	C₂×D₇₁	—	8	2	no	—
166	C₄×C₇₁	—	4	2	no	—
167	C₄×C₇₁	—	4	4	no	—
168	C₄×C₇₁	—	8	2	no	—
169	C₄×C₇₁	#442	1	1	yes	characteristic
170	C₄×C₇₁	#442	1	1	yes	characteristic
171	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	Dic₇₁	—	4	2	no	—
174	C₂×#2180	—	71	1	no	maximal, Sylow
175	C₂×C₇₁	—	2	1	no	—
176	C₂×C₇₁	—	2	1	no	—
177	C₂×C₇₁	#2180	1	2	yes	complemented
178	C₂×C₇₁	C₂×#442	1	1	yes	characteristic
179	D₇₁	—	8	2	no	—
180	C₂×#510	—	71	1	no	—
181	C₂×#444	—	71	1	no	—
182	C₂×#441	D₇₁	1	1	yes	characteristic, complemented
183	#2180	—	71	8	no	—
184	C₂×#365	—	71	1	no	—
185	C₂×#362	—	71	1	no	—
186	C₂×#360	—	71	1	no	—
187	C₂×#347	—	71	1	no	—
188	C₇₁	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	71	1	no	—
190	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
191	C₂×#131	—	142	1	no	—
192	C₂×C₄×Q₈	—	2	1	no	—
193	#510	—	71	4	no	—
194	#510	—	71	4	no	—
195	C₂×Q₃₂	—	71	2	no	—
196	C₂×SD₃₂	—	71	2	no	—
197	C₂×M₃₂	—	71	1	no	—
198	C₂×M₃₂	—	71	1	no	—
199	#444	—	71	4	no	—
200	#444	—	142	2	no	—
201	#441	—	2	2	no	—
202	#441	C₂×D₇₁	1	4	yes	complemented
203	C₂×#108	—	71	1	no	—
204	C₂×#106	C₂×D₇₁	1	1	yes	characteristic
205	C₂×#104	—	142	1	no	—
206	C₂×#103	—	2	1	no	—
207	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
208	#365	—	71	4	no	—
209	#362	—	71	4	no	—
210	#360	—	71	4	no	—
211	#347	—	71	4	no	—
212	C₂×#41	—	142	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	#135	—	71	2	no	—
215	#135	—	71	2	no	—
216	#135	—	142	1	no	—
217	#135	—	142	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	71	1	no	—
222	C₂×Q₁₆	C₄×D₇₁	1	2	yes	—
223	C₂×Q₁₆	C₄×D₇₁	1	2	yes	—
224	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
225	C₂×SD₁₆	—	142	1	no	—
226	C₂×D₈	—	71	1	no	—
227	#131	—	284	2	no	—
228	#131	—	142	4	no	—
229	C₂×M₁₆	—	284	1	no	—
230	C₂×M₁₆	—	142	1	no	—
231	C₂²×C₈	—	284	1	no	—
232	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	4	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	4	1	no	—
238	C₂×C₄²	—	4	1	no	—
239	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
240	Q₃₂	—	71	8	no	—
241	SD₃₂	—	71	8	no	—
242	M₃₂	—	71	2	no	—
243	M₃₂	—	71	2	no	—
244	M₃₂	—	71	2	no	—
245	M₃₂	—	142	1	no	—
246	C₂×C₁₆	—	71	2	no	—
247	C₂×C₁₆	—	142	1	no	—
248	#108	—	71	2	no	—
249	#108	—	142	1	no	—
250	#106	—	2	1	no	—
251	#106	C₂²×D₇₁	1	2	yes	—
252	#104	—	142	4	no	—
253	#103	—	2	4	no	—
254	#97	C₂²⋊D₇₁	1	2	yes	—
255	#97	C₂²×D₇₁	1	2	yes	—
256	#41	—	284	1	no	—
257	#41	—	284	1	no	—
258	#41	—	142	2	no	—
259	#41	—	142	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	142	1	no	—
267	C₂×D₄	—	284	1	no	—
268	C₂×D₄	—	142	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	284	1	no	—
273	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	71	2	no	—
278	Q₁₆	—	142	1	no	—
279	Q₁₆	C₂×C₄×D₇₁	1	4	yes	—
280	SD₁₆	—	142	2	no	—
281	SD₁₆	—	142	2	no	—
282	D₈	—	71	2	no	—
283	D₈	—	142	1	no	—
284	M₁₆	—	284	4	no	—
285	M₁₆	—	284	1	no	—
286	M₁₆	—	142	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	284	4	no	—
