Subgroups — #3288×C₁₇₃ #36351073

Order 22144.

464 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#577×C₁₇₃	C₂	1	2	yes	maximal, complemented
2	#576×C₁₇₃	C₂	1	2	yes	maximal, complemented
3	#496×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#472×C₁₇₃	C₂	1	2	yes	maximal, complemented
5	#471×C₁₇₃	C₂	1	2	yes	maximal, complemented
6	#455×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#454×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#453×C₁₇₃	C₂	1	2	yes	maximal, complemented
9	#444×C₁₇₃	C₂	1	2	yes	maximal, complemented
10	2₋¹⁺⁴×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
11	H(5,𝔽₂)×C₁₇₃	C₂²	1	1	yes	characteristic
12	H(5,𝔽₂)×C₁₇₃	C₂²	1	2	yes	complemented
13	#137×C₁₇₃	—	2	2	no	—
14	#137×C₁₇₃	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)×C₁₇₃	—	2	2	no	—
16	AGL(1,ℤ/8)×C₁₇₃	—	2	2	no	—
17	AGL(1,ℤ/8)×C₁₇₃	—	2	2	no	—
18	AGL(1,ℤ/8)×C₁₇₃	C₂²	1	2	yes	—
19	AGL(1,ℤ/8)×C₁₇₃	C₂²	1	2	yes	complemented
20	AGL(1,ℤ/8)×C₁₇₃	C₂²	1	2	yes	complemented
21	#135×C₁₇₃	—	2	2	no	—
22	#135×C₁₇₃	—	2	2	no	—
23	#135×C₁₇₃	C₂²	1	2	yes	—
24	C₂×SD₁₆×C₁₇₃	—	2	2	no	—
25	C₂×SD₁₆×C₁₇₃	C₂²	1	2	yes	complemented
26	C₂×D₈×C₁₇₃	—	2	2	no	—
27	C₂×D₈×C₁₇₃	C₂²	1	2	yes	complemented
28	#131×C₁₇₃	C₂²	1	2	yes	complemented
29	#131×C₁₇₃	C₂²	1	2	yes	complemented
30	#127×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
31	#124×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
32	#108×C₁₇₃	C₂²	1	2	yes	complemented
33	#104×C₁₇₃	C₂²	1	2	yes	complemented
34	#104×C₁₇₃	C₂²	1	2	yes	complemented
35	#101×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
36	#100×C₁₇₃	C₂²	1	2	yes	complemented
37	#100×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
38	#97×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
39	#41×C₁₇₃	—	2	2	no	—
40	#41×C₁₇₃	—	2	2	no	—
41	#41×C₁₇₃	—	2	2	no	—
42	#41×C₁₇₃	—	2	2	no	—
43	#41×C₁₇₃	—	4	1	no	—
44	#41×C₁₇₃	C₂³	1	2	yes	—
45	#41×C₁₇₃	C₂³	1	2	yes	—
46	C₂×Q₈×C₁₇₃	—	2	2	no	—
47	C₂×Q₈×C₁₇₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₇₃	—	2	2	no	—
49	C₂×D₄×C₁₇₃	—	2	2	no	—
50	C₂×D₄×C₁₇₃	—	2	2	no	—
51	C₂×D₄×C₁₇₃	—	4	1	no	—
52	C₂×D₄×C₁₇₃	—	4	1	no	—
53	C₂×D₄×C₁₇₃	—	4	2	no	—
54	C₂×D₄×C₁₇₃	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₁₇₃	C₂³	1	2	yes	—
56	Q₁₆×C₁₇₃	—	2	2	no	—
57	Q₁₆×C₁₇₃	D₄	1	2	yes	complemented
58	SD₁₆×C₁₇₃	—	2	2	no	—
59	SD₁₆×C₁₇₃	—	2	2	no	—
60	SD₁₆×C₁₇₃	—	2	2	no	—
61	SD₁₆×C₁₇₃	—	2	2	no	—
62	SD₁₆×C₁₇₃	—	2	2	no	—
63	SD₁₆×C₁₇₃	—	2	2	no	—
64	SD₁₆×C₁₇₃	D₄	1	2	yes	complemented
65	SD₁₆×C₁₇₃	D₄	1	2	yes	complemented
