Subgroups — C₄×C₄⋊C₄×D₁₇₃ #36355896

Order 22144.

928 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₇₃	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₇₃	C₂	1	2	yes	maximal
3	C₂×C₄²×D₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₇₃	C₂	1	2	yes	maximal, complemented
5	C₄×#14583805	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#26011342	C₂	1	2	yes	maximal, complemented
8	#95×D₁₇₃	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₇₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
15	C₂×#14583805	C₂²	1	4	yes	—
16	C₄⋊C₄×D₁₇₃	C₄	1	16	yes	direct product factor
17	C₂×C₄×Dic₁₇₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
22	C₄²×D₁₇₃	—	2	8	no	—
23	C₄²×D₁₇₃	C₄	1	8	yes	complemented
24	#19149586	C₂²	1	2	yes	—
25	#19149586	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
29	#95×C₁₇₃	C₂²	1	2	yes	—
30	C₂³×D₁₇₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
33	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×D₁₇₃	—	2	8	no	—
36	C₂×C₄×D₁₇₃	—	2	4	no	—
37	C₂×C₄×D₁₇₃	—	2	8	no	—
38	C₂×C₄×D₁₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₇₃	C₂×C₄	1	16	yes	—
40	C₂×C₄×D₁₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₁₇₃	C₂×C₄	1	8	yes	—
42	C₂×C₄×D₁₇₃	C₂×C₄	1	4	yes	—
43	C₂×C₄×D₁₇₃	D₄	1	4	yes	—
44	C₂×C₄×D₁₇₃	Q₈	1	4	yes	—
45	C₄⋊Dic₁₇₃	C₂×C₄	1	8	yes	complemented
46	#14583805	C₂×C₄	1	16	yes	complemented
47	C₄×Dic₁₇₃	—	2	8	no	—
48	C₄×Dic₁₇₃	—	2	4	no	—
49	C₄×Dic₁₇₃	C₂×C₄	1	4	yes	complemented
50	C₄×Dic₁₇₃	C₂×C₄	1	4	yes	complemented
51	C₄×Dic₁₇₃	C₂×C₄	1	4	yes	complemented
52	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₇₃	C₂×C₄	1	8	yes	complemented
57	C₄²×C₁₇₃	—	2	4	no	—
58	C₄²×C₁₇₃	C₂×C₄	1	4	yes	complemented
59	C₂²×D₁₇₃	C₂²×C₄	1	2	yes	—
60	C₂²×D₁₇₃	C₂²×C₄	1	4	yes	—
61	C₂²×D₁₇₃	C₂×D₄	1	2	yes	—
62	C₂²×D₁₇₃	C₂×Q₈	1	2	yes	—
63	C₂²×D₁₇₃	#41	1	4	yes	—
64	C₂×Dic₁₇₃	—	2	4	no	—
65	C₂×Dic₁₇₃	—	2	2	no	—
66	C₂×Dic₁₇₃	—	2	4	no	—
67	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
68	C₂×Dic₁₇₃	C₂²×C₄	1	8	yes	—
69	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
70	C₂×Dic₁₇₃	C₂²×C₄	1	4	yes	—
71	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₇₃	C₂×D₄	1	2	yes	—
73	C₂×Dic₁₇₃	C₂×Q₈	1	2	yes	—
74	C₄×D₁₇₃	—	2	8	no	—
75	C₄×D₁₇₃	—	2	32	no	—
76	C₄×D₁₇₃	C₄²	1	16	yes	complemented
77	C₄×D₁₇₃	C₄⋊C₄	1	16	yes	direct product factor
78	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₄×C₁₇₃	—	2	4	no	—
80	C₂×C₄×C₁₇₃	—	2	2	no	—
81	C₂×C₄×C₁₇₃	—	2	4	no	—
82	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₇₃	C₂²×C₄	1	8	yes	—
84	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
89	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂²×C₁₇₃	C₂³×C₄	1	2	yes	—
91	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
