Subgroups — #36370323

Order 22144.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012559	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012558	C₂	1	1	yes	maximal, characteristic
3	#26012530	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#10621924	C₂	1	4	yes	maximal, complemented
5	#26012501	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012484	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012483	C₂	1	1	yes	maximal, characteristic
8	#26012454	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₁₇₃	C₂	1	1	yes	maximal, characteristic
10	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#10621924	—	2	2	no	—
12	Q₈×#10621924	C₂²	1	4	yes	complemented
13	D₄×#10621924	—	2	2	no	—
14	D₄×#10621924	C₂²	1	4	yes	—
15	#19149758	—	2	2	no	—
16	#19149758	C₂²	1	1	yes	characteristic
17	#19149758	C₂²	1	1	yes	characteristic
18	#19149758	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#10621924	—	2	2	no	—
20	#19149748	C₂²	1	2	yes	complemented
21	#19149747	C₂²	1	2	yes	complemented
22	#19149742	C₂²	1	4	yes	complemented
23	#19149741	C₂²	1	4	yes	—
24	#19149733	C₂²	1	2	yes	complemented
25	C₈×#10621924	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₁₇₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₇₃	C₂²	1	1	yes	characteristic
28	C₂×#14583815	C₄	1	1	yes	characteristic, complemented
29	C₂×#14583814	C₄	1	1	yes	characteristic, complemented
30	C₂×#14583813	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₁₇₃	C₄	1	2	yes	complemented
32	SD₁₆×D₁₇₃	C₄	1	2	yes	complemented
33	SD₁₆×D₁₇₃	C₂²	1	2	yes	—
34	SD₁₆×D₁₇₃	C₂²	1	2	yes	—
35	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
37	C₂²×#10621924	—	2	4	no	—
38	#14583836	—	2	1	no	—
39	#14583835	—	2	1	no	—
40	D₁₇₃⋊C₈	C₂³	1	2	yes	—
41	C₄⋊#10621924	—	2	4	no	—
42	C₄⋊#10621924	D₄	1	2	yes	—
43	C₄⋊#10621924	C₂³	1	2	yes	—
44	C₄⋊#10621924	C₂³	1	2	yes	—
45	C₂²⋊#10621924	—	2	4	no	—
46	C₂²⋊#10621924	—	2	1	no	—
47	C₂²⋊#10621924	—	2	1	no	—
48	C₂²⋊#10621924	D₄	1	2	yes	—
49	C₄×#10621924	—	2	4	no	—
50	C₄×#10621924	—	2	1	no	—
51	C₄×#10621924	—	2	1	no	—
52	C₄×#10621924	C₂³	1	2	yes	—
53	C₂³×D₁₇₃	—	2	1	no	—
54	C₂×Dic₃₄₆	—	2	1	no	—
55	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₁₇₃	—	2	1	no	—
57	Q₈×D₁₇₃	—	2	1	no	—
58	Q₈×D₁₇₃	C₂×C₄	1	2	yes	complemented
59	Q₈×D₁₇₃	C₂³	1	2	yes	—
60	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
62	D₄×D₁₇₃	—	2	1	no	—
63	D₄×D₁₇₃	—	2	1	no	—
64	D₄×D₁₇₃	C₂×C₄	1	2	yes	—
65	D₄×D₁₇₃	C₂³	1	2	yes	—
66	C₂×C₄×D₁₇₃	—	2	1	no	—
67	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
68	#14583815	C₂×C₄	1	2	yes	complemented
69	#14583815	C₂×C₄	1	2	yes	complemented
