Subgroups — #36356168

Order 22144.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149636	C₂	1	1	yes	maximal, characteristic
2	#26011382	C₂	1	2	yes	maximal
3	#26011380	C₂	1	2	yes	maximal
4	#26011375	C₂	1	2	yes	maximal, complemented
5	#26011373	C₂	1	2	yes	maximal, complemented
6	#26011360	C₂	1	1	yes	maximal, characteristic, complemented
7	#26011356	C₂	1	1	yes	maximal, characteristic, complemented
8	#26011348	C₂	1	1	yes	maximal, characteristic
9	#26011348	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
13	#19149636	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
16	C₂×#14583805	C₂²	1	2	yes	—
17	C₂×#14583805	C₂²	1	2	yes	—
18	C₂×#14583805	C₂²	1	1	yes	characteristic
19	C₂×#14583805	C₂²	1	1	yes	characteristic
20	C₂×#14583803	C₂²	1	2	yes	—
21	C₂×#14583803	C₂²	1	2	yes	—
22	C₂×#14583803	C₂²	1	1	yes	characteristic
23	C₂×#14583803	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
26	#19149586	C₂²	1	2	yes	—
27	#19149586	C₂²	1	2	yes	—
28	#19149586	C₂²	1	2	yes	—
29	#19149586	C₂²	1	2	yes	—
30	#19149586	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
35	#95×C₁₇₃	C₂²	1	2	yes	—
36	C₂³×D₁₇₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
38	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
39	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₇₃	—	2	2	no	—
43	C₂×C₄×D₁₇₃	—	2	2	no	—
44	C₂×C₄×D₁₇₃	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₁₇₃	—	2	2	no	—
46	C₄⋊Dic₁₇₃	—	2	2	no	—
47	#14583805	—	4	2	no	—
48	#14583805	—	4	2	no	—
49	#14583805	C₂×C₄	1	4	yes	—
50	#14583805	C₂×C₄	1	4	yes	—
51	#14583803	—	4	2	no	—
52	#14583803	—	4	2	no	—
53	#14583803	C₂×C₄	1	4	yes	—
54	#14583803	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₇₃	—	4	2	no	—
56	C₄×Dic₁₇₃	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₇₃	—	2	2	no	—
63	C₄⋊C₄×C₁₇₃	—	2	2	no	—
64	C₄²×C₁₇₃	—	4	2	no	—
65	C₄²×C₁₇₃	C₂×C₄	1	4	yes	—
66	C₂²×D₁₇₃	—	2	1	no	—
67	C₂²×D₁₇₃	—	2	1	no	—
68	C₂²×D₁₇₃	—	2	1	no	—
69	C₂²×D₁₇₃	—	2	1	no	—
70	C₂²×D₁₇₃	—	2	1	no	—
71	C₂²×D₁₇₃	—	2	1	no	—
72	C₂²×D₁₇₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₇₃	—	2	1	no	—
74	C₂×Dic₁₇₃	—	2	1	no	—
75	C₂×Dic₁₇₃	—	2	1	no	—
76	C₂×Dic₁₇₃	—	2	1	no	—
77	C₂×Dic₁₇₃	—	2	1	no	—
78	C₂×Dic₁₇₃	—	2	1	no	—
79	C₂×Dic₁₇₃	—	2	2	no	—
80	C₂×Dic₁₇₃	—	2	2	no	—
81	C₂×Dic₁₇₃	—	2	2	no	—
82	C₂×Dic₁₇₃	—	2	2	no	—
83	C₂×Dic₁₇₃	—	2	2	no	—
84	C₂×Dic₁₇₃	—	2	2	no	—
85	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
88	C₄×D₁₇₃	—	2	8	no	—
89	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₇₃	—	2	1	no	—
91	C₂×C₄×C₁₇₃	—	2	1	no	—
92	C₂×C₄×C₁₇₃	—	2	1	no	—
93	C₂×C₄×C₁₇₃	—	2	1	no	—
