Subgroups — C₂×#435×C₁₇₃ #36350712

Order 22144.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#435×C₁₇₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₇₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₇₃	—	2	4	no	—
13	C₂×M₁₆×C₁₇₃	C₂²	1	4	yes	complemented
14	C₂²×C₈×C₁₇₃	—	2	4	no	—
15	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₇₃	C₂²	1	4	yes	complemented
20	C₄×D₄×C₁₇₃	C₄	1	4	yes	—
21	C₄×D₄×C₁₇₃	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₇₃	C₄	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₁₇₃	C₄	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₁₇₃	C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₁₇₃	C₂²	1	4	yes	complemented
27	C₂²⋊C₈×C₁₇₃	C₂²	1	4	yes	complemented
28	C₂²⋊C₈×C₁₇₃	C₂²	1	4	yes	complemented
29	#97×C₁₇₃	C₂²	1	4	yes	complemented
30	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₁₇₃	—	2	4	no	—
33	C₂×D₄×C₁₇₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₇₃	—	2	2	no	—
35	C₂²×C₄×C₁₇₃	—	2	2	no	—
36	C₂²×C₄×C₁₇₃	—	2	2	no	—
37	C₂²×C₄×C₁₇₃	—	2	2	no	—
38	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
47	M₁₆×C₁₇₃	—	2	8	no	—
48	C₂×C₈×C₁₇₃	—	2	2	no	—
49	C₂×C₈×C₁₇₃	—	2	2	no	—
50	C₂×C₈×C₁₇₃	—	2	2	no	—
51	C₂×C₈×C₁₇₃	—	2	8	no	—
52	C₂×C₈×C₁₇₃	D₄	1	4	yes	complemented
53	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
54	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
55	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
58	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
59	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
60	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
61	C₄²×C₁₇₃	C₂³	1	2	yes	—
62	C₂³×C₁₇₃	—	2	2	no	—
63	C₂³×C₁₇₃	—	2	2	no	—
64	C₂³×C₁₇₃	—	2	2	no	—
65	C₂³×C₁₇₃	—	2	2	no	—
66	C₂³×C₁₇₃	—	2	2	no	—
67	C₂³×C₁₇₃	M₁₆	1	4	yes	complemented
68	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
71	D₄×C₁₇₃	—	2	8	no	—
72	C₂×C₄×C₁₇₃	—	2	2	no	—
73	C₂×C₄×C₁₇₃	—	2	2	no	—
74	C₂×C₄×C₁₇₃	—	2	2	no	—
75	C₂×C₄×C₁₇₃	—	2	2	no	—
76	C₂×C₄×C₁₇₃	—	2	4	no	—
77	C₂×C₄×C₁₇₃	—	2	4	no	—
78	C₂×C₄×C₁₇₃	—	2	2	no	—
79	C₂×C₄×C₁₇₃	—	2	2	no	—
80	C₂×C₄×C₁₇₃	—	2	2	no	—
81	C₂×C₄×C₁₇₃	—	2	2	no	—
82	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₇₃	#41	1	2	yes	—
89	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₇₃	—	2	2	no	—
91	C₈×C₁₇₃	—	2	2	no	—
92	C₈×C₁₇₃	—	2	2	no	—
93	C₈×C₁₇₃	C₂×D₄	1	4	yes	complemented
94	C₂²×C₁₇₃	—	2	2	no	—
95	C₂²×C₁₇₃	—	2	2	no	—
96	C₂²×C₁₇₃	—	2	8	no	—
97	C₂²×C₁₇₃	—	2	2	no	—
98	C₂²×C₁₇₃	—	2	2	no	—
99	C₂²×C₁₇₃	—	2	2	no	—
100	C₂²×C₁₇₃	—	4	4	no	—
101	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
102	C₂²×C₁₇₃	C₂×M₁₆	1	2	yes	—
103	C₂²×C₁₇₃	C₂×M₁₆	1	4	yes	complemented
104	C₂²×C₁₇₃	#131	1	2	yes	—
105	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₁₇₃	—	2	2	no	—
