Subgroups — C₂×#22586450 #31912376

Order 18176.

614 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
3	#22586450	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₁	—	2	1	no	—
6	C₂×#14556068	—	2	1	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	C₄	1	4	yes	complemented
9	C₂×#14556068	C₂²	1	2	yes	—
10	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
11	C₂×#14556062	—	2	1	no	—
12	C₂×#14556062	—	2	1	no	—
13	C₂×#14556062	—	2	1	no	—
14	C₂×#14556062	C₄	1	2	yes	—
15	C₂×#14556062	C₄	1	2	yes	—
16	C₂×#14556062	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₇₁	—	2	1	no	—
18	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#108×C₇₁	—	2	1	no	—
20	C₂×#108×C₇₁	—	2	2	no	—
21	C₂×#108×C₇₁	C₄	1	4	yes	complemented
22	C₂×#108×C₇₁	C₂²	1	2	yes	—
23	C₂³×Dic₇₁	—	4	1	no	—
24	C₂×C₄⋊Dic₇₁	—	2	2	no	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	4	1	no	—
28	C₂×C₄⋊Dic₇₁	—	4	1	no	—
29	C₂×C₄⋊Dic₇₁	—	4	1	no	—
30	#14556068	—	2	4	no	—
31	#14556068	—	2	2	no	—
32	#14556068	C₂×C₄	1	4	yes	complemented
33	C₂×C₇₁⋊C₁₆	—	2	4	no	—
34	C₂×C₇₁⋊C₁₆	—	2	2	no	—
35	#14556062	—	2	2	no	—
36	#14556062	—	2	2	no	—
37	#14556062	—	2	2	no	—
38	#14556062	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₁	—	2	4	no	—
41	C₂×M₁₆×C₇₁	—	2	2	no	—
42	C₂×M₁₆×C₇₁	—	4	1	no	—
43	C₂×M₁₆×C₇₁	—	4	2	no	—
44	C₂²×C₈×C₇₁	—	4	1	no	—
45	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	D₄	1	2	yes	—
53	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
54	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
55	#108×C₇₁	—	2	4	no	—
56	#108×C₇₁	—	2	2	no	—
57	#108×C₇₁	C₂×C₄	1	4	yes	complemented
58	C₂²×Dic₇₁	—	4	2	no	—
59	C₂²×Dic₇₁	—	4	2	no	—
60	C₂²×Dic₇₁	—	4	2	no	—
61	C₂²×Dic₇₁	—	8	4	no	—
62	C₄⋊Dic₇₁	—	2	4	no	—
63	C₄⋊Dic₇₁	—	4	2	no	—
64	C₄⋊Dic₇₁	—	4	2	no	—
65	C₄⋊Dic₇₁	—	4	2	no	—
66	C₇₁⋊C₁₆	—	2	4	no	—
67	C₂⁴×C₇₁	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₇₁	—	2	4	no	—
79	M₁₆×C₇₁	—	4	4	no	—
80	M₁₆×C₇₁	—	4	2	no	—
81	C₂×C₈×C₇₁	—	4	4	no	—
82	C₂×C₈×C₇₁	—	4	2	no	—
83	C₂×C₈×C₇₁	C₄²	1	2	yes	—
84	C₂×C₈×C₇₁	C₄²	1	2	yes	—
85	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
101	C₂³×C₇₁	—	2	2	no	—
102	C₂³×C₇₁	—	2	2	no	—
103	C₂³×C₇₁	#102	1	1	yes	characteristic
104	C₂³×C₇₁	#102	1	1	yes	characteristic
105	C₂³×C₇₁	#103	1	2	yes	—
106	C₂³×C₇₁	#106	1	2	yes	—
107	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#95	1	2	yes	—
109	C₂×C₄×C₇₁	#95	1	2	yes	—
110	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#102	1	2	yes	—
113	C₂×C₄×C₇₁	#102	1	2	yes	—
114	C₂×C₄×C₇₁	#103	1	2	yes	—
115	C₂×C₄×C₇₁	#103	1	2	yes	—
116	C₂×C₄×C₇₁	#103	1	2	yes	—
117	C₂×C₄×C₇₁	#106	1	2	yes	—
118	C₂×C₄×C₇₁	#107	1	2	yes	—
119	C₂×C₄×C₇₁	#107	1	2	yes	—
120	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₇₁	—	4	4	no	—
127	C₂×Dic₇₁	—	8	4	no	—
128	C₂×Dic₇₁	—	8	4	no	—
129	C₂×Dic₇₁	—	8	4	no	—
130	C₈×C₇₁	—	4	4	no	—
131	C₈×C₇₁	#95	1	2	yes	—
132	C₈×C₇₁	#95	1	2	yes	—
133	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	2	4	no	—
142	C₂²×C₇₁	—	2	4	no	—
143	C₂²×C₇₁	#340	1	2	yes	—
144	C₂²×C₇₁	#340	1	2	yes	—
145	C₂²×C₇₁	#361	1	2	yes	—
146	C₂²×C₇₁	#362	1	2	yes	—
147	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
148	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