66	D₈×C₁₇₃	—	2	2	no	—
67	D₈×C₁₇₃	—	2	2	no	—
68	D₈×C₁₇₃	—	2	2	no	—
69	D₈×C₁₇₃	—	2	2	no	—
70	D₈×C₁₇₃	—	2	2	no	—
71	D₈×C₁₇₃	D₄	1	2	yes	complemented
72	M₁₆×C₁₇₃	—	2	2	no	—
73	M₁₆×C₁₇₃	—	2	2	no	—
74	M₁₆×C₁₇₃	C₂³	1	2	yes	—
75	M₁₆×C₁₇₃	C₂³	1	2	yes	—
76	C₂×C₈×C₁₇₃	—	2	2	no	—
77	C₂×C₈×C₁₇₃	—	2	2	no	—
78	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
79	C₂²⋊C₄×C₁₇₃	—	4	1	no	—
80	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
81	C₂³×C₁₇₃	—	2	1	no	—
82	C₂³×C₁₇₃	—	2	2	no	—
83	C₂³×C₁₇₃	—	4	2	no	—
84	C₂³×C₁₇₃	—	4	1	no	—
85	Q₈×C₁₇₃	—	2	2	no	—
86	Q₈×C₁₇₃	—	4	1	no	—
87	Q₈×C₁₇₃	C₂×D₄	1	2	yes	—
88	Q₈×C₁₇₃	C₂×D₄	1	2	yes	—
89	D₄×C₁₇₃	—	2	2	no	—
90	D₄×C₁₇₃	—	2	2	no	—
91	D₄×C₁₇₃	—	2	2	no	—
92	D₄×C₁₇₃	—	2	2	no	—
93	D₄×C₁₇₃	—	2	2	no	—
94	D₄×C₁₇₃	—	4	2	no	—
95	D₄×C₁₇₃	—	4	2	no	—
96	D₄×C₁₇₃	—	4	2	no	—
97	D₄×C₁₇₃	—	4	1	no	—
98	D₄×C₁₇₃	—	8	2	no	—
99	D₄×C₁₇₃	C₂×D₄	1	2	yes	—
100	D₄×C₁₇₃	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₇₃	—	2	2	no	—
102	C₂×C₄×C₁₇₃	—	2	2	no	—
103	C₂×C₄×C₁₇₃	—	2	1	no	—
104	C₂×C₄×C₁₇₃	—	2	1	no	—
105	C₂×C₄×C₁₇₃	—	4	1	no	—
106	C₂×C₄×C₁₇₃	—	4	2	no	—
107	C₂×C₄×C₁₇₃	—	4	2	no	—
108	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₇₃	—	2	2	no	—
110	C₈×C₁₇₃	—	2	2	no	—
111	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
112	C₈×C₁₇₃	C₂×D₄	1	2	yes	—
113	C₂²×C₁₇₃	—	2	2	no	—
114	C₂²×C₁₇₃	—	2	2	no	—
115	C₂²×C₁₇₃	—	4	1	no	—
116	C₂²×C₁₇₃	—	4	2	no	—
117	C₂²×C₁₇₃	—	8	2	no	—
118	C₂²×C₁₇₃	—	8	1	no	—
119	C₂²×C₁₇₃	—	16	2	no	—
120	C₂²×C₁₇₃	—	16	1	no	—
121	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₁₇₃	—	2	2	no	—
123	C₄×C₁₇₃	—	2	2	no	—
124	C₄×C₁₇₃	—	4	1	no	—
125	C₄×C₁₇₃	—	4	1	no	—
126	C₄×C₁₇₃	C₂²×D₄	1	2	yes	—
127	C₂×C₁₇₃	—	2	1	no	—
128	C₂×C₁₇₃	—	4	2	no	—
129	C₂×C₁₇₃	—	4	2	no	—
130	C₂×C₁₇₃	—	8	2	no	—
131	C₂×C₁₇₃	—	8	1	no	—
132	C₂×C₁₇₃	D₄²	1	1	yes	characteristic
133	C₁₇₃	#3288	1	1	yes	characteristic, direct product factor, Sylow
134	#3288	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	#577	C₂×C₁₇₃	1	2	yes	complemented
136	#576	C₂×C₁₇₃	1	2	yes	complemented
137	#496	C₂×C₁₇₃	1	1	yes	characteristic, complemented
138	#472	C₂×C₁₇₃	1	2	yes	complemented
139	#471	C₂×C₁₇₃	1	2	yes	complemented
140	#455	C₂×C₁₇₃	1	1	yes	characteristic, complemented
141	#454	C₂×C₁₇₃	1	1	yes	characteristic, complemented
142	#453	C₂×C₁₇₃	1	2	yes	complemented
143	#444	C₂×C₁₇₃	1	2	yes	complemented
144	2₋¹⁺⁴	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
145	H(5,𝔽₂)	C₂²×C₁₇₃	1	1	yes	characteristic
146	H(5,𝔽₂)	C₂²×C₁₇₃	1	2	yes	complemented
