Subgroups — C₂×#22586417 #31912288

Order 18176.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586417	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
8	C₂²×C₇₁⋊C₁₆	—	2	2	no	—
9	C₂²×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
10	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₈×Dic₇₁	—	2	2	no	—
12	C₂×C₈×Dic₇₁	—	2	2	no	—
13	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	—	2	2	no	—
17	C₂²×C₁₆×C₇₁	—	2	2	no	—
18	C₂²×C₁₆×C₇₁	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
20	C₂³×Dic₇₁	C₈	1	2	yes	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₂²×C₇₁⋊C₈	—	2	2	no	—
23	C₂²×C₇₁⋊C₈	C₈	1	4	yes	—
24	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₇₁	—	2	2	no	—
26	C₂×C₄×Dic₇₁	—	2	2	no	—
27	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₇₁⋊C₁₆	—	2	8	no	—
30	C₂×C₇₁⋊C₁₆	—	2	2	no	—
31	C₂×C₇₁⋊C₁₆	—	2	2	no	—
32	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₇₁	—	2	4	no	—
34	C₈×Dic₇₁	—	2	2	no	—
35	C₈×Dic₇₁	—	2	2	no	—
36	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₁	D₄	1	2	yes	—
41	C₂²×C₈×C₇₁	D₄	1	2	yes	—
42	C₂²×C₈×C₇₁	D₄	1	2	yes	—
43	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
44	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₇₁	—	2	8	no	—
46	C₂×C₁₆×C₇₁	—	2	2	no	—
47	C₂×C₁₆×C₇₁	—	2	2	no	—
48	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
49	C₂²×Dic₇₁	—	2	4	no	—
50	C₂²×Dic₇₁	—	2	4	no	—
51	C₂²×Dic₇₁	C₁₆	1	8	yes	complemented
52	C₂²×Dic₇₁	C₂×C₈	1	4	yes	—
53	C₂×C₇₁⋊C₈	—	2	8	no	—
54	C₂×C₇₁⋊C₈	—	2	2	no	—
55	C₂×C₇₁⋊C₈	—	2	2	no	—
56	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
57	C₄×Dic₇₁	—	2	4	no	—
58	C₄×Dic₇₁	—	2	2	no	—
59	C₄×Dic₇₁	—	2	2	no	—
60	C₇₁⋊C₁₆	—	2	4	no	—
61	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
63	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
68	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₄²	1	2	yes	—
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
74	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
77	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
83	C₁₆×C₇₁	—	2	4	no	—
84	C₂³×C₇₁	C₄×C₈	1	2	yes	—
85	C₂³×C₇₁	C₂²⋊C₈	1	2	yes	—
86	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
87	C₂³×C₇₁	C₂×C₁₆	1	4	yes	—
88	C₂³×C₇₁	M₃₂	1	4	yes	—
89	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#95	1	4	yes	—
91	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₇₁	#97	1	4	yes	—
93	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₇₁	—	2	16	no	—
104	C₂×Dic₇₁	—	2	4	no	—
105	C₂×Dic₇₁	—	2	4	no	—
106	C₂×Dic₇₁	C₂×C₁₆	1	8	yes	complemented
107	C₈×C₇₁	#95	1	4	yes	—
108	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₇₁⋊C₈	—	2	4	no	—
113	C₂²×C₇₁	#336	1	4	yes	—
114	C₂²×C₇₁	C₄×C₁₆	1	4	yes	—
115	C₂²×C₇₁	#346	1	4	yes	—
116	C₂²×C₇₁	C₂²⋊C₁₆	1	8	yes	—
117	C₂²×C₇₁	C₄⋊C₁₆	1	8	yes	—
118	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₇₁	C₂²×C₁₆	1	2	yes	—
122	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
123	C₄×C₇₁	#336	1	4	yes	—
124	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
125	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
126	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
127	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
128	Dic₇₁	—	2	8	no	—
129	C₂×#1374	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	#1374	1	8	yes	complemented
131	C₂×C₇₁	C₂×#336	1	1	yes	characteristic
132	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
133	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
134	C₂×C₇₁	C₂×C₂²⋊C₁₆	1	2	yes	—
135	C₂×C₇₁	C₂×C₄⋊C₁₆	1	2	yes	—
136	C₂³×C₁₆	—	71	1	no	—
137	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
138	C₂²×C₄×C₈	—	71	1	no	—
139	#1374	—	71	4	no	—
140	C₇₁	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₈	—	71	1	no	—
142	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
143	C₂²×C₄²	—	71	1	no	—
144	C₂²×C₁₆	—	2	2	no	—
145	C₂²×C₁₆	—	2	2	no	—
146	C₂²×C₁₆	—	71	4	no	—
147	C₂²×C₁₆	—	71	2	no	—
148	C₂²×C₁₆	—	142	2	no	—
149	C₂²×C₁₆	—	142	2	no	—
150	C₂²×C₁₆	Dic₇₁	1	4	yes	complemented
151	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
152	C₂×C₄×C₈	—	71	2	no	—
153	C₂×C₄×C₈	—	71	2	no	—
154	C₂×C₄×C₈	—	142	2	no	—
155	C₂×C₄×C₈	—	142	2	no	—
156	C₂³×C₄	—	71	2	no	—
157	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
158	C₂²×C₈	—	71	4	no	—
159	C₂²×C₈	—	71	2	no	—
160	C₂²×C₈	—	142	2	no	—
161	C₂²×C₈	—	142	2	no	—
162	C₂²×C₈	Dic₁₄₂	1	2	yes	—
163	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₇₁	1	2	yes	—
165	C₂²×C₈	D₂₈₄	1	2	yes	—
166	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