94	C₂×C₄×C₁₇₃	—	2	2	no	—
95	C₂×C₄×C₁₇₃	—	2	2	no	—
96	C₂×C₄×C₁₇₃	—	2	2	no	—
97	C₂×C₄×C₁₇₃	—	2	2	no	—
98	C₂×C₄×C₁₇₃	—	2	2	no	—
99	C₂×C₄×C₁₇₃	—	2	2	no	—
100	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₇₃	#41	1	2	yes	—
104	C₂×C₄×C₁₇₃	#41	1	2	yes	—
105	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂²×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×D₁₇₃	—	2	2	no	—
113	C₂×D₁₇₃	—	2	2	no	—
114	C₂×D₁₇₃	—	2	2	no	—
115	C₂×D₁₇₃	—	4	1	no	—
116	C₂×D₁₇₃	—	4	1	no	—
117	C₂×D₁₇₃	—	4	1	no	—
118	C₂×D₁₇₃	—	4	1	no	—
119	C₄×C₁₇₃	—	2	2	no	—
120	C₄×C₁₇₃	—	2	2	no	—
121	C₄×C₁₇₃	—	4	2	no	—
122	C₄×C₁₇₃	—	4	2	no	—
123	C₄×C₁₇₃	#117	1	4	yes	—
124	Dic₁₇₃	—	2	2	no	—
125	Dic₁₇₃	—	2	2	no	—
126	Dic₁₇₃	—	2	2	no	—
127	Dic₁₇₃	—	4	2	no	—
128	Dic₁₇₃	—	4	2	no	—
129	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
130	C₂×C₁₇₃	#520	1	1	yes	characteristic
131	C₂×C₁₇₃	#520	1	1	yes	characteristic
132	C₂×C₁₇₃	#540	1	1	yes	characteristic
133	C₂×C₁₇₃	#541	1	1	yes	characteristic
134	C₂×C₁₇₃	#542	1	1	yes	characteristic
135	C₂×C₁₇₃	#542	1	1	yes	characteristic
136	D₁₇₃	—	4	2	no	—
137	C₁₇₃	#2314	1	1	yes	characteristic, complemented, Sylow
138	#2314	—	173	1	no	maximal, Sylow
139	C₂×#117	—	173	1	no	—
140	#388	—	173	2	no	—
141	#388	—	173	2	no	—
142	#387	—	173	2	no	—
143	#387	—	173	2	no	—
144	#384	—	173	1	no	—
145	#382	—	173	1	no	—
146	#380	—	173	1	no	—
147	#380	—	173	1	no	—
148	C₄×C₄⋊C₄	—	173	1	no	—
149	C₄×C₄⋊C₄	D₁₇₃	1	1	yes	characteristic, complemented
150	C₂³×C₄	—	173	1	no	—
151	#117	—	173	8	no	—
152	C₂×C₄⋊C₄	—	173	1	no	—
153	C₂×C₄⋊C₄	—	173	1	no	—
154	C₂×C₄⋊C₄	—	173	1	no	—
155	C₂×C₄⋊C₄	—	173	1	no	—
156	C₂×C₄⋊C₄	—	173	2	no	—
157	C₂×C₄⋊C₄	—	173	2	no	—
158	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	173	1	no	—
161	C₂×C₂²⋊C₄	—	173	1	no	—
162	C₂×C₂²⋊C₄	—	173	2	no	—
163	C₂×C₂²⋊C₄	—	173	2	no	—
164	C₂×C₄²	—	173	1	no	—
165	C₂×C₄²	—	173	2	no	—
166	C₂×C₄²	C₂×D₁₇₃	1	2	yes	—
167	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
168	#95	—	173	2	no	—
169	#95	—	173	2	no	—
170	#95	—	173	2	no	—
171	#95	—	173	2	no	—
172	#95	—	173	2	no	—
173	#95	C₂×D₁₇₃	1	2	yes	—
174	C₂⁴	—	173	1	no	—
175	C₂²×C₄	—	346	2	no	—
176	C₂²×C₄	—	346	2	no	—
177	C₂²×C₄	—	173	2	no	—
178	C₂²×C₄	—	173	2	no	—
179	C₂²×C₄	—	173	1	no	—
180	C₂²×C₄	—	173	1	no	—
181	C₂²×C₄	—	173	1	no	—
182	C₂²×C₄	—	173	4	no	—
183	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	346	2	no	—
191	C₄⋊C₄	—	346	2	no	—
192	C₄⋊C₄	—	173	4	no	—
193	C₄⋊C₄	—	173	4	no	—
194	C₄⋊C₄	—	692	2	no	—
195	C₄⋊C₄	—	692	2	no	—
196	C₂²⋊C₄	—	173	4	no	—
197	C₂²⋊C₄	—	173	4	no	—
198	C₂²⋊C₄	—	692	2	no	—
