Subgroups — #3331×C₁₇₃ #36351116

Order 22144.

388 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#487×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#483×C₁₇₃	C₂	1	2	yes	maximal, complemented
5	#476×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#470×C₁₇₃	C₂	1	2	yes	maximal, complemented
7	#462×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₁₇₃	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₁₇₃	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₇₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₁₇₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
18	#125×C₁₇₃	—	2	1	no	—
19	#123×C₁₇₃	—	2	2	no	—
20	C₂²⋊Q₈×C₁₇₃	—	2	2	no	—
21	C₂²⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
22	C₂²⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
23	#121×C₁₇₃	—	2	1	no	—
24	C₄×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₇₃	—	2	1	no	—
26	C₄×D₄×C₁₇₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₇₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
30	#106×C₁₇₃	C₂²	1	1	yes	characteristic
31	#106×C₁₇₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
33	#103×C₁₇₃	C₂²	1	2	yes	—
34	#103×C₁₇₃	C₂²	1	2	yes	—
35	#103×C₁₇₃	C₂²	1	1	yes	characteristic
36	#103×C₁₇₃	C₂²	1	2	yes	—
37	#102×C₁₇₃	C₂²	1	1	yes	characteristic
38	#102×C₁₇₃	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₁₇₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
41	#41×C₁₇₃	—	2	4	no	—
42	#41×C₁₇₃	—	4	2	no	—
43	C₂×Q₈×C₁₇₃	—	2	1	no	—
44	C₂×Q₈×C₁₇₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₇₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₇₃	—	2	2	no	—
47	C₂×D₄×C₁₇₃	D₄	1	1	yes	characteristic
48	C₂×D₄×C₁₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₇₃	—	2	2	no	—
50	C₂²×C₄×C₁₇₃	—	2	2	no	—
51	C₂²×C₄×C₁₇₃	—	2	1	no	—
52	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
53	Q₁₆×C₁₇₃	—	4	2	no	—
54	SD₁₆×C₁₇₃	—	2	2	no	—
55	M₁₆×C₁₇₃	—	4	2	no	—
56	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
58	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇₃	—	2	1	no	—
60	C₄⋊C₄×C₁₇₃	—	2	1	no	—
61	C₄⋊C₄×C₁₇₃	—	2	1	no	—
62	C₄⋊C₄×C₁₇₃	—	2	2	no	—
63	C₄⋊C₄×C₁₇₃	—	2	2	no	—
64	C₄⋊C₄×C₁₇₃	—	2	1	no	—
65	C₄⋊C₄×C₁₇₃	—	2	2	no	—
66	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₁₇₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
71	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
72	C₂²⋊C₄×C₁₇₃	—	2	1	no	—
73	C₂²⋊C₄×C₁₇₃	D₄	1	2	yes	—
74	C₄²×C₁₇₃	—	2	1	no	—
75	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
76	C₂³×C₁₇₃	—	2	1	no	—
77	C₂³×C₁₇₃	C₂×D₄	1	2	yes	—
78	Q₈×C₁₇₃	—	2	1	no	—
79	Q₈×C₁₇₃	—	2	2	no	—
80	Q₈×C₁₇₃	—	2	1	no	—
81	Q₈×C₁₇₃	—	2	2	no	—
82	Q₈×C₁₇₃	—	8	1	no	—
83	D₄×C₁₇₃	—	2	2	no	—
84	D₄×C₁₇₃	—	2	1	no	—
85	D₄×C₁₇₃	—	4	4	no	—