147	#137	—	2	2	no	—
148	#137	C₂²×C₁₇₃	1	2	yes	complemented
149	AGL(1,ℤ/8)	—	2	2	no	—
150	AGL(1,ℤ/8)	—	2	2	no	—
151	AGL(1,ℤ/8)	—	2	2	no	—
152	AGL(1,ℤ/8)	C₂²×C₁₇₃	1	2	yes	—
153	AGL(1,ℤ/8)	C₂²×C₁₇₃	1	2	yes	complemented
154	AGL(1,ℤ/8)	C₂²×C₁₇₃	1	2	yes	complemented
155	#135	—	2	2	no	—
156	#135	—	2	2	no	—
157	#135	C₂²×C₁₇₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	C₂²×C₁₇₃	1	2	yes	complemented
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	C₂²×C₁₇₃	1	2	yes	complemented
162	#131	C₂²×C₁₇₃	1	2	yes	complemented
163	#131	C₂²×C₁₇₃	1	2	yes	complemented
164	#127	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
165	#124	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
166	#108	C₂²×C₁₇₃	1	2	yes	complemented
167	#104	C₂²×C₁₇₃	1	2	yes	complemented
168	#104	C₂²×C₁₇₃	1	2	yes	complemented
169	#101	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
170	#100	C₂²×C₁₇₃	1	2	yes	complemented
171	#100	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
172	#97	C₂²×C₁₇₃	1	1	yes	characteristic, complemented
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	4	1	no	—
178	#41	C₂³×C₁₇₃	1	2	yes	—
179	#41	C₂³×C₁₇₃	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂³×C₁₇₃	1	1	yes	characteristic
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	C₂³×C₁₇₃	1	1	yes	characteristic
189	C₂×D₄	C₂³×C₁₇₃	1	2	yes	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	D₄×C₁₇₃	1	2	yes	complemented
192	SD₁₆	—	2	2	no	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	D₄×C₁₇₃	1	2	yes	complemented
199	SD₁₆	D₄×C₁₇₃	1	2	yes	complemented
200	D₈	—	2	2	no	—
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	2	2	no	—
204	D₈	—	2	2	no	—
205	D₈	D₄×C₁₇₃	1	2	yes	complemented
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₂³×C₁₇₃	1	2	yes	—
209	M₁₆	C₂³×C₁₇₃	1	2	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₄²	C₂³×C₁₇₃	1	1	yes	characteristic
215	C₂³	—	2	1	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	1	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	4	1	no	—
221	Q₈	C₂×D₄×C₁₇₃	1	2	yes	—
222	Q₈	C₂×D₄×C₁₇₃	1	2	yes	—
223	D₄	—	2	2	no	—
224	D₄	—	2	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	1	no	—
232	D₄	—	8	2	no	—
233	D₄	C₂×D₄×C₁₇₃	1	2	yes	—
234	D₄	C₂×D₄×C₁₇₃	1	2	yes	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	C₂⁴×C₁₇₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	C₂×D₄×C₁₇₃	1	2	yes	—
246	C₈	C₂×D₄×C₁₇₃	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	16	2	no	—
254	C₂²	—	16	1	no	—
