Subgroups — #36362670

Order 22144.

432 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14583825	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011615	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×#13063436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₁₇₃	C₂	1	2	yes	maximal, complemented
6	#26011554	C₂	1	2	yes	maximal, complemented
7	#26011542	C₂	1	1	yes	maximal, characteristic, complemented
8	#26011541	C₂	1	2	yes	maximal, complemented
9	C₂×#19149608	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₈×D₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583825	C₄	1	1	yes	characteristic
13	C₂²×#13063436	C₂²	1	2	yes	—
14	C₂×C₈×D₁₇₃	C₂²	1	2	yes	—
15	C₄×Dic₃₄₆	C₂²	1	1	yes	characteristic
16	#19149644	C₄	1	1	yes	characteristic
17	D₄×Dic₁₇₃	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₇₃	C₂²	1	2	yes	—
19	#19149639	C₄	1	2	yes	—
20	#19149637	C₄	1	1	yes	characteristic
21	#19149635	C₄	1	2	yes	—
22	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
23	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊#13063436	C₂²	1	1	yes	characteristic
25	#19149608	C₂²	1	2	yes	—
26	#19149608	C₂²	1	1	yes	characteristic
27	#19149608	C₂²	1	2	yes	—
28	#19149608	C₂²	1	2	yes	—
29	#19149608	C₂²	1	1	yes	characteristic
30	C₂²⋊#13063436	C₂²	1	2	yes	—
31	#19149591	C₂²	1	1	yes	characteristic
32	#19149591	C₂²	1	1	yes	characteristic
33	C₄×#13063436	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₇₃	C₂²	1	2	yes	—
35	C₈×Dic₁₇₃	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
37	C₄×D₄×C₁₇₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₇₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₇₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
41	#14583825	C₈	1	8	yes	complemented
42	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
49	C₂×#13063436	—	4	2	no	—
50	C₂×#13063436	C₂³	1	1	yes	characteristic
51	C₂×#13063436	C₂³	1	2	yes	—
52	C₂×#13063436	C₂³	1	1	yes	characteristic
53	C₈×D₁₇₃	—	4	2	no	—
54	C₄⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
55	#14583805	C₂×C₄	1	2	yes	—
56	#14583805	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
58	#14583803	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₇₃	C₂³	1	2	yes	—
64	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
67	C₂×C₈×C₁₇₃	—	4	2	no	—
68	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
70	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
73	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
74	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
75	Dic₃₄₆	C₂×C₈	1	4	yes	complemented
76	C₂²⋊D₁₇₃	C₂×C₈	1	8	yes	complemented
77	C₂×Dic₁₇₃	—	4	2	no	—
78	C₂×Dic₁₇₃	C₂×C₈	1	8	yes	complemented