290	C₂×C₈	—	284	1	no	—
291	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
294	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
295	C₂×C₈	#11765865	1	1	yes	characteristic
296	C₂×C₈	#11765865	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	4	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	4	4	no	—
303	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
304	C₁₆	—	71	4	no	—
305	C₁₆	—	142	2	no	—
306	C₂³	—	284	1	no	—
307	C₂³	#14556154	1	1	yes	characteristic
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	284	1	no	—
313	Q₈	—	142	2	no	—
314	D₄	—	284	2	no	—
315	D₄	—	284	2	no	—
316	D₄	—	284	1	no	—
317	D₄	—	142	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	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	—	8	2	no	—
325	C₂×C₄	—	284	1	no	—
326	C₂×C₄	—	284	2	no	—
327	C₂×C₄	—	284	2	no	—
328	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
330	C₂×C₄	#14556153	1	1	yes	characteristic
331	C₂×C₄	#14556153	1	1	yes	characteristic
332	C₂×C₄	#14556154	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	284	2	no	—
336	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
339	C₈	C₂×#11765865	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	284	2	no	—
342	C₂²	—	568	2	no	—
343	C₂²	#17986109	1	1	yes	characteristic
344	C₂²	#17986109	1	1	yes	characteristic
345	C₂²	C₂×#14556154	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	8	2	no	—
349	C₄	—	284	2	no	—
350	C₄	#17986109	1	1	yes	characteristic
351	C₄	#17986109	1	1	yes	characteristic
352	C₄	C₂×C₄×C₂²⋊D₇₁	1	1	yes	characteristic
353	C₄	C₂×#14556153	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	568	2	no	—
357	C₂	#22590229	1	2	yes	direct product factor
358	C₂	C₂×#17986109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17986109
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×#441×C₇₁	C₂
socle	C₂²×C₇₁	#442
prime radical	C₂×#14556150	C₂²
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×#441	D₇₁
hypocenter	C₇₁	C₂×#2180
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#441

Complements

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

N	H ≅ G/N	type
3. #22590229	C₂	direct (N × H)
4. C₂×#17985821	C₂	semidirect
5. C₂×#17985815	C₂	semidirect
7. C₂×#17985777	C₂	semidirect
8. C₂×#441×C₇₁	C₂	semidirect
11. #17986440	C₄	semidirect
13. C₂×#14556074	C₄	semidirect
14. C₂×#14556071	C₄	semidirect
21. #17985821	C₂²	semidirect
22. #17985815	C₂²	semidirect
24. #17985777	C₂²	semidirect
28. #441×C₇₁	C₂²	semidirect
45. #14556074	C₂×C₄	semidirect
46. #14556071	C₂×C₄	semidirect
177. C₂×C₇₁	#2180	semidirect
182. C₂×#441	D₇₁	semidirect
188. C₇₁	C₂×#2180	semidirect
202. #441	C₂×D₇₁	semidirect
357. C₂	#22590229	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×#17986440	2
C₂×#17986122	2
#22590229	2
C₂×#17985821	2
C₂×#17985815	2
C₂×#17985808	2
C₂×#17985777	2
C₂×#441×C₇₁	2
C₂×#2180	71

Subgroups — C2×#22590229 #32150883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590229 #32150883