93	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
94	C₂×D₁₇₃	C₂×C₄²	1	4	yes	—
95	C₂×D₁₇₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×D₁₇₃	#117	1	4	yes	—
97	C₂×D₁₇₃	C₄×D₄	1	8	yes	—
98	C₂×D₁₇₃	C₄×Q₈	1	8	yes	—
99	C₄×C₁₇₃	—	2	2	no	—
100	C₄×C₁₇₃	—	2	8	no	—
101	C₄×C₁₇₃	C₂×C₄²	1	4	yes	complemented
102	C₄×C₁₇₃	C₂×C₄⋊C₄	1	4	yes	complemented
103	Dic₁₇₃	—	2	2	no	—
104	Dic₁₇₃	—	2	8	no	—
105	Dic₁₇₃	C₂×C₄²	1	4	yes	complemented
106	Dic₁₇₃	C₂×C₄⋊C₄	1	4	yes	complemented
107	C₂×C₁₇₃	C₂²×C₄²	1	1	yes	characteristic
108	C₂×C₁₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
110	C₂×C₁₇₃	C₂×C₄×D₄	1	2	yes	—
111	C₂×C₁₇₃	C₂×C₄×Q₈	1	2	yes	—
112	D₁₇₃	C₄×C₄⋊C₄	1	8	yes	direct product factor
113	C₁₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×C₄⋊C₄	—	173	1	no	maximal, Sylow
115	C₂²×C₄⋊C₄	—	173	2	no	—
116	C₂²×C₄²	—	173	2	no	—
117	C₂²×C₄²	—	173	1	no	—
118	C₄×C₄⋊C₄	—	173	2	no	—
119	C₄×C₄⋊C₄	—	173	1	no	—
120	C₄×C₄⋊C₄	—	173	2	no	—
121	C₄×C₄⋊C₄	—	173	2	no	—
122	C₄×C₄⋊C₄	D₁₇₃	1	1	yes	characteristic, direct product factor
123	C₂×#95	—	173	2	no	—
124	C₂³×C₄	—	173	4	no	—
125	C₂³×C₄	—	173	1	no	—
126	C₂³×C₄	—	173	1	no	—
127	C₂³×C₄	—	173	1	no	—
128	C₂×C₄⋊C₄	—	173	4	no	—
129	C₂×C₄⋊C₄	—	173	2	no	—
130	C₂×C₄⋊C₄	—	173	16	no	—
131	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
132	C₂×C₄²	—	346	8	no	—
133	C₂×C₄²	—	173	4	no	—
134	C₂×C₄²	—	173	2	no	—
135	C₂×C₄²	—	173	1	no	—
136	C₂×C₄²	—	173	1	no	—
137	C₂×C₄²	—	173	1	no	—
138	C₂×C₄²	—	173	8	no	—
139	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
140	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
141	#95	—	173	4	no	—
142	#95	—	173	2	no	—
143	#95	C₂×D₁₇₃	1	2	yes	—
144	C₂⁴	—	173	1	no	—
145	C₂²×C₄	—	346	8	no	—
146	C₂²×C₄	—	346	8	no	—
147	C₂²×C₄	—	346	4	no	—
148	C₂²×C₄	—	173	1	no	—
149	C₂²×C₄	—	173	4	no	—
150	C₂²×C₄	—	173	4	no	—
151	C₂²×C₄	—	173	16	no	—
152	C₂²×C₄	—	173	1	no	—
153	C₂²×C₄	—	173	1	no	—
154	C₂²×C₄	—	173	4	no	—
155	C₂²×C₄	—	173	4	no	—
156	C₂²×C₄	—	173	8	no	—
157	C₂²×C₄	—	173	4	no	—
158	C₂²×C₄	—	173	4	no	—
159	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
161	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
163	C₄⋊C₄	—	173	16	no	—
164	C₄⋊C₄	—	173	8	no	—
165	C₄⋊C₄	C₄×D₁₇₃	1	8	yes	direct product factor
166	C₄²	—	2	4	no	—
167	C₄²	—	346	8	no	—
168	C₄²	—	346	4	no	—
169	C₄²	—	173	4	no	—
170	C₄²	—	173	4	no	—
171	C₄²	—	173	4	no	—
172	C₄²	C₄×D₁₇₃	1	4	yes	complemented
173	C₂³	—	173	2	no	—
174	C₂³	—	173	4	no	—
175	C₂³	—	173	2	no	—
176	C₂³	—	173	2	no	—
177	C₂³	—	173	4	no	—
178	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	346	8	no	—
183	C₂×C₄	—	346	32	no	—
184	C₂×C₄	—	346	4	no	—
185	C₂×C₄	—	346	2	no	—
186	C₂×C₄	—	346	4	no	—