79	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
84	C₄×D₁₇₃	—	2	1	no	—
85	C₄×D₁₇₃	—	2	1	no	—
86	C₄×D₁₇₃	—	4	1	no	—
87	C₄×D₁₇₃	C₂×C₈	1	4	yes	complemented
88	#13063436	—	2	1	no	—
89	#13063436	—	2	2	no	—
90	#13063436	—	2	1	no	—
91	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
92	D₄×C₁₇₃	C₂×C₈	1	4	yes	complemented
93	C₂×C₄×C₁₇₃	—	2	2	no	—
94	C₂×C₄×C₁₇₃	—	2	2	no	—
95	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₇₃	—	2	1	no	—
100	C₈×C₁₇₃	—	2	2	no	—
101	C₈×C₁₇₃	—	2	1	no	—
102	C₂²×C₁₇₃	—	2	2	no	—
103	C₂²×C₁₇₃	—	2	2	no	—
104	C₂²×C₁₇₃	C₂²×C₈	1	4	yes	—
105	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₇₃	—	2	1	no	—
107	C₂×D₁₇₃	—	2	1	no	—
108	C₂×D₁₇₃	C₂²×C₈	1	2	yes	—
109	C₄×C₁₇₃	—	2	2	no	—
110	C₄×C₁₇₃	—	2	1	no	—
111	C₄×C₁₇₃	C₂²×C₈	1	2	yes	—
112	C₄×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₁₇₃	—	2	1	no	—
115	Dic₁₇₃	—	2	1	no	—
116	Dic₁₇₃	—	2	2	no	—
117	Dic₁₇₃	C₂²×C₈	1	2	yes	—
118	Dic₁₇₃	C₂²×C₈	1	4	yes	—
119	C₂×C₁₇₃	—	2	4	no	—
120	C₂×C₁₇₃	#520	1	1	yes	characteristic
121	C₂×C₁₇₃	C₂³×C₈	1	1	yes	characteristic
122	C₂×C₁₇₃	#568	1	1	yes	characteristic
123	D₁₇₃	—	2	2	no	—
124	C₁₇₃	#2988	1	1	yes	characteristic, complemented, Sylow
125	#2988	—	173	1	no	maximal, Sylow
126	C₄×#41	—	173	1	no	—
127	C₈×Q₈	—	173	1	no	—
128	C₈×Q₈	—	173	1	no	—
129	C₈×D₄	—	173	2	no	—
130	C₈×D₄	—	173	1	no	—
131	C₈×D₄	—	173	2	no	—
132	C₈×D₄	D₁₇₃	1	1	yes	characteristic, complemented
133	#431	—	173	2	no	—
134	#431	—	173	1	no	—
135	C₂×C₄⋊C₈	—	173	2	no	—
136	C₂×C₄⋊C₈	—	173	1	no	—
137	C₂×#41	—	173	1	no	—
138	C₂²×C₈	—	173	2	no	—
139	C₂²×C₈	—	173	2	no	—
140	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
141	C₄×Q₈	—	173	1	no	—
142	C₄×Q₈	—	173	1	no	—
143	C₄×D₄	—	173	1	no	—
144	C₄×D₄	—	173	2	no	—
145	C₄×D₄	—	173	2	no	—
146	C₄×D₄	C₂×D₁₇₃	1	1	yes	characteristic
147	#117	—	173	1	no	—
148	#117	—	173	2	no	—
149	C₂×C₄²	—	173	2	no	—
150	C₂×C₄²	—	173	1	no	—
151	C₄⋊C₈	—	173	2	no	—
152	C₄⋊C₈	—	173	1	no	—
153	C₄⋊C₈	—	173	2	no	—
154	C₄⋊C₈	—	173	2	no	—
155	C₄⋊C₈	—	173	1	no	—
156	C₄⋊C₈	—	173	1	no	—
157	C₄⋊C₈	C₂×D₁₇₃	1	1	yes	characteristic
158	C₂²⋊C₈	—	173	2	no	—
159	C₂²⋊C₈	—	173	1	no	—
160	C₂²⋊C₈	—	173	1	no	—
161	C₂²⋊C₈	C₂×D₁₇₃	1	2	yes	—
162	C₄×C₈	—	173	2	no	—
163	C₄×C₈	—	173	2	no	—
164	C₄×C₈	—	173	1	no	—
165	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
166	#41	—	173	8	no	—
167	C₂×Q₈	—	173	1	no	—
168	C₂×D₄	—	173	2	no	—
169	C₂×D₄	C₄×D₁₇₃	1	1	yes	characteristic
170	C₂²×C₄	—	173	1	no	—
171	C₂²×C₄	—	173	2	no	—
172	C₂²×C₄	—	173	1	no	—
173	C₂²×C₄	—	173	2	no	—
174	C₂²×C₄	—	173	1	no	—
175	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	173	1	no	—
178	C₂×C₈	—	173	2	no	—
179	C₂×C₈	—	173	1	no	—