149	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
150	C₄×C₇₁	#340	1	2	yes	—
151	C₄×C₇₁	#340	1	2	yes	—
152	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
153	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
154	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
155	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
156	Dic₇₁	—	8	4	no	—
157	C₂×#1387	—	71	1	no	maximal, Sylow
158	C₂×C₇₁	—	2	2	no	—
159	C₂×C₇₁	—	2	1	no	—
160	C₂×C₇₁	—	2	1	no	—
161	C₂×C₇₁	#1387	1	4	yes	complemented
162	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
163	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
164	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
165	C₂²×M₃₂	—	71	1	no	—
166	C₂²×#108	D₇₁	1	1	yes	characteristic
167	C₂²×#107	—	71	1	no	—
168	#1387	—	71	4	no	—
169	C₇₁	C₂×#1387	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	—	2	1	no	—
171	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	—	142	1	no	—
173	C₂×M₃₂	—	71	4	no	—
174	C₂×M₃₂	—	71	2	no	—
175	C₂×M₃₂	—	142	1	no	—
176	C₂×M₃₂	—	142	2	no	—
177	C₂²×C₁₆	—	142	1	no	—
178	C₂×#108	—	2	1	no	—
179	C₂×#108	—	2	2	no	—
180	C₂×#108	Dic₇₁	1	4	yes	complemented
181	C₂×#108	C₂×D₇₁	1	2	yes	—
182	C₂×#107	—	71	2	no	—
183	C₂×#107	—	71	2	no	—
184	C₂×#107	—	71	2	no	—
185	C₂×#107	—	142	1	no	—
186	C₂×#107	—	142	1	no	—
187	C₂×#107	—	142	1	no	—
188	C₂³×C₄	—	284	1	no	—
189	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	1	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	4	1	no	—
195	C₂²×C₈	Dic₁₄₂	1	2	yes	—
196	C₂²×C₈	C₄×D₇₁	1	2	yes	—
197	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
199	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
200	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
201	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
202	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
204	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	284	1	no	—
206	C₂×C₄⋊C₄	—	284	1	no	—
207	C₂×C₄⋊C₄	—	284	1	no	—
208	C₂×C₄⋊C₄	—	142	2	no	—
209	C₂×C₄⋊C₄	—	142	2	no	—
210	C₂×C₄⋊C₄	—	142	2	no	—
211	M₃₂	—	71	4	no	—
212	M₃₂	—	142	4	no	—
213	M₃₂	—	142	2	no	—
214	C₂×C₁₆	—	142	4	no	—
215	C₂×C₁₆	—	142	2	no	—
216	#108	—	2	4	no	—
217	#108	—	2	2	no	—
218	#108	C₂×Dic₇₁	1	4	yes	complemented
219	#107	—	71	4	no	—
220	#107	—	142	2	no	—
221	#107	—	142	2	no	—
222	#107	—	142	2	no	—
223	C₂⁴	#11765852	1	1	yes	characteristic
224	C₂²×C₄	—	284	2	no	—
225	C₂²×C₄	—	284	2	no	—
226	C₂²×C₄	—	284	2	no	—
227	C₂²×C₄	—	568	4	no	—
228	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
231	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
232	C₂²×C₄	#11765850	1	1	yes	characteristic
233	C₂²×C₄	#11765850	1	1	yes	characteristic
234	C₂²×C₄	#11765852	1	1	yes	characteristic
235	C₂²×C₄	#11765853	1	2	yes	—
236	C₂²×C₄	#11765855	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	4	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
244	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
245	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
246	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
247	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
248	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
249	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
