Subgroups — C₂×#22586171 #31886868

Order 18176.

678 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	C₂×#17986052	C₂	1	2	yes	maximal
2	#22586171	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17986052	—	2	2	no	—
5	#17986052	—	2	2	no	—
6	#17986052	C₄	1	4	yes	—
7	#17986052	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₇₁⋊C₈	C₄	1	4	yes	—
11	C₂³×C₈×C₇₁	C₄	1	2	yes	—
12	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#97×C₇₁	—	2	2	no	—
14	C₂×#97×C₇₁	—	2	2	no	—
15	C₂×#97×C₇₁	C₄	1	2	yes	—
16	C₂×#97×C₇₁	C₂²	1	2	yes	—
17	C₂²×C₇₁⋊C₈	—	2	4	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
22	C₄×C₇₁⋊C₈	—	2	4	no	—
23	C₄×C₇₁⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	—	2	4	no	—
30	C₂²×C₈×C₇₁	C₈	1	8	yes	complemented
31	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	D₄	1	2	yes	—
37	C₂×C₄²×C₇₁	D₄	1	2	yes	—
38	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	#97×C₇₁	—	2	4	no	—
41	#97×C₇₁	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	2	8	no	—
43	C₂×C₇₁⋊C₈	—	4	8	no	—
44	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	—	4	2	no	—
46	C₂²×C₄×C₇₁	—	4	2	no	—
47	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₇₁	M₁₆	1	4	yes	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	2	8	no	—
65	C₂×C₈×C₇₁	—	2	4	no	—
66	C₂×C₈×C₇₁	—	2	4	no	—
67	C₂×C₈×C₇₁	C₂×C₈	1	8	yes	complemented
68	C₄²×C₇₁	C₄²	1	2	yes	—
69	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₁	C₂×D₄	1	2	yes	—
76	C₂³×C₇₁	—	2	2	no	—
77	C₂³×C₇₁	—	2	2	no	—
78	C₂³×C₇₁	#95	1	1	yes	characteristic
79	C₂³×C₇₁	#95	1	1	yes	characteristic
80	C₂³×C₇₁	C₄⋊C₈	1	4	yes	—
81	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₁	—	2	4	no	—
83	C₂×C₄×C₇₁	—	2	4	no	—
84	C₂×C₄×C₇₁	—	4	2	no	—
85	C₂×C₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	—	4	1	no	—
87	C₂×C₄×C₇₁	—	4	1	no	—
88	C₂×C₄×C₇₁	—	4	1	no	—
89	C₂×C₄×C₇₁	—	4	1	no	—
90	C₂×C₄×C₇₁	#95	1	2	yes	—
91	C₂×C₄×C₇₁	#95	1	2	yes	—
92	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₇₁	#97	1	2	yes	—
97	C₂×C₄×C₇₁	#97	1	2	yes	—
98	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₇₁	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
101	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
107	C₈×C₇₁	—	2	8	no	—
108	C₇₁⋊C₈	—	4	8	no	—
109	C₂²×C₇₁	—	2	1	no	—
110	C₂²×C₇₁	—	2	1	no	—
111	C₂²×C₇₁	—	2	1	no	—
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	8	no	—
114	C₂²×C₇₁	#336	1	2	yes	—
115	C₂²×C₇₁	#336	1	2	yes	—
116	C₂²×C₇₁	H(3,ℤ/4)	1	2	yes	—
117	C₂²×C₇₁	#338	1	2	yes	—
118	C₂²×C₇₁	C₂×#95	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
120	C₄×C₇₁	—	2	2	no	—
121	C₄×C₇₁	—	2	2	no	—
122	C₄×C₇₁	—	2	2	no	—
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	#336	1	2	yes	—
125	C₄×C₇₁	#336	1	2	yes	—
126	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
127	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
128	C₄×C₇₁	C₂×C₂²⋊C₈	1	2	yes	—
129	C₂×#1275	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	—	2	2	no	—
131	C₂×C₇₁	—	2	2	no	—
132	C₂×C₇₁	#1275	1	4	yes	complemented
133	C₂×C₇₁	C₂×#336	1	1	yes	characteristic
134	C₂×C₇₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#338	1	1	yes	characteristic
136	C₂×#431	—	71	2	no	—
137	C₂²×#97	D₇₁	1	1	yes	characteristic
138	#1275	—	71	4	no	—
139	C₇₁	C₂×#1275	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₈	Dic₇₁	1	2	yes	—
141	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
142	#431	—	71	4	no	—
143	#431	—	71	4	no	—
144	#431	—	142	2	no	—
145	#431	—	142	2	no	—
146	C₂×C₄⋊C₈	—	142	2	no	—
147	C₂×C₂²⋊C₈	—	142	2	no	—
148	C₂×#97	—	2	2	no	—
149	C₂×#97	—	2	2	no	—
150	C₂×#97	Dic₇₁	1	2	yes	—
151	C₂×#97	C₂×D₇₁	1	2	yes	—
152	C₂×C₄×C₈	—	71	4	no	—
153	C₂³×C₄	Dic₁₄₂	1	1	yes	characteristic