107	C₄×C₁₇₃	—	2	2	no	—
108	C₄×C₁₇₃	—	2	2	no	—
109	C₄×C₁₇₃	—	2	2	no	—
110	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
111	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₁₇₃	—	2	4	no	—
114	C₂×C₁₇₃	—	4	2	no	—
115	C₂×C₁₇₃	#435	1	4	yes	direct product factor
116	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂×C₁₇₃	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
119	C₁₇₃	C₂×#435	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×#435	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×M₁₆	C₂×C₁₇₃	1	1	yes	characteristic, complemented
122	C₂³×C₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
123	C₂×C₄×D₄	C₂×C₁₇₃	1	1	yes	characteristic
124	#435	C₂×C₁₇₃	1	8	yes	direct product factor
125	C₂×C₄⋊C₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
126	C₂×C₂²⋊C₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
127	C₂×C₂²⋊C₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
128	C₂×#97	C₂×C₁₇₃	1	1	yes	characteristic, complemented
129	C₂²×D₄	C₄×C₁₇₃	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
131	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₂²×C₁₇₃	1	4	yes	complemented
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²×C₁₇₃	1	4	yes	complemented
140	C₄×D₄	C₄×C₁₇₃	1	4	yes	—
141	C₄×D₄	C₂²×C₁₇₃	1	4	yes	—
142	C₂×C₄⋊C₄	C₄×C₁₇₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	C₄×C₁₇₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₄×C₁₇₃	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₁₇₃	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₁₇₃	1	4	yes	complemented
147	C₂²⋊C₈	C₂²×C₁₇₃	1	4	yes	complemented
148	C₂²⋊C₈	C₂²×C₁₇₃	1	4	yes	complemented
149	#97	C₂²×C₁₇₃	1	4	yes	complemented
150	C₂⁴	C₂×C₄×C₁₇₃	1	1	yes	characteristic
151	C₂⁴	C₂×C₄×C₁₇₃	1	1	yes	characteristic
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	C₂×C₄×C₁₇₃	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×C₄×C₁₇₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×C₁₇₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₄×C₁₇₃	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×C₁₇₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₄×C₁₇₃	1	2	yes	—
163	C₂²×C₄	C₂×C₄×C₁₇₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	D₄×C₁₇₃	1	4	yes	complemented
173	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
174	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
175	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
176	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
177	C₄⋊C₄	C₂×C₄×C₁₇₃	1	4	yes	—
178	C₂²⋊C₄	C₂×C₄×C₁₇₃	1	4	yes	—
179	C₂²⋊C₄	C₂×C₄×C₁₇₃	1	4	yes	—
180	C₄²	C₂×C₄×C₁₇₃	1	2	yes	—
181	C₄²	C₂³×C₁₇₃	1	2	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	M₁₆×C₁₇₃	1	4	yes	complemented