250	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
251	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
252	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
253	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
254	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
260	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
261	C₄⋊C₄	—	284	2	no	—
262	C₄⋊C₄	—	284	2	no	—
263	C₄⋊C₄	—	284	2	no	—
264	C₄⋊C₄	—	142	4	no	—
265	C₁₆	—	142	4	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#14556048	1	1	yes	characteristic
269	C₂³	#14556048	1	1	yes	characteristic
270	C₂³	#14556052	1	2	yes	—
271	C₂³	#14556059	1	2	yes	—
272	C₂³	C₂×#11765852	1	1	yes	characteristic
273	C₂×C₄	—	284	4	no	—
274	C₂×C₄	—	568	4	no	—
275	C₂×C₄	—	568	4	no	—
276	C₂×C₄	—	568	4	no	—
277	C₂×C₄	C₇₁⋊#95	1	2	yes	—
278	C₂×C₄	C₇₁⋊#95	1	2	yes	—
279	C₂×C₄	#14556048	1	1	yes	characteristic
280	C₂×C₄	#14556048	1	1	yes	characteristic
281	C₂×C₄	#14556049	1	2	yes	—
282	C₂×C₄	#14556049	1	2	yes	—
283	C₂×C₄	#14556051	1	2	yes	—
284	C₂×C₄	#14556051	1	2	yes	—
285	C₂×C₄	#14556052	1	2	yes	—
286	C₂×C₄	#14556059	1	2	yes	—
287	C₂×C₄	#14556061	1	2	yes	—
288	C₂×C₄	#14556061	1	2	yes	—
289	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
291	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
293	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
294	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	C₇₁⋊#95	1	2	yes	—
297	C₈	C₇₁⋊#95	1	2	yes	—
298	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
299	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
300	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
301	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#17985761	1	2	yes	—
309	C₂²	#17985761	1	2	yes	—
310	C₂²	#17985811	1	2	yes	—
311	C₂²	#17985814	1	2	yes	—
312	C₂²	C₂×#14556048	1	1	yes	characteristic
313	C₂²	C₂×#14556052	1	1	yes	characteristic
314	C₂²	C₂×#14556059	1	1	yes	characteristic
315	C₄	—	568	4	no	—
316	C₄	#17985761	1	2	yes	—
317	C₄	#17985761	1	2	yes	—
318	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
319	C₄	C₂×#14556049	1	1	yes	characteristic
320	C₄	C₂×#14556051	1	1	yes	characteristic
321	C₄	C₂×#14556061	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	#22586450	1	4	yes	direct product factor
326	C₂	C₂×#17985761	1	1	yes	characteristic
327	C₂	C₂×#17985811	1	1	yes	characteristic
328	C₂	C₂×#17985814	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556048
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂²×#108×C₇₁	C₂
socle	C₂³×C₇₁	#102
prime radical	C₂⁴×C₇₁	SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂²×#108	D₇₁
hypocenter	C₇₁	C₂×#1387
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #22586450	C₂	direct (N × H)
8. C₂×#14556068	C₄	semidirect
21. C₂×#108×C₇₁	C₄	semidirect
32. #14556068	C₂×C₄	semidirect
57. #108×C₇₁	C₂×C₄	semidirect
161. C₂×C₇₁	#1387	semidirect
169. C₇₁	C₂×#1387	semidirect
180. C₂×#108	Dic₇₁	semidirect
218. #108	C₂×Dic₇₁	semidirect
325. C₂	#22586450	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14556068	2
C₂²×#14556062	2
#22586450	2
C₂²×#108×C₇₁	2
C₂×#1387	71

Subgroups — C2×#22586450 #31912376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586450 #31912376