Subgroups — #36370010

Order 22144.

568 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012436	C₂	1	2	yes	maximal, complemented
2	#26012435	C₂	1	2	yes	maximal, complemented
3	#26012434	C₂	1	2	yes	maximal
4	#26012433	C₂	1	2	yes	maximal
5	C₄×C₄⋊#10621924	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊#10621924	C₂	1	2	yes	maximal
7	#26012427	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₁₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#10621924	C₂²	1	4	yes	—
10	C₂×C₂²⋊#10621924	C₂²	1	4	yes	—
11	C₂×C₄×#10621924	C₂²	1	4	yes	—
12	(C₂×#10621924)⋊C₄	C₂²	1	4	yes	—
13	(C₂×#10621924)⋊C₄	C₂²	1	4	yes	—
14	(C₂×#10621924)⋊C₄	C₂²	1	4	yes	—
15	(C₂×#10621924)⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×Dic₁₇₃	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×Dic₁₇₃	C₄	1	1	yes	characteristic, complemented
22	C₄²×D₁₇₃	C₄	1	1	yes	characteristic, complemented
23	C₄²×D₁₇₃	C₄	1	1	yes	characteristic, complemented
24	C₄²×D₁₇₃	C₄	1	1	yes	characteristic, complemented
25	C₄²×D₁₇₃	C₄	1	1	yes	characteristic, complemented
26	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
27	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
28	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
29	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₇₃	C₄	1	1	yes	characteristic, complemented
31	C₂²×#10621924	C₂³	1	8	yes	—
32	C₄⋊#10621924	—	2	8	no	—
33	C₂²⋊#10621924	—	2	8	no	—
34	C₄×#10621924	—	2	8	no	—
35	C₂³×D₁₇₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₇₃	—	2	4	no	—
40	C₂×C₄×D₁₇₃	—	2	4	no	—
41	C₂×C₄×D₁₇₃	—	2	4	no	—
42	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
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×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
54	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
73	C₂×#10621924	—	2	8	no	—
74	C₂×#10621924	—	2	8	no	—
75	C₂×#10621924	—	2	8	no	—
76	C₂²×D₁₇₃	—	2	2	no	—
77	C₂²×D₁₇₃	—	2	2	no	—
78	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₇₃	#41	1	4	yes	—
80	C₂²×D₁₇₃	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₇₃	—	2	2	no	—
82	C₂×Dic₁₇₃	—	2	2	no	—
83	C₂×Dic₁₇₃	—	2	2	no	—
84	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₁₇₃	—	2	2	no	—
91	C₄×D₁₇₃	—	2	2	no	—
92	C₄×D₁₇₃	—	2	2	no	—
93	C₄×D₁₇₃	—	2	2	no	—
94	C₄×D₁₇₃	—	2	2	no	—
95	C₄×D₁₇₃	—	2	2	no	—
96	C₄×D₁₇₃	—	2	2	no	—
97	C₄×D₁₇₃	—	2	2	no	—
98	C₄×D₁₇₃	—	2	2	no	—
99	C₄×D₁₇₃	—	2	2	no	—
100	C₄×D₁₇₃	#41	1	4	yes	—
101	C₄×D₁₇₃	#41	1	4	yes	—
102	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇₃	—	2	2	no	—
104	C₂×C₄×C₁₇₃	—	2	2	no	—
105	C₂×C₄×C₁₇₃	—	2	2	no	—
106	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₁₇₃	—	2	1	no	—