180	C₂×C₈	—	692	2	no	—
181	C₂×C₈	—	692	2	no	—
182	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
184	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
185	C₄⋊C₄	—	173	2	no	—
186	C₄⋊C₄	—	173	2	no	—
187	C₄⋊C₄	—	173	1	no	—
188	C₄⋊C₄	C₄×D₁₇₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	173	2	no	—
190	C₂²⋊C₄	—	173	2	no	—
191	C₂²⋊C₄	C₄×D₁₇₃	1	2	yes	—
192	C₄²	—	173	2	no	—
193	C₄²	—	173	1	no	—
194	C₄²	—	173	1	no	—
195	C₄²	—	173	2	no	—
196	C₄²	—	173	2	no	—
197	C₄²	—	173	1	no	—
198	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
199	C₂³	—	173	1	no	—
200	C₂³	C₂×C₄×D₁₇₃	1	2	yes	—
201	Q₈	—	173	4	no	—
202	D₄	—	173	8	no	—
203	D₄	C₈×D₁₇₃	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	346	1	no	—
207	C₂×C₄	—	346	1	no	—
208	C₂×C₄	—	173	1	no	—
209	C₂×C₄	—	173	1	no	—
210	C₂×C₄	—	173	2	no	—
211	C₂×C₄	—	173	8	no	—
212	C₂×C₄	—	173	1	no	—
213	C₂×C₄	—	173	2	no	—
214	C₂×C₄	—	173	4	no	—
215	C₂×C₄	—	692	1	no	—
216	C₂×C₄	—	692	2	no	—
217	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	346	1	no	—
225	C₈	—	346	2	no	—
226	C₈	—	346	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	346	1	no	—
230	C₂²	—	346	1	no	—
231	C₂²	—	173	2	no	—
232	C₂²	C₂×C₈×D₁₇₃	1	4	yes	—
233	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	346	1	no	—
237	C₄	—	346	1	no	—
238	C₄	—	346	2	no	—
239	C₄	—	173	2	no	—
240	C₄	—	173	4	no	—
241	C₄	C₂×C₈×D₁₇₃	1	2	yes	—
242	C₄	#19149724	1	1	yes	characteristic
243	C₄	#19149726	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	346	2	no	—
246	C₂	#26011970	1	1	yes	characteristic
247	C₂	C₂²×C₈×D₁₇₃	1	1	yes	characteristic
248	C₂	#26012256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇₃
commutator	C₂×C₁₇₃	C₂³×C₈
Fitting	C₈×D₄×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂³×C₄
prime radical	C₂×#14583825	C₄
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	C₈×D₄	D₁₇₃
hypocenter	C₁₇₃	#2988
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₃₄₆	C₂	semidirect
3. #26011615	C₂	semidirect
4. D₄×#13063436	C₂	semidirect
5. C₈×C₂²⋊D₁₇₃	C₂	semidirect
6. #26011554	C₂	semidirect
7. #26011542	C₂	semidirect
8. #26011541	C₂	semidirect
9. C₂×#19149608	C₂	semidirect
10. C₄⋊C₈×D₁₇₃	C₂	semidirect
11. C₈×D₄×C₁₇₃	C₂	semidirect
41. #14583825	C₈	semidirect
75. Dic₃₄₆	C₂×C₈	semidirect
76. C₂²⋊D₁₇₃	C₂×C₈	semidirect
78. C₂×Dic₁₇₃	C₂×C₈	semidirect
87. C₄×D₁₇₃	C₂×C₈	semidirect
92. D₄×C₁₇₃	C₂×C₈	semidirect
124. C₁₇₃	#2988	semidirect
132. C₈×D₄	D₁₇₃	semidirect
203. D₄	C₈×D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14583825	2
C₈×Dic₃₄₆	2
#26011615	2
D₄×#13063436	2
C₈×C₂²⋊D₁₇₃	2
#26011554	2
#26011542	2
#26011541	2
C₂×#19149608	2
C₄⋊C₈×D₁₇₃	2
C₈×D₄×C₁₇₃	2
#2988	173