Subgroups — D₁₇₃⋊#444 #36370334

Order 22144.

464 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇₃⋊#131	C₂	1	2	yes	maximal, complemented
2	#26012538	C₂	1	2	yes	maximal, complemented
3	#26012537	C₂	1	2	yes	maximal, complemented
4	#26012536	C₂	1	4	yes	maximal, complemented
5	D₁₇₃⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₇₃⋊#104	C₂	1	2	yes	maximal, complemented
7	C₂×#19149733	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₇₃	C₂	1	1	yes	maximal, characteristic
9	#19149767	C₂²	1	4	yes	complemented
10	#19149766	—	2	4	no	—
11	#19149766	C₂²	1	4	yes	complemented
12	D₁₇₃⋊M₁₆	—	2	2	no	—
13	D₁₇₃⋊M₁₆	C₂²	1	2	yes	—
14	C₂×D₁₇₃⋊C₈	—	2	2	no	—
15	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10621924	C₂²	1	1	yes	characteristic
17	#19149750	C₂²	1	2	yes	complemented
18	#19149749	C₂²	1	2	yes	complemented
19	#19149744	C₂²	1	4	yes	complemented
20	#19149743	C₂²	1	4	yes	complemented
21	#19149733	C₂²	1	2	yes	complemented
22	#19149733	C₂²	1	2	yes	complemented
23	#41×D₁₇₃	C₂²	1	2	yes	—
24	#19149702	C₂²	1	1	yes	characteristic
25	#19149701	C₄	1	1	yes	characteristic, complemented
26	#19149700	C₂²	1	2	yes	—
27	#19149699	C₄	1	2	yes	complemented
28	#19149698	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₇₃	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₇₃	C₄	1	2	yes	complemented
31	D₈×D₁₇₃	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
33	#135×C₁₇₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×#10621924	—	2	1	no	—
35	#14583836	—	2	4	no	—
36	#14583836	C₂³	1	4	yes	—
37	#14583835	—	2	4	no	—
38	#14583835	—	2	2	no	—
39	D₁₇₃⋊C₈	—	2	4	no	—
40	D₁₇₃⋊C₈	—	2	2	no	—
41	D₁₇₃⋊C₈	D₄	1	2	yes	—
42	D₁₇₃⋊C₈	C₂³	1	2	yes	—
43	C₂×#13063443	—	2	4	no	—
44	C₂×#13063443	D₄	1	2	yes	—
45	C₄×#10621924	—	2	1	no	—
46	C₄×#10621924	C₂³	1	2	yes	—
47	#14583827	C₂³	1	2	yes	—
48	#14583826	—	2	2	no	—
49	#14583826	C₂×C₄	1	2	yes	—
50	#14583825	—	2	2	no	—
51	#14583825	C₂³	1	2	yes	—
52	Q₈×D₁₇₃	C₂×C₄	1	2	yes	—
53	D₄×D₁₇₃	—	2	2	no	—
54	D₄×D₁₇₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₁₇₃	—	2	2	no	—
56	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
57	Dic₆₉₂	C₂×C₄	1	1	yes	characteristic, complemented
58	#14583816	C₂×C₄	1	2	yes	complemented
59	#14583815	C₂×C₄	1	2	yes	complemented
60	#14583814	C₂×C₄	1	2	yes	complemented
61	#14583813	C₂×C₄	1	2	yes	complemented
62	D₁₃₈₄	C₂×C₄	1	1	yes	characteristic, complemented
63	#14583811	C₂×C₄	1	2	yes	complemented
64	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
65	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
68	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
69	#41×C₁₇₃	C₂×C₄	1	2	yes	—
70	Q₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
72	D₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
74	C₂×#10621924	—	2	2	no	—
75	C₂×#10621924	—	4	2	no	—
76	#13063443	—	2	4	no	—
77	#13063443	C₂×D₄	1	4	yes	—
78	C₂²×D₁₇₃	—	2	2	no	—
79	C₂²×D₁₇₃	#41	1	1	yes	characteristic
80	Dic₃₄₆	—	2	2	no	—
81	Dic₃₄₆	C₂²×C₄	1	2	yes	—
82	D₆₉₂	—	2	2	no	—
83	D₆₉₂	C₂²×C₄	1	2	yes	—
84	C₂²⋊D₁₇₃	—	2	2	no	—
85	C₂²⋊D₁₇₃	—	2	2	no	—
86	C₂²⋊D₁₇₃	—	2	2	no	—
87	C₂×Dic₁₇₃	—	2	2	no	—