70	#14583814	C₂×C₄	1	2	yes	complemented
71	#14583814	C₂×C₄	1	2	yes	complemented
72	#14583813	C₂×C₄	1	2	yes	complemented
73	#14583813	C₂×C₄	1	2	yes	complemented
74	C₂×#13063436	C₂×C₄	1	1	yes	characteristic, complemented
75	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
76	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
77	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
78	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
82	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
83	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂×#10621924	—	2	2	no	—
85	C₂×#10621924	—	2	2	no	—
86	C₂×#10621924	—	8	4	no	—
87	C₂×#10621924	C₂×D₄	1	4	yes	—
88	#13063443	—	2	2	no	—
89	C₂²×D₁₇₃	—	2	1	no	—
90	C₂²×D₁₇₃	—	2	2	no	—
91	C₂²×D₁₇₃	—	2	2	no	—
92	C₂²×D₁₇₃	—	8	2	no	—
93	C₂²×D₁₇₃	C₂×D₄	1	1	yes	characteristic
94	Dic₃₄₆	—	2	1	no	—
95	Dic₃₄₆	—	2	2	no	—
96	Dic₃₄₆	—	2	2	no	—
97	Dic₃₄₆	C₂²×C₄	1	2	yes	—
98	D₆₉₂	—	2	1	no	—
99	D₆₉₂	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₁₇₃	—	2	2	no	—
101	C₂²⋊D₁₇₃	—	2	2	no	—
102	C₂×Dic₁₇₃	—	2	1	no	—
103	C₂×Dic₁₇₃	#41	1	1	yes	characteristic
104	C₄×D₁₇₃	—	2	2	no	—
105	C₄×D₁₇₃	—	2	2	no	—
106	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
108	C₄×D₁₇₃	#41	1	1	yes	characteristic
109	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
110	#13063436	C₂²×C₄	1	1	yes	characteristic
111	#13063436	C₂²×C₄	1	1	yes	characteristic
112	C₂³×C₁₇₃	—	2	1	no	—
113	Q₈×C₁₇₃	—	2	1	no	—
114	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
115	D₄×C₁₇₃	—	2	1	no	—
116	D₄×C₁₇₃	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₇₃	—	2	1	no	—
118	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₁₇₃	—	2	2	no	—
122	C₂²×C₁₇₃	—	8	1	no	—
123	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×D₁₇₃	—	2	2	no	—
125	C₂×D₁₇₃	—	2	2	no	—
126	C₂×D₁₇₃	—	4	4	no	—
127	C₂×D₁₇₃	—	8	2	no	—
128	C₂×D₁₇₃	—	8	1	no	—
129	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
130	C₂×D₁₇₃	C₂²×D₄	1	1	yes	characteristic
131	#10621924	—	2	4	no	—
132	#10621924	—	4	2	no	—
133	#10621924	—	4	2	no	—
134	C₄×C₁₇₃	—	2	2	no	—
135	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
137	Dic₁₇₃	—	2	2	no	—
138	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
139	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₁₇₃	—	2	1	no	—
141	C₂×C₁₇₃	—	4	2	no	—
142	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₇₃	—	2	1	no	—
144	D₁₇₃	—	4	2	no	—
145	D₁₇₃	#577	1	2	yes	—
146	C₁₇₃	#2949	1	1	yes	characteristic, complemented, Sylow
147	#2949	—	173	1	no	maximal, Sylow
148	C₂²×SD₁₆	—	173	1	no	—
149	#519	—	173	1	no	—
150	#518	—	173	1	no	—
151	#440	—	173	4	no	—
152	C₄×SD₁₆	—	173	4	no	—