167	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
168	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
169	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
170	C₂×C₄²	—	71	2	no	—
171	C₂×C₄²	—	71	2	no	—
172	C₂×C₄²	—	142	2	no	—
173	C₂×C₄²	—	142	2	no	—
174	C₂×C₁₆	—	2	8	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	71	4	no	—
178	C₂×C₁₆	—	142	8	no	—
179	C₂×C₁₆	—	142	2	no	—
180	C₂×C₁₆	—	142	2	no	—
181	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	complemented
182	C₄×C₈	—	142	4	no	—
183	C₄×C₈	—	142	2	no	—
184	C₄×C₈	—	142	2	no	—
185	C₂⁴	C₈×D₇₁	1	1	yes	characteristic
186	C₂²×C₄	—	71	8	no	—
187	C₂²×C₄	—	71	4	no	—
188	C₂²×C₄	—	142	4	no	—
189	C₂²×C₄	—	142	4	no	—
190	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
191	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
192	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
193	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
194	C₂²×C₄	C₈×D₇₁	1	2	yes	—
195	C₂²×C₄	#11765848	1	2	yes	—
196	C₂²×C₄	#11765848	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
198	C₂×C₈	—	71	4	no	—
199	C₂×C₈	—	142	8	no	—
200	C₂×C₈	—	142	2	no	—
201	C₂×C₈	—	142	2	no	—
202	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
203	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
204	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
205	C₂×C₈	C₂²⋊Dic₇₁	1	4	yes	—
206	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
207	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
208	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
209	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
214	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
215	C₄²	—	142	4	no	—
216	C₄²	—	142	2	no	—
217	C₄²	—	142	2	no	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	142	4	no	—
220	C₂³	C₈×Dic₇₁	1	2	yes	—
221	C₂³	(C₂×D₇₁)⋊C₈	1	2	yes	—
222	C₂³	Dic₇₁⋊C₈	1	2	yes	—
223	C₂³	C₁₆×D₇₁	1	4	yes	—
224	C₂³	#14556067	1	4	yes	—
225	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
226	C₂×C₄	—	71	8	no	—
227	C₂×C₄	—	142	4	no	—
228	C₂×C₄	—	142	4	no	—
229	C₂×C₄	—	142	16	no	—
230	C₂×C₄	C₇₁⋊#95	1	4	yes	—
231	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
232	C₂×C₄	#14556038	1	4	yes	—
233	C₂×C₄	(C₂×D₇₁)⋊C₈	1	4	yes	—
234	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
235	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
236	C₂×C₄	Dic₇₁⋊C₈	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
241	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
242	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
243	C₈	—	142	4	no	—
244	C₈	C₇₁⋊#95	1	4	yes	—
245	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
246	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
247	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
248	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
249	C₂²	#17985753	1	4	yes	—
250	C₂²	C₁₆×Dic₇₁	1	4	yes	—
251	C₂²	#17985774	1	4	yes	—
252	C₂²	(C₂×D₇₁)⋊C₁₆	1	8	yes	—
253	C₂²	Dic₇₁⋊C₁₆	1	8	yes	—
254	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
255	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
256	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
257	C₂²	C₂×C₁₆×D₇₁	1	2	yes	—
258	C₂²	C₂×#14556067	1	2	yes	—
259	C₄	—	142	8	no	—
260	C₄	#17985753	1	4	yes	—
261	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
262	C₄	C₂×#14556038	1	1	yes	characteristic
263	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
264	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
265	C₂	#22586417	1	8	yes	direct product factor
266	C₂	C₂×#17985753	1	1	yes	characteristic
267	C₂	C₂×C₁₆×Dic₇₁	1	1	yes	characteristic
268	C₂	C₂×#17985774	1	1	yes	characteristic
269	C₂	C₂×(C₂×D₇₁)⋊C₁₆	1	2	yes	—
270	C₂	C₂×Dic₇₁⋊C₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂³×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#1374
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22586417	C₂	direct (N × H)
9. C₂²×C₇₁⋊C₁₆	C₄	semidirect
18. C₂²×C₁₆×C₇₁	C₄	semidirect
32. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
48. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
51. C₂²×Dic₇₁	C₁₆	semidirect
106. C₂×Dic₇₁	C₂×C₁₆	semidirect
130. C₂×C₇₁	#1374	semidirect
140. C₇₁	C₂×#1374	semidirect
150. C₂²×C₁₆	Dic₇₁	semidirect
181. C₂×C₁₆	C₂×Dic₇₁	semidirect
265. C₂	#22586417	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂²×C₈×Dic₇₁	2
#22586417	2
C₂³×C₁₆×C₇₁	2
C₂×#1374	71

Subgroups — C2×#22586417 #31912288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586417 #31912288