188	C₂³	C₂²×C₄×C₁₇₃	1	2	yes	—
189	C₂³	C₂²×C₄×C₁₇₃	1	2	yes	—
190	C₂³	C₂²×C₄×C₁₇₃	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
203	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
204	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
205	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
206	C₂×C₄	C₂²×C₄×C₁₇₃	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
208	C₂×C₄	#41×C₁₇₃	1	2	yes	—
209	C₂×C₄	C₂⁴×C₁₇₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	C₂×D₄×C₁₇₃	1	4	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₄×D₄×C₁₇₃	1	2	yes	—
222	C₂²	C₂×M₁₆×C₁₇₃	1	2	yes	—
223	C₂²	C₂×M₁₆×C₁₇₃	1	4	yes	complemented
224	C₂²	#131×C₁₇₃	1	2	yes	—
225	C₂²	C₂³×C₄×C₁₇₃	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	C₄×D₄×C₁₇₃	1	2	yes	—
231	C₄	C₂²×D₄×C₁₇₃	1	1	yes	characteristic
232	C₄	C₂×#41×C₁₇₃	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	4	2	no	—
235	C₂	#435×C₁₇₃	1	4	yes	direct product factor
236	C₂	C₂×C₄×D₄×C₁₇₃	1	1	yes	characteristic
237	C₂	C₂²×M₁₆×C₁₇₃	1	1	yes	characteristic
238	C₂	C₂×#131×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₇₃	C₂³
commutator	C₂²	C₂³×C₄×C₁₇₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇₃
socle	C₂³×C₁₇₃	C₂²×C₄
prime radical	C₂²×D₄×C₁₇₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₁₆×C₁₇₃	C₂	semidirect
2. C₂³×C₈×C₁₇₃	C₂	semidirect
4. #435×C₁₇₃	C₂	direct (N × H)
5. C₂×C₄⋊C₈×C₁₇₃	C₂	semidirect
6. C₂×C₂²⋊C₈×C₁₇₃	C₂	semidirect
7. C₂×C₂²⋊C₈×C₁₇₃	C₂	semidirect
8. C₂×#97×C₁₇₃	C₂	semidirect
13. C₂×M₁₆×C₁₇₃	C₂²	semidirect
19. C₂²×C₈×C₁₇₃	C₂²	semidirect
26. C₄⋊C₈×C₁₇₃	C₂²	semidirect
27. C₂²⋊C₈×C₁₇₃	C₂²	semidirect
28. C₂²⋊C₈×C₁₇₃	C₂²	semidirect
29. #97×C₁₇₃	C₂²	semidirect
52. C₂×C₈×C₁₇₃	D₄	semidirect
67. C₂³×C₁₇₃	M₁₆	semidirect
93. C₈×C₁₇₃	C₂×D₄	semidirect
103. C₂²×C₁₇₃	C₂×M₁₆	semidirect
115. C₂×C₁₇₃	#435	direct (N × H)
119. C₁₇₃	C₂×#435	direct (N × H)
120. C₂×#435	C₁₇₃	direct (N × H)
121. C₂²×M₁₆	C₂×C₁₇₃	semidirect
122. C₂³×C₈	C₂×C₁₇₃	semidirect
124. #435	C₂×C₁₇₃	direct (N × H)
125. C₂×C₄⋊C₈	C₂×C₁₇₃	semidirect
126. C₂×C₂²⋊C₈	C₂×C₁₇₃	semidirect
127. C₂×C₂²⋊C₈	C₂×C₁₇₃	semidirect
128. C₂×#97	C₂×C₁₇₃	semidirect
133. C₂×M₁₆	C₂²×C₁₇₃	semidirect
139. C₂²×C₈	C₂²×C₁₇₃	semidirect
146. C₄⋊C₈	C₂²×C₁₇₃	semidirect
147. C₂²⋊C₈	C₂²×C₁₇₃	semidirect
148. C₂²⋊C₈	C₂²×C₁₇₃	semidirect
149. #97	C₂²×C₁₇₃	semidirect
172. C₂×C₈	D₄×C₁₇₃	semidirect
187. C₂³	M₁₆×C₁₇₃	semidirect
213. C₈	C₂×D₄×C₁₇₃	semidirect
223. C₂²	C₂×M₁₆×C₁₇₃	semidirect
235. C₂	#435×C₁₇₃	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×M₁₆×C₁₇₃	2
C₂³×C₈×C₁₇₃	2
C₂×C₄×D₄×C₁₇₃	2
#435×C₁₇₃	2
C₂×C₄⋊C₈×C₁₇₃	2
C₂×C₂²⋊C₈×C₁₇₃	2
C₂×#97×C₁₇₃	2
C₂×#435	173

Subgroups — C2×#435×C173 #36350712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#435×C₁₇₃ #36350712