153	#427	—	173	1	no	—
154	#419	—	173	1	no	—
155	#418	—	173	1	no	—
156	#405	—	173	1	no	—
157	C₂²×Q₈	—	173	1	no	—
158	C₂²×D₄	—	173	1	no	—
159	C₂×SD₁₆	—	173	1	no	—
160	C₂×SD₁₆	—	173	1	no	—
161	C₂×SD₁₆	—	173	2	no	—
162	C₂×SD₁₆	—	173	2	no	—
163	C₂×SD₁₆	—	173	2	no	—
164	C₂×SD₁₆	—	173	2	no	—
165	C₂×SD₁₆	—	173	1	no	—
166	C₂×SD₁₆	#10621924	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	173	1	no	—
168	C₂²×C₈	—	173	1	no	—
169	C₄×Q₈	—	346	2	no	—
170	C₄×Q₈	—	173	4	no	—
171	C₄×D₄	—	346	2	no	—
172	C₄×D₄	—	173	4	no	—
173	#117	—	346	2	no	—
174	#117	—	173	1	no	—
175	#117	—	173	1	no	—
176	#117	—	173	1	no	—
177	C₂×C₂²⋊C₄	—	346	2	no	—
178	#107	—	173	2	no	—
179	#106	—	173	2	no	—
180	#103	—	173	4	no	—
181	#102	—	173	4	no	—
182	#97	—	173	2	no	—
183	C₄×C₈	—	173	2	no	—
184	C₂⁴	—	346	1	no	—
185	C₂×Q₈	—	346	1	no	—
186	C₂×Q₈	—	346	1	no	—
187	C₂×Q₈	—	346	1	no	—
188	C₂×Q₈	—	173	2	no	—
189	C₂×Q₈	—	173	2	no	—
190	C₂×Q₈	—	173	1	no	—
191	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic, complemented
192	C₂×D₄	—	346	1	no	—
193	C₂×D₄	—	346	1	no	—
194	C₂×D₄	—	346	1	no	—
195	C₂×D₄	—	173	2	no	—
196	C₂×D₄	—	173	2	no	—
197	C₂×D₄	—	173	1	no	—
198	C₂×D₄	C₂×#10621924	1	1	yes	characteristic
199	C₂²×C₄	—	346	1	no	—
200	C₂²×C₄	—	346	4	no	—
201	C₂²×C₄	—	173	1	no	—
202	SD₁₆	—	173	2	no	—
203	SD₁₆	—	173	2	no	—
204	SD₁₆	—	173	2	no	—
205	SD₁₆	—	173	2	no	—
206	SD₁₆	—	173	2	no	—
207	SD₁₆	—	173	2	no	—
208	SD₁₆	C₂×#10621924	1	2	yes	complemented
209	SD₁₆	C₂×#10621924	1	2	yes	complemented
210	M₁₆	—	346	1	no	—
211	M₁₆	—	346	1	no	—
212	C₂×C₈	—	173	2	no	—
213	C₂×C₈	—	173	1	no	—
214	C₂×C₈	—	173	1	no	—
215	C₂×C₈	—	173	1	no	—
216	C₂×C₈	—	173	1	no	—
217	C₂×C₈	—	173	1	no	—
218	C₂×C₈	C₂×#10621924	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	346	4	no	—
220	C₄⋊C₄	—	173	2	no	—
221	C₄⋊C₄	—	173	2	no	—
222	C₄⋊C₄	—	173	2	no	—
223	C₂²⋊C₄	—	346	1	no	—
224	C₂²⋊C₄	—	346	1	no	—
225	C₂²⋊C₄	—	346	4	no	—
226	C₂²⋊C₄	—	173	2	no	—
227	C₄²	—	346	4	no	—
228	C₄²	—	346	1	no	—
229	C₄²	—	346	1	no	—
230	C₄²	—	173	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	346	2	no	—
233	C₂³	—	346	2	no	—
234	C₂³	—	346	1	no	—
235	C₂³	—	1384	2	no	—
236	C₂³	—	173	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	346	2	no	—
239	Q₈	—	346	2	no	—
240	Q₈	—	346	1	no	—
241	Q₈	—	173	2	no	—
242	Q₈	C₂²×#10621924	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	346	2	no	—
245	D₄	—	346	2	no	—
246	D₄	—	346	1	no	—
247	D₄	—	173	2	no	—
248	D₄	C₂²×#10621924	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	346	2	no	—
251	C₂×C₄	—	346	2	no	—
252	C₂×C₄	—	346	1	no	—
253	C₂×C₄	—	346	2	no	—
254	C₂×C₄	—	346	2	no	—
255	C₂×C₄	—	1384	4	no	—