255	C₂²	H(5,𝔽₂)×C₁₇₃	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	C₂²×D₄×C₁₇₃	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	4	2	no	—
264	C₂	—	8	2	no	—
265	C₂	—	8	1	no	—
266	C₂	D₄²×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₇₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇₃ ◁ C₂×C₄×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #577×C₁₇₃	C₂	semidirect
2. #576×C₁₇₃	C₂	semidirect
3. #496×C₁₇₃	C₂	semidirect
4. #472×C₁₇₃	C₂	semidirect
5. #471×C₁₇₃	C₂	semidirect
6. #455×C₁₇₃	C₂	semidirect
7. #454×C₁₇₃	C₂	semidirect
8. #453×C₁₇₃	C₂	semidirect
9. #444×C₁₇₃	C₂	semidirect
10. 2₋¹⁺⁴×C₁₇₃	C₂²	semidirect
12. H(5,𝔽₂)×C₁₇₃	C₂²	semidirect
14. #137×C₁₇₃	C₂²	semidirect
19. AGL(1,ℤ/8)×C₁₇₃	C₂²	semidirect
20. AGL(1,ℤ/8)×C₁₇₃	C₂²	semidirect
25. C₂×SD₁₆×C₁₇₃	C₂²	semidirect
27. C₂×D₈×C₁₇₃	C₂²	semidirect
28. #131×C₁₇₃	C₂²	semidirect
29. #131×C₁₇₃	C₂²	semidirect
30. #127×C₁₇₃	C₂²	semidirect
31. #124×C₁₇₃	C₂²	semidirect
32. #108×C₁₇₃	C₂²	semidirect
33. #104×C₁₇₃	C₂²	semidirect
34. #104×C₁₇₃	C₂²	semidirect
35. #101×C₁₇₃	C₂²	semidirect
36. #100×C₁₇₃	C₂²	semidirect
37. #100×C₁₇₃	C₂²	semidirect
38. #97×C₁₇₃	C₂²	semidirect
57. Q₁₆×C₁₇₃	D₄	semidirect
64. SD₁₆×C₁₇₃	D₄	semidirect
65. SD₁₆×C₁₇₃	D₄	semidirect
71. D₈×C₁₇₃	D₄	semidirect
133. C₁₇₃	#3288	direct (N × H)
134. #3288	C₁₇₃	direct (N × H)
135. #577	C₂×C₁₇₃	semidirect
136. #576	C₂×C₁₇₃	semidirect
137. #496	C₂×C₁₇₃	semidirect
138. #472	C₂×C₁₇₃	semidirect
139. #471	C₂×C₁₇₃	semidirect
140. #455	C₂×C₁₇₃	semidirect
141. #454	C₂×C₁₇₃	semidirect
142. #453	C₂×C₁₇₃	semidirect
143. #444	C₂×C₁₇₃	semidirect
144. 2₋¹⁺⁴	C₂²×C₁₇₃	semidirect
146. H(5,𝔽₂)	C₂²×C₁₇₃	semidirect
148. #137	C₂²×C₁₇₃	semidirect
153. AGL(1,ℤ/8)	C₂²×C₁₇₃	semidirect
154. AGL(1,ℤ/8)	C₂²×C₁₇₃	semidirect
159. C₂×SD₁₆	C₂²×C₁₇₃	semidirect
161. C₂×D₈	C₂²×C₁₇₃	semidirect
162. #131	C₂²×C₁₇₃	semidirect
163. #131	C₂²×C₁₇₃	semidirect
164. #127	C₂²×C₁₇₃	semidirect
165. #124	C₂²×C₁₇₃	semidirect
166. #108	C₂²×C₁₇₃	semidirect
167. #104	C₂²×C₁₇₃	semidirect
168. #104	C₂²×C₁₇₃	semidirect
169. #101	C₂²×C₁₇₃	semidirect
170. #100	C₂²×C₁₇₃	semidirect
171. #100	C₂²×C₁₇₃	semidirect
172. #97	C₂²×C₁₇₃	semidirect
191. Q₁₆	D₄×C₁₇₃	semidirect
198. SD₁₆	D₄×C₁₇₃	semidirect
199. SD₁₆	D₄×C₁₇₃	semidirect
205. D₈	D₄×C₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#577×C₁₇₃	2
#576×C₁₇₃	2
#496×C₁₇₃	2
#472×C₁₇₃	2
#471×C₁₇₃	2
#455×C₁₇₃	2
#454×C₁₇₃	2
#453×C₁₇₃	2
#444×C₁₇₃	2
#3288	173

Subgroups — #3288×C173 #36351073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3288×C₁₇₃ #36351073