154	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	—	142	4	no	—
160	C₂²×C₈	C₇₁⋊C₈	1	8	yes	complemented
161	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
162	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₇₁	1	2	yes	—
164	C₂×C₄²	C₄×D₇₁	1	2	yes	—
165	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₇₁	1	2	yes	—
169	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
170	C₄⋊C₈	—	142	4	no	—
171	C₄⋊C₈	—	142	4	no	—
172	C₂²⋊C₈	—	142	4	no	—
173	C₂²⋊C₈	—	142	4	no	—
174	#97	—	2	4	no	—
175	#97	—	2	4	no	—
176	C₄×C₈	—	71	8	no	—
177	C₄×C₈	—	142	4	no	—
178	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
182	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
183	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
184	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
185	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
187	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
188	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
189	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₇₁⋊C₈	1	4	yes	—
193	C₂²×C₄	#11765849	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	284	8	no	—
202	C₂×C₈	—	142	8	no	—
203	C₂×C₈	C₂×C₇₁⋊C₈	1	8	yes	complemented
204	C₄²	C₄×Dic₇₁	1	2	yes	—
205	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
206	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
207	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
208	C₄²	Dic₇₁⋊C₄	1	2	yes	—
209	C₄²	Dic₇₁⋊C₄	1	2	yes	—
210	C₄²	C₂×C₄×D₇₁	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₇₁⋊#95	1	1	yes	characteristic
215	C₂³	C₇₁⋊#95	1	1	yes	characteristic
216	C₂³	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
217	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	C₇₁⋊#95	1	2	yes	—
227	C₂×C₄	C₇₁⋊#95	1	2	yes	—
228	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
229	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
230	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
231	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
232	C₂×C₄	#14556039	1	2	yes	—
233	C₂×C₄	#14556039	1	2	yes	—
234	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	4	yes	—
235	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	4	yes	—
236	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
238	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
241	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
242	C₂×C₄	C₂×#11765849	1	2	yes	—
243	C₈	—	2	8	no	—
244	C₈	—	284	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	#17985754	1	2	yes	—
251	C₂²	#17985754	1	2	yes	—
252	C₂²	#17985756	1	2	yes	—
253	C₂²	C₇₁⋊#338	1	2	yes	—
254	C₂²	C₂×C₇₁⋊#95	1	1	yes	characteristic
255	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	#17985754	1	2	yes	—
261	C₄	#17985754	1	2	yes	—
262	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
263	C₄	C₂×#14556039	1	1	yes	characteristic
264	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	—	2	2	no	—
267	C₂	#22586171	1	4	yes	direct product factor
268	C₂	C₂×#17985754	1	1	yes	characteristic
269	C₂	C₂×#17985756	1	1	yes	characteristic
270	C₂	C₂×C₇₁⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂²×#97×C₇₁	C₂
socle	C₂³×C₇₁	#95
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂²×#97	D₇₁
hypocenter	C₇₁	C₂×#1275
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #22586171	C₂	direct (N × H)
30. C₂²×C₈×C₇₁	C₈	semidirect
67. C₂×C₈×C₇₁	C₂×C₈	semidirect
132. C₂×C₇₁	#1275	semidirect
139. C₇₁	C₂×#1275	semidirect
160. C₂²×C₈	C₇₁⋊C₈	semidirect
203. C₂×C₈	C₂×C₇₁⋊C₈	semidirect
267. C₂	#22586171	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×#17986052	2
#22586171	2
C₂²×#97×C₇₁	2
C₂×#1275	71

Subgroups — C2×#22586171 #31886868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586171 #31886868