256	C₂×C₄	—	173	4	no	—
257	C₂×C₄	—	173	1	no	—
258	C₂×C₄	—	173	1	no	—
259	C₂×C₄	—	173	1	no	—
260	C₂×C₄	—	173	1	no	—
261	C₂×C₄	—	173	1	no	—
262	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
263	C₈	—	346	2	no	—
264	C₈	—	173	1	no	—
265	C₈	—	173	1	no	—
266	C₈	C₂²×#10621924	1	1	yes	characteristic
267	C₈	C₂²×#10621924	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	346	2	no	—
271	C₂²	—	346	2	no	—
272	C₂²	—	1384	1	no	—
273	C₂²	—	1384	2	no	—
274	C₂²	—	173	1	no	—
275	C₂²	—	173	1	no	—
276	C₂²	—	692	4	no	—
277	C₂²	D₄×#10621924	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	346	4	no	—
280	C₄	—	346	2	no	—
281	C₄	—	173	1	no	—
282	C₄	—	173	1	no	—
283	C₄	—	692	2	no	—
284	C₄	—	692	2	no	—
285	C₄	D₄×#10621924	1	1	yes	characteristic
286	C₄	C₂³×#10621924	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	346	1	no	—
290	C₂	—	173	2	no	—
291	C₂	—	692	2	no	—
292	C₂	C₂×D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10621924
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₄	C₂³×#10621924
Fitting	C₂×SD₁₆×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	C₂×D₄×D₁₇₃	C₂²
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	C₂×SD₁₆	#10621924
hypocenter	C₁₇₃	#2949
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #26012559	C₂	semidirect
3. #26012530	C₂	semidirect
4. SD₁₆×#10621924	C₂	semidirect
5. #26012501	C₂	semidirect
6. #26012484	C₂	semidirect
8. #26012454	C₂	semidirect
12. Q₈×#10621924	C₂²	semidirect
20. #19149748	C₂²	semidirect
21. #19149747	C₂²	semidirect
22. #19149742	C₂²	semidirect
24. #19149733	C₂²	semidirect
25. C₈×#10621924	C₂²	semidirect
28. C₂×#14583815	C₄	semidirect
29. C₂×#14583814	C₄	semidirect
30. C₂×#14583813	C₄	semidirect
31. SD₁₆×D₁₇₃	C₄	semidirect
32. SD₁₆×D₁₇₃	C₄	semidirect
36. C₂×SD₁₆×C₁₇₃	C₄	semidirect
55. C₂×Dic₃₄₆	C₂×C₄	semidirect
58. Q₈×D₁₇₃	C₂×C₄	semidirect
68. #14583815	C₂×C₄	semidirect
69. #14583815	C₂×C₄	semidirect
70. #14583814	C₂×C₄	semidirect
71. #14583814	C₂×C₄	semidirect
72. #14583813	C₂×C₄	semidirect
73. #14583813	C₂×C₄	semidirect
74. C₂×#13063436	C₂×C₄	semidirect
75. C₈×D₁₇₃	C₂×C₄	semidirect
76. C₈×D₁₇₃	C₂×C₄	semidirect
79. C₂×Q₈×C₁₇₃	C₂×C₄	semidirect
81. SD₁₆×C₁₇₃	C₂×C₄	semidirect
82. SD₁₆×C₁₇₃	C₂×C₄	semidirect
83. C₂×C₈×C₁₇₃	C₂×C₄	semidirect
146. C₁₇₃	#2949	semidirect
166. C₂×SD₁₆	#10621924	semidirect
191. C₂×Q₈	C₂×#10621924	semidirect
208. SD₁₆	C₂×#10621924	semidirect
209. SD₁₆	C₂×#10621924	semidirect
218. C₂×C₈	C₂×#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012559	2
#26012558	2
#26012530	2
SD₁₆×#10621924	2
#26012501	2
#26012484	2
#26012483	2
#26012454	2
C₂×SD₁₆×D₁₇₃	2
#2949	173