113	C₂²×C₁₇₃	—	2	1	no	—
114	C₂²×C₁₇₃	#117	1	2	yes	—
115	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₇₃	—	2	1	no	—
117	C₂×D₁₇₃	—	2	1	no	—
118	C₂×D₁₇₃	—	2	8	no	—
119	C₂×D₁₇₃	#117	1	2	yes	—
120	C₂×D₁₇₃	C₂×#41	1	2	yes	—
121	C₂×D₁₇₃	C₂×#41	1	2	yes	—
122	C₂×D₁₇₃	C₂×#41	1	2	yes	—
123	#10621924	—	4	8	no	—
124	C₄×C₁₇₃	—	2	1	no	—
125	C₄×C₁₇₃	—	2	1	no	—
126	C₄×C₁₇₃	—	2	1	no	—
127	C₄×C₁₇₃	—	2	1	no	—
128	C₄×C₁₇₃	#117	1	2	yes	—
129	C₄×C₁₇₃	#117	1	2	yes	—
130	Dic₁₇₃	—	2	1	no	—
131	Dic₁₇₃	—	2	1	no	—
132	Dic₁₇₃	—	2	1	no	—
133	Dic₁₇₃	—	2	1	no	—
134	Dic₁₇₃	#117	1	2	yes	—
135	Dic₁₇₃	#117	1	2	yes	—
136	C₂×C₁₇₃	—	2	2	no	—
137	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
138	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
139	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
140	D₁₇₃	—	2	2	no	—
141	D₁₇₃	#529	1	4	yes	—
142	C₁₇₃	#2283	1	1	yes	characteristic, complemented, Sylow
143	#2283	—	173	1	no	maximal, Sylow
144	C₂²×C₄²	—	173	1	no	—
145	#383	—	173	2	no	—
146	#382	—	173	2	no	—
147	#381	—	173	2	no	—
148	#380	—	173	2	no	—
149	C₄×C₄⋊C₄	—	173	2	no	—
150	C₄×C₂²⋊C₄	—	173	2	no	—
151	#376	—	173	2	no	—
152	C₂³×C₄	—	173	1	no	—
153	C₂³×C₄	—	173	1	no	—
154	C₂³×C₄	—	173	1	no	—
155	C₂×C₄⋊C₄	—	173	4	no	—
156	C₂×C₂²⋊C₄	—	173	4	no	—
157	C₂×C₄²	—	173	4	no	—
158	C₂×C₄²	—	173	1	no	—
159	C₂×C₄²	—	173	1	no	—
160	C₂×C₄²	—	173	1	no	—
161	C₂×C₄²	—	173	1	no	—
162	C₂×C₄²	—	173	1	no	—
163	C₂×C₄²	—	173	1	no	—
164	C₂×C₄²	—	173	1	no	—
165	C₂×C₄²	—	173	1	no	—
166	C₂×C₄²	—	173	1	no	—
167	C₂×C₄²	—	173	1	no	—
168	C₂×C₄²	—	173	1	no	—
169	C₂×C₄²	#10621924	1	1	yes	characteristic, complemented
170	#95	—	173	4	no	—
171	#95	—	173	4	no	—
172	#95	—	173	4	no	—
173	#95	—	173	4	no	—
174	C₂⁴	—	173	1	no	—
175	C₂²×C₄	—	346	4	no	—
176	C₂²×C₄	—	346	4	no	—
177	C₂²×C₄	—	346	4	no	—
178	C₂²×C₄	—	173	8	no	—
179	C₂²×C₄	—	173	1	no	—
180	C₂²×C₄	—	173	1	no	—
181	C₂²×C₄	—	173	1	no	—
182	C₂²×C₄	—	173	1	no	—
183	C₂²×C₄	—	173	1	no	—
184	C₂²×C₄	—	173	1	no	—
185	C₂²×C₄	—	173	1	no	—
186	C₂²×C₄	—	173	1	no	—
187	C₂²×C₄	—	173	1	no	—
188	C₂²×C₄	—	173	1	no	—
189	C₂²×C₄	—	173	1	no	—
190	C₂²×C₄	—	173	1	no	—
191	C₂²×C₄	—	173	1	no	—
192	C₂²×C₄	—	173	1	no	—
193	C₂²×C₄	—	173	1	no	—
194	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
195	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
196	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
197	C₄⋊C₄	—	346	8	no	—
198	C₂²⋊C₄	—	346	8	no	—
199	C₄²	—	346	8	no	—
200	C₄²	—	173	1	no	—
201	C₄²	—	173	1	no	—
202	C₄²	—	173	1	no	—
203	C₄²	—	173	1	no	—
204	C₄²	—	173	1	no	—
205	C₄²	—	173	1	no	—
206	C₄²	—	173	1	no	—
207	C₄²	—	173	1	no	—
208	C₄²	—	173	1	no	—
209	C₄²	—	173	1	no	—