187	C₂×C₄	—	173	2	no	—
188	C₂×C₄	—	173	8	no	—
189	C₂×C₄	—	173	2	no	—
190	C₂×C₄	—	173	2	no	—
191	C₂×C₄	—	173	4	no	—
192	C₂×C₄	—	173	2	no	—
193	C₂×C₄	—	173	2	no	—
194	C₂×C₄	—	173	16	no	—
195	C₂×C₄	—	173	16	no	—
196	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
197	C₂×C₄	C₂×C₄×D₁₇₃	1	8	yes	—
198	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₇₃	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
201	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
202	C₂×C₄	Q₈×D₁₇₃	1	2	yes	—
203	C₂²	—	173	4	no	—
204	C₂²	—	173	4	no	—
205	C₂²	—	173	4	no	—
206	C₂²	—	173	8	no	—
207	C₂²	—	173	8	no	—
208	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
209	C₂²	C₂²×C₄×D₁₇₃	1	2	yes	—
210	C₂²	C₂×D₄×D₁₇₃	1	1	yes	characteristic
211	C₂²	C₂×Q₈×D₁₇₃	1	1	yes	characteristic
212	C₂²	#41×D₁₇₃	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	8	no	—
215	C₄	—	346	2	no	—
216	C₄	—	346	8	no	—
217	C₄	—	173	4	no	—
218	C₄	—	173	4	no	—
219	C₄	C₄²×D₁₇₃	1	4	yes	complemented
220	C₄	C₄⋊C₄×D₁₇₃	1	4	yes	direct product factor
221	C₂	—	173	8	no	—
222	C₂	C₂×C₄²×D₁₇₃	1	1	yes	characteristic
223	C₂	C₂×C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
224	C₂	#117×D₁₇₃	1	1	yes	characteristic
225	C₂	C₄×D₄×D₁₇₃	1	2	yes	—
226	C₂	C₄×Q₈×D₁₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₇₃
commutator	C₂×C₁₇₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₇₃
Fitting	C₄×C₄⋊C₄×C₁₇₃	C₂
socle, cosocle	C₂³×C₁₇₃	C₂⁴
prime radical	C₂³×D₁₇₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₇₃
hypocenter	C₁₇₃	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₁₇₃	C₂	semidirect
5. C₄×#14583805	C₂	semidirect
6. C₄×C₄⋊Dic₁₇₃	C₂	semidirect
7. #26011342	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₇₃	C₂	semidirect
16. C₄⋊C₄×D₁₇₃	C₄	direct (N × H)
23. C₄²×D₁₇₃	C₄	semidirect
45. C₄⋊Dic₁₇₃	C₂×C₄	semidirect
46. #14583805	C₂×C₄	semidirect
49. C₄×Dic₁₇₃	C₂×C₄	semidirect
50. C₄×Dic₁₇₃	C₂×C₄	semidirect
51. C₄×Dic₁₇₃	C₂×C₄	semidirect
56. C₄⋊C₄×C₁₇₃	C₂×C₄	semidirect
58. C₄²×C₁₇₃	C₂×C₄	semidirect
76. C₄×D₁₇₃	C₄²	semidirect
77. C₄×D₁₇₃	C₄⋊C₄	direct (N × H)
101. C₄×C₁₇₃	C₂×C₄²	semidirect
102. C₄×C₁₇₃	C₂×C₄⋊C₄	semidirect
105. Dic₁₇₃	C₂×C₄²	semidirect
106. Dic₁₇₃	C₂×C₄⋊C₄	semidirect
112. D₁₇₃	C₄×C₄⋊C₄	direct (N × H)
113. C₁₇₃	C₂×C₄×C₄⋊C₄	semidirect
122. C₄×C₄⋊C₄	D₁₇₃	direct (N × H)
165. C₄⋊C₄	C₄×D₁₇₃	direct (N × H)
172. C₄²	C₄×D₁₇₃	semidirect
219. C₄	C₄²×D₁₇₃	semidirect
220. C₄	C₄⋊C₄×D₁₇₃	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×C₄⋊C₄×D₁₇₃	2
C₂×C₄²×D₁₇₃	2
C₄⋊C₄×Dic₁₇₃	2
C₄×#14583805	2
C₄×C₄⋊Dic₁₇₃	2
#26011342	2
#95×D₁₇₃	2
C₄×C₄⋊C₄×C₁₇₃	2
C₂×C₄×C₄⋊C₄	173

Subgroups — C4×C4⋊C4×D173 #36355896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊C₄×D₁₇₃ #36355896