88	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
89	C₄×D₁₇₃	—	2	2	no	—
90	C₄×D₁₇₃	—	2	2	no	—
91	C₄×D₁₇₃	—	2	2	no	—
92	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
94	C₄×D₁₇₃	#41	1	1	yes	characteristic
95	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
96	#13063436	C₂²×C₄	1	1	yes	characteristic
97	#13063436	C₂²×C₄	1	1	yes	characteristic
98	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
99	D₄×C₁₇₃	—	2	2	no	—
100	D₄×C₁₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₇₃	—	2	2	no	—
102	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₁₇₃	—	2	2	no	—
106	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×D₁₇₃	—	2	2	no	—
108	C₂×D₁₇₃	—	2	2	no	—
109	C₂×D₁₇₃	—	4	4	no	—
110	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
111	C₂×D₁₇₃	C₂×#41	1	1	yes	characteristic
112	#10621924	—	4	2	no	—
113	C₄×C₁₇₃	—	2	2	no	—
114	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
115	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
116	Dic₁₇₃	—	2	2	no	—
117	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
118	Dic₁₇₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₁₇₃	—	2	1	no	—
120	C₂×C₁₇₃	—	4	2	no	—
121	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
122	D₁₇₃	—	2	1	no	—
123	D₁₇₃	—	4	2	no	—
124	D₁₇₃	#444	1	2	yes	complemented
125	C₁₇₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
126	C₂×#444	—	173	1	no	maximal, Sylow
127	C₂×#135	—	173	1	no	—
128	C₂×#131	—	173	2	no	—
129	#444	—	173	4	no	—
130	#444	—	173	2	no	—
131	#444	—	173	2	no	—
132	C₂×#108	—	173	1	no	—
133	C₂×#104	—	173	2	no	—
134	C₂×#97	—	173	1	no	—
135	C₂×#41	—	173	2	no	—
136	#135	—	173	2	no	—
137	#135	—	173	1	no	—
138	#135	—	173	2	no	—
139	#135	—	173	1	no	—
140	#135	—	173	1	no	—
141	#135	#10621924	1	1	yes	characteristic, complemented
142	C₂×Q₁₆	—	173	1	no	—
143	C₂×SD₁₆	—	173	2	no	—
144	C₂×D₈	—	173	1	no	—
145	#131	—	346	4	no	—
146	#131	—	173	4	no	—
147	#131	—	173	4	no	—
148	C₂×M₁₆	—	346	2	no	—
149	C₂×M₁₆	—	173	2	no	—
150	C₂²×C₈	—	346	2	no	—
151	C₂²×C₈	—	173	1	no	—
152	C₂²×C₈	—	173	1	no	—
153	C₂×C₄²	—	173	1	no	—
154	#108	—	173	2	no	—
155	#108	—	173	2	no	—
156	#104	—	173	4	no	—
157	#104	—	173	4	no	—
158	#97	—	173	2	no	—
159	#97	—	173	2	no	—
160	#41	—	346	2	no	—
161	#41	—	346	2	no	—
162	#41	—	173	2	no	—
163	#41	—	173	2	no	—
164	#41	—	173	2	no	—
165	#41	C₂×#10621924	1	2	yes	—
166	C₂×Q₈	—	173	2	no	—
167	C₂×D₄	—	346	2	no	—
168	C₂×D₄	—	173	2	no	—
169	C₂²×C₄	—	346	2	no	—
170	C₂²×C₄	—	346	1	no	—
171	C₂²×C₄	—	173	1	no	—
172	Q₁₆	—	173	2	no	—
173	Q₁₆	—	173	1	no	—
174	Q₁₆	C₂×#10621924	1	1	yes	characteristic, complemented
175	SD₁₆	—	173	2	no	—
176	SD₁₆	—	173	2	no	—
177	SD₁₆	—	173	2	no	—
178	SD₁₆	C₂×#10621924	1	2	yes	complemented
179	D₈	—	173	2	no	—
180	D₈	—	173	1	no	—
181	D₈	C₂×#10621924	1	1	yes	characteristic, complemented
182	M₁₆	—	346	2	no	—
183	M₁₆	—	346	4	no	—
184	M₁₆	—	346	4	no	—
185	M₁₆	—	173	4	no	—
186	C₂×C₈	—	346	2	no	—
187	C₂×C₈	—	346	4	no	—
188	C₂×C₈	—	346	4	no	—
189	C₂×C₈	—	173	2	no	—
190	C₂×C₈	—	173	2	no	—
191	C₂×C₈	—	173	2	no	—
192	C₂×C₈	—	173	1	no	—
193	C₂×C₈	—	173	1	no	—
194	C₂×C₈	—	173	1	no	—
195	C₂×C₈	—	173	1	no	—
196	C₂×C₈	—	173	1	no	—