210	C₄²	—	173	1	no	—
211	C₄²	—	173	1	no	—
212	C₄²	C₂×#10621924	1	1	yes	characteristic
213	C₄²	C₂×#10621924	1	1	yes	characteristic
214	C₄²	C₂×#10621924	1	1	yes	characteristic
215	C₄²	C₂×#10621924	1	1	yes	characteristic
216	C₂³	—	346	2	no	—
217	C₂³	—	346	2	no	—
218	C₂³	—	173	4	no	—
219	C₂³	—	173	1	no	—
220	C₂³	—	173	1	no	—
221	C₂³	C₂²×#10621924	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	346	2	no	—
226	C₂×C₄	—	346	2	no	—
227	C₂×C₄	—	346	2	no	—
228	C₂×C₄	—	346	2	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	—	346	2	no	—
236	C₂×C₄	—	346	2	no	—
237	C₂×C₄	—	346	2	no	—
238	C₂×C₄	—	346	8	no	—
239	C₂×C₄	—	346	8	no	—
240	C₂×C₄	—	346	8	no	—
241	C₂×C₄	—	173	1	no	—
242	C₂×C₄	—	173	1	no	—
243	C₂×C₄	—	173	1	no	—
244	C₂×C₄	—	173	1	no	—
245	C₂×C₄	—	173	1	no	—
246	C₂×C₄	—	173	1	no	—
247	C₂×C₄	—	173	4	no	—
248	C₂×C₄	—	173	4	no	—
249	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
250	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
251	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
252	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
253	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
254	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	346	8	no	—
258	C₂²	—	346	1	no	—
259	C₂²	—	346	1	no	—
260	C₂²	—	173	2	no	—
261	C₂²	—	173	2	no	—
262	C₂²	—	173	2	no	—
263	C₂²	—	173	2	no	—
264	C₂²	#19149758	1	2	yes	—
265	C₂²	C₂³×#10621924	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	346	1	no	—
271	C₄	—	346	1	no	—
272	C₄	—	346	1	no	—
273	C₄	—	346	1	no	—
274	C₄	—	173	2	no	—
275	C₄	—	173	2	no	—
276	C₄	—	692	8	no	—
277	C₄	#19149758	1	2	yes	—
278	C₄	#19149758	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	346	2	no	—
281	C₂	—	173	4	no	—
282	C₂	C₂×#19149758	1	1	yes	characteristic
283	C₂	C₂×#19149758	1	1	yes	characteristic
284	C₂	C₂×#19149758	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Frattini	C₂²	C₂³×#10621924
Fitting	C₂×C₄²×C₁₇₃	C₄
prime radical	C₂³×D₁₇₃	C₂³
cosocle	C₂²×D₁₇₃	C₂⁴
hypercenter	C₂×C₄²	#10621924
hypocenter	C₁₇₃	#2283
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #26012436	C₂	semidirect
2. #26012435	C₂	semidirect
5. C₄×C₄⋊#10621924	C₂	semidirect
7. #26012427	C₂	semidirect
19. C₂×C₄×Dic₁₇₃	C₄	semidirect
20. C₂×C₄×Dic₁₇₃	C₄	semidirect
21. C₂×C₄×Dic₁₇₃	C₄	semidirect
22. C₄²×D₁₇₃	C₄	semidirect
23. C₄²×D₁₇₃	C₄	semidirect
24. C₄²×D₁₇₃	C₄	semidirect
25. C₄²×D₁₇₃	C₄	semidirect
30. C₂×C₄²×C₁₇₃	C₄	semidirect
142. C₁₇₃	#2283	semidirect
169. C₂×C₄²	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012436	2
#26012435	2
#26012434	2
#26012433	2
C₄×C₄⋊#10621924	2
C₄×C₂²⋊#10621924	2
#26012427	2
C₂×C₄²×D₁₇₃	2
#2283	173