86	D₄×C₁₇₃	#41	1	2	yes	—
87	C₂×C₄×C₁₇₃	—	2	2	no	—
88	C₂×C₄×C₁₇₃	—	2	2	no	—
89	C₂×C₄×C₁₇₃	—	2	1	no	—
90	C₂×C₄×C₁₇₃	—	2	1	no	—
91	C₂×C₄×C₁₇₃	—	2	1	no	—
92	C₂×C₄×C₁₇₃	—	2	2	no	—
93	C₂×C₄×C₁₇₃	—	2	2	no	—
94	C₂×C₄×C₁₇₃	—	4	4	no	—
95	C₂×C₄×C₁₇₃	—	4	2	no	—
96	C₂×C₄×C₁₇₃	—	8	2	no	—
97	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
101	C₈×C₁₇₃	—	2	1	no	—
102	C₈×C₁₇₃	—	2	1	no	—
103	C₈×C₁₇₃	—	2	2	no	—
104	C₂²×C₁₇₃	—	2	2	no	—
105	C₂²×C₁₇₃	—	2	2	no	—
106	C₂²×C₁₇₃	—	2	2	no	—
107	C₂²×C₁₇₃	—	2	2	no	—
108	C₂²×C₁₇₃	—	4	1	no	—
109	C₂²×C₁₇₃	—	4	1	no	—
110	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₇₃	—	2	2	no	—
112	C₄×C₁₇₃	—	2	1	no	—
113	C₄×C₁₇₃	—	4	2	no	—
114	C₄×C₁₇₃	—	4	1	no	—
115	C₄×C₁₇₃	—	4	1	no	—
116	C₄×C₁₇₃	—	4	1	no	—
117	C₄×C₁₇₃	—	4	2	no	—
118	C₄×C₁₇₃	#137	1	2	yes	—
119	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
120	C₄×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₁₇₃	—	4	2	no	—
122	C₂×C₁₇₃	—	4	2	no	—
123	C₂×C₁₇₃	#546	1	1	yes	characteristic
124	C₂×C₁₇₃	C₂×#137	1	1	yes	characteristic
125	C₂×C₁₇₃	#577	1	1	yes	characteristic
126	C₁₇₃	#3331	1	1	yes	characteristic, direct product factor, Sylow
127	#3331	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	#554	C₂×C₁₇₃	1	1	yes	characteristic, complemented
129	#547	C₂×C₁₇₃	1	1	yes	characteristic
130	#487	C₂×C₁₇₃	1	1	yes	characteristic, complemented
131	#483	C₂×C₁₇₃	1	2	yes	complemented
132	#476	C₂×C₁₇₃	1	1	yes	characteristic, complemented
133	#470	C₂×C₁₇₃	1	2	yes	complemented
134	#462	C₂×C₁₇₃	1	1	yes	characteristic, complemented
135	#452	C₂×C₁₇₃	1	2	yes	complemented
136	C₄×SD₁₆	C₂×C₁₇₃	1	1	yes	characteristic, complemented
137	#436	C₂×C₁₇₃	1	1	yes	characteristic, complemented
138	#417	C₂×C₁₇₃	1	2	yes	complemented
139	C₂×#41	C₂²×C₁₇₃	1	2	yes	—
140	C₂×Q₁₆	C₂²×C₁₇₃	1	2	yes	—
141	C₂×SD₁₆	C₂²×C₁₇₃	1	1	yes	characteristic
142	C₂×M₁₆	C₂²×C₁₇₃	1	2	yes	—
143	C₄⋊Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
144	C₄⋊Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
145	#125	—	2	1	no	—
146	#123	—	2	2	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	C₂²⋊Q₈	C₂²×C₁₇₃	1	2	yes	—
149	C₂²⋊Q₈	C₂²×C₁₇₃	1	2	yes	—
150	#121	—	2	1	no	—
151	C₄×Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	C₂²×C₁₇₃	1	1	yes	characteristic
154	C₄×D₄	C₂²×C₁₇₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
157	#106	C₂²×C₁₇₃	1	1	yes	characteristic
158	#106	C₂²×C₁₇₃	1	2	yes	—
159	C₄⋊C₈	C₂²×C₁₇₃	1	1	yes	characteristic
160	#103	C₂²×C₁₇₃	1	2	yes	—
161	#103	C₂²×C₁₇₃	1	2	yes	—
162	#103	C₂²×C₁₇₃	1	1	yes	characteristic
163	#103	C₂²×C₁₇₃	1	2	yes	—
164	#102	C₂²×C₁₇₃	1	1	yes	characteristic
165	#102	C₂²×C₁₇₃	1	2	yes	—
166	C₂²⋊C₈	C₂²×C₁₇₃	1	2	yes	—
167	C₄×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
168	#41	—	2	4	no	—