197	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
198	C₄²	—	346	1	no	—
199	C₄²	—	173	2	no	—
200	C₂³	—	346	2	no	—
201	C₂³	—	173	1	no	—
202	Q₈	—	346	2	no	—
203	Q₈	—	173	2	no	—
204	Q₈	C₂²×#10621924	1	2	yes	—
205	D₄	—	2	2	no	—
206	D₄	—	346	2	no	—
207	D₄	—	346	2	no	—
208	D₄	—	346	2	no	—
209	D₄	—	346	2	no	—
210	D₄	—	173	2	no	—
211	D₄	C₂²×#10621924	1	2	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	346	2	no	—
214	C₂×C₄	—	346	2	no	—
215	C₂×C₄	—	346	2	no	—
216	C₂×C₄	—	346	2	no	—
217	C₂×C₄	—	346	2	no	—
218	C₂×C₄	—	173	1	no	—
219	C₂×C₄	—	173	1	no	—
220	C₂×C₄	—	173	1	no	—
221	C₂×C₄	—	173	1	no	—
222	C₂×C₄	—	173	1	no	—
223	C₂×C₄	—	692	2	no	—
224	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
225	C₈	—	346	4	no	—
226	C₈	—	173	4	no	—
227	C₈	—	173	1	no	—
228	C₈	—	173	1	no	—
229	C₈	C₂²×#10621924	1	1	yes	characteristic
230	C₈	C₂²×#10621924	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	346	2	no	—
233	C₂²	—	346	2	no	—
234	C₂²	—	173	1	no	—
235	C₂²	—	173	1	no	—
236	C₂²	—	692	4	no	—
237	C₂²	D₄×#10621924	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	346	2	no	—
240	C₄	—	173	1	no	—
241	C₄	—	173	1	no	—
242	C₄	—	692	2	no	—
243	C₄	D₄×#10621924	1	1	yes	characteristic
244	C₄	C₂³×#10621924	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	4	2	no	—
247	C₂	—	346	1	no	—
248	C₂	—	173	2	no	—
249	C₂	—	692	2	no	—
250	C₂	C₂×D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10621924
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₄	C₂³×#10621924
Fitting	#135×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	#135×D₁₇₃	C₂
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	#135	#10621924
hypocenter	C₁₇₃	C₂×#444
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. D₁₇₃⋊#131	C₂	semidirect
2. #26012538	C₂	semidirect
3. #26012537	C₂	semidirect
4. #26012536	C₂	semidirect
5. D₁₇₃⋊#108	C₂	semidirect
6. D₁₇₃⋊#104	C₂	semidirect
7. C₂×#19149733	C₂	semidirect
9. #19149767	C₂²	semidirect
11. #19149766	C₂²	semidirect
17. #19149750	C₂²	semidirect
18. #19149749	C₂²	semidirect
19. #19149744	C₂²	semidirect
20. #19149743	C₂²	semidirect
21. #19149733	C₂²	semidirect
22. #19149733	C₂²	semidirect
25. #19149701	C₄	semidirect
27. #19149699	C₄	semidirect
29. Q₁₆×D₁₇₃	C₄	semidirect
30. SD₁₆×D₁₇₃	C₄	semidirect
31. D₈×D₁₇₃	C₄	semidirect
33. #135×C₁₇₃	C₄	semidirect
57. Dic₆₉₂	C₂×C₄	semidirect
58. #14583816	C₂×C₄	semidirect
59. #14583815	C₂×C₄	semidirect
60. #14583814	C₂×C₄	semidirect
61. #14583813	C₂×C₄	semidirect
62. D₁₃₈₄	C₂×C₄	semidirect
63. #14583811	C₂×C₄	semidirect
70. Q₁₆×C₁₇₃	C₂×C₄	semidirect
71. SD₁₆×C₁₇₃	C₂×C₄	semidirect
72. D₈×C₁₇₃	C₂×C₄	semidirect
124. D₁₇₃	#444	semidirect
125. C₁₇₃	C₂×#444	semidirect
141. #135	#10621924	semidirect
174. Q₁₆	C₂×#10621924	semidirect
178. SD₁₆	C₂×#10621924	semidirect
181. D₈	C₂×#10621924	semidirect

Maximal subgroups

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

subgroup	index
D₁₇₃⋊#131	2
#26012538	2
#26012537	2
#26012536	2
D₁₇₃⋊#108	2
D₁₇₃⋊#104	2
C₂×#19149733	2
#135×D₁₇₃	2
C₂×#444	173

Subgroups — D173⋊#444 #36370334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇₃⋊#444 #36370334