199	C₂²⋊C₄	—	692	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	173	4	no	—
202	C₄²	—	692	2	no	—
203	C₄²	C₄×D₁₇₃	1	4	yes	—
204	C₂³	—	346	1	no	—
205	C₂³	—	346	1	no	—
206	C₂³	—	346	1	no	—
207	C₂³	—	346	1	no	—
208	C₂³	—	346	1	no	—
209	C₂³	—	346	1	no	—
210	C₂³	—	173	2	no	—
211	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	346	1	no	—
223	C₂×C₄	—	346	1	no	—
224	C₂×C₄	—	346	1	no	—
225	C₂×C₄	—	346	1	no	—
226	C₂×C₄	—	346	1	no	—
227	C₂×C₄	—	346	1	no	—
228	C₂×C₄	—	346	8	no	—
229	C₂×C₄	—	346	2	no	—
230	C₂×C₄	—	346	2	no	—
231	C₂×C₄	—	346	2	no	—
232	C₂×C₄	—	346	2	no	—
233	C₂×C₄	—	346	2	no	—
234	C₂×C₄	—	346	2	no	—
235	C₂×C₄	—	173	2	no	—
236	C₂×C₄	—	173	2	no	—
237	C₂×C₄	—	173	2	no	—
238	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
239	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
240	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
241	C₂×C₄	#14583826	1	2	yes	—
242	C₂×C₄	#14583826	1	2	yes	—
243	C₂²	—	346	2	no	—
244	C₂²	—	346	2	no	—
245	C₂²	—	346	2	no	—
246	C₂²	—	692	1	no	—
247	C₂²	—	692	1	no	—
248	C₂²	—	692	1	no	—
249	C₂²	—	692	1	no	—
250	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
251	C₂²	C₂×#14583826	1	1	yes	characteristic
252	C₂²	C₂×#14583826	1	1	yes	characteristic
253	C₂²	#19149724	1	1	yes	characteristic
254	C₂²	#19149725	1	1	yes	characteristic
255	C₂²	#19149726	1	1	yes	characteristic
256	C₂²	#19149727	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	346	2	no	—
262	C₄	—	346	2	no	—
263	C₄	—	346	2	no	—
264	C₄	—	692	2	no	—
265	C₄	—	692	2	no	—
266	C₄	#19149636	1	4	yes	—
267	C₂	—	692	2	no	—
268	C₂	C₂×#19149636	1	1	yes	characteristic
269	C₂	#26011971	1	1	yes	characteristic
270	C₂	#26011972	1	1	yes	characteristic
271	C₂	#26012092	1	1	yes	characteristic
272	C₂	#26012099	1	1	yes	characteristic
273	C₂	#26012106	1	1	yes	characteristic
274	C₂	#26012113	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator	C₂²×C₁₇₃	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₇₃	C₂
socle, cosocle	C₂³×C₁₇₃	C₂⁴
prime radical	C₂³×D₁₇₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₇₃
hypocenter	C₁₇₃	#2314
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #26011375	C₂	semidirect
5. #26011373	C₂	semidirect
6. #26011360	C₂	semidirect
7. #26011356	C₂	semidirect
10. C₄×C₄⋊Dic₁₇₃	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₇₃	C₂	semidirect
13. #19149636	C₄	semidirect
137. C₁₇₃	#2314	semidirect
149. C₄×C₄⋊C₄	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149636	2
#26011382	2
#26011380	2
#26011375	2
#26011373	2
#26011360	2
#26011356	2
#26011348	2
C₄×C₄⋊Dic₁₇₃	2
C₄×C₄⋊C₄×C₁₇₃	2
#2314	173