169	#41	—	4	2	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	C₂³×C₁₇₃	1	2	yes	—
172	C₂×Q₈	C₂³×C₁₇₃	1	1	yes	characteristic
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	D₄×C₁₇₃	1	1	yes	characteristic
175	C₂×D₄	C₂³×C₁₇₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
180	Q₁₆	—	4	2	no	—
181	SD₁₆	—	2	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	C₂³×C₁₇₃	1	1	yes	characteristic
184	C₂×C₈	C₂³×C₁₇₃	1	1	yes	characteristic
185	C₂×C₈	C₂³×C₁₇₃	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	D₄×C₁₇₃	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₁₇₃	1	1	yes	characteristic
195	C₄⋊C₄	C₂³×C₁₇₃	1	2	yes	—
196	C₄⋊C₄	C₂³×C₁₇₃	1	1	yes	characteristic
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	D₄×C₁₇₃	1	2	yes	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂³×C₁₇₃	1	1	yes	characteristic
203	C₂³	—	2	1	no	—
204	C₂³	C₂×D₄×C₁₇₃	1	2	yes	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	1	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	8	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	4	4	no	—
213	D₄	#41×C₁₇₃	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₂×D₄×C₁₇₃	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
226	C₂×C₄	C₂×D₄×C₁₇₃	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₁₇₃	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	C₂²×D₄×C₁₇₃	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	#137×C₁₇₃	1	2	yes	—
246	C₄	C₂×#41×C₁₇₃	1	1	yes	characteristic
247	C₄	H(5,𝔽₂)×C₁₇₃	1	1	yes	characteristic
248	C₂	—	4	2	no	—
249	C₂	—	4	2	no	—
250	C₂	#546×C₁₇₃	1	1	yes	characteristic
251	C₂	C₂×#137×C₁₇₃	1	1	yes	characteristic
252	C₂	#577×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇₃
prime radical	#547×C₁₇₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇₃ ◁ C₄×D₄×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #554×C₁₇₃	C₂	semidirect
3. #487×C₁₇₃	C₂	semidirect
4. #483×C₁₇₃	C₂	semidirect
5. #476×C₁₇₃	C₂	semidirect
6. #470×C₁₇₃	C₂	semidirect
7. #462×C₁₇₃	C₂	semidirect
8. #452×C₁₇₃	C₂	semidirect
9. C₄×SD₁₆×C₁₇₃	C₂	semidirect
10. #436×C₁₇₃	C₂	semidirect
11. #417×C₁₇₃	C₂	semidirect
126. C₁₇₃	#3331	direct (N × H)
127. #3331	C₁₇₃	direct (N × H)
128. #554	C₂×C₁₇₃	semidirect
130. #487	C₂×C₁₇₃	semidirect
131. #483	C₂×C₁₇₃	semidirect
132. #476	C₂×C₁₇₃	semidirect
133. #470	C₂×C₁₇₃	semidirect
134. #462	C₂×C₁₇₃	semidirect
135. #452	C₂×C₁₇₃	semidirect
136. C₄×SD₁₆	C₂×C₁₇₃	semidirect
137. #436	C₂×C₁₇₃	semidirect
138. #417	C₂×C₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#554×C₁₇₃	2
#547×C₁₇₃	2
#487×C₁₇₃	2
#483×C₁₇₃	2
#476×C₁₇₃	2
#470×C₁₇₃	2
#462×C₁₇₃	2
#452×C₁₇₃	2
C₄×SD₁₆×C₁₇₃	2
#436×C₁₇₃	2
#417×C₁₇₃	2
#3331	173

Subgroups — #3331×C173 #36351116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3331×C₁₇₃ #36351116