Subgroups — #36362370

Order 22144.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149707	C₂	1	1	yes	maximal, characteristic
2	#26011959	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19149641	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011602	C₂	1	2	yes	maximal, complemented
5	#26011590	C₂	1	2	yes	maximal, complemented
6	#26011578	C₂	1	2	yes	maximal, complemented
7	#26011573	C₂	1	2	yes	maximal, complemented
8	#26011495	C₂	1	1	yes	maximal, characteristic, complemented
9	#26011490	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583826	C₂²	1	1	yes	characteristic
13	C₂²×D₆₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₇₃	—	2	1	no	—
15	#19149707	C₄	1	8	yes	complemented
16	C₂×#14583815	C₂²	1	2	yes	—
17	C₂×D₁₃₈₄	C₂²	1	2	yes	—
18	C₂×#14583810	C₂²	1	1	yes	characteristic
19	#19149644	C₂²	1	1	yes	characteristic, complemented
20	#19149644	C₂²	1	1	yes	characteristic, complemented
21	#19149641	—	2	1	no	—
22	#19149641	—	2	1	no	—
23	#19149641	C₂²	1	2	yes	—
24	#19149641	C₂²	1	2	yes	—
25	#19149641	C₂²	1	1	yes	characteristic, complemented
26	#19149641	C₂²	1	1	yes	characteristic, complemented
27	#19149639	—	2	1	no	—
28	#19149639	—	2	1	no	—
29	#19149636	—	2	1	no	—
30	C₂×#14583803	—	2	1	no	—
31	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
32	C₄²×D₁₇₃	—	2	1	no	—
33	#19149602	C₂²	1	2	yes	complemented
34	#19149599	C₂²	1	2	yes	—
35	#19149599	C₂²	1	2	yes	—
36	#19149599	C₂²	1	2	yes	complemented
37	#19149589	C₂²	1	2	yes	complemented
38	C₈×Dic₁₇₃	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
40	#117×C₁₇₃	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
42	#107×C₁₇₃	C₂²	1	2	yes	complemented
43	#106×C₁₇₃	C₂²	1	2	yes	complemented
44	C₂³×D₁₇₃	—	2	1	no	—
45	#14583826	—	2	2	no	—
46	#14583826	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₃₄₆	C₂³	1	1	yes	characteristic
48	C₂×D₆₉₂	—	2	1	no	—
49	C₂×D₆₉₂	—	2	2	no	—
50	C₂×D₆₉₂	C₂×C₄	1	4	yes	complemented
51	C₂×D₆₉₂	C₂³	1	2	yes	—
52	C₂×D₆₉₂	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
54	C₂²×Dic₁₇₃	D₄	1	1	yes	characteristic
55	C₂²×Dic₁₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₇₃	—	2	1	no	—
57	C₂×C₄×D₁₇₃	—	2	1	no	—
58	C₂×C₄×D₁₇₃	—	2	2	no	—
59	C₂×C₄×D₁₇₃	—	2	2	no	—
60	C₂×C₄×D₁₇₃	—	2	1	no	—
61	C₂×C₄×D₁₇₃	—	4	4	no	—
62	#14583815	C₂×C₄	1	8	yes	complemented
63	D₁₃₈₄	C₂×C₄	1	8	yes	complemented
64	#14583810	—	2	2	no	—
65	C₂×#13063436	C₂³	1	2	yes	—
66	#14583805	—	2	1	no	—
67	#14583805	—	2	1	no	—
68	#14583803	—	2	2	no	—
69	#14583803	—	2	2	no	—
70	#14583803	—	2	1	no	—
71	#14583803	—	2	1	no	—
72	C₄×Dic₁₇₃	—	2	1	no	—
73	C₄×Dic₁₇₃	—	2	1	no	—
74	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
75	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
76	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₇₃	—	2	1	no	—
79	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₇₃	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₇₃	—	2	1	no	—
83	C₄⋊C₄×C₁₇₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
86	C₂²⋊C₄×C₁₇₃	—	2	1	no	—
87	C₄²×C₁₇₃	—	2	1	no	—
88	C₂²×D₁₇₃	—	2	4	no	—
89	C₂²×D₁₇₃	—	2	1	no	—
90	C₂²×D₁₇₃	—	2	2	no	—
91	C₂²×D₁₇₃	—	4	2	no	—
92	C₂²×D₁₇₃	—	4	2	no	—
93	Dic₃₄₆	—	2	1	no	—
94	Dic₃₄₆	C₂²×C₄	1	2	yes	complemented
95	D₆₉₂	—	2	4	no	—
96	D₆₉₂	—	2	2	no	—
97	D₆₉₂	—	2	1	no	—
98	D₆₉₂	C₂²×C₄	1	4	yes	—
99	D₆₉₂	C₂²×C₄	1	2	yes	complemented
100	C₂²⋊D₁₇₃	—	2	4	no	—
101	C₂×Dic₁₇₃	—	2	1	no	—
102	C₂×Dic₁₇₃	—	2	2	no	—
103	C₂×Dic₁₇₃	—	4	1	no	—
104	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
108	C₄×D₁₇₃	—	2	4	no	—
109	C₄×D₁₇₃	—	4	2	no	—
110	C₄×D₁₇₃	—	4	4	no	—
111	C₄×D₁₇₃	—	4	2	no	—
112	C₄×D₁₇₃	—	4	4	no	—
113	#13063436	—	2	2	no	—
114	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇₃	—	2	2	no	—
116	C₂×C₄×C₁₇₃	—	2	1	no	—
117	C₂×C₄×C₁₇₃	—	2	1	no	—
118	C₂×C₄×C₁₇₃	—	4	2	no	—
119	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇₃	#41	1	2	yes	—
122	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₇₃	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₁₇₃	—	2	1	no	—
125	C₂²×C₁₇₃	—	2	1	no	—
126	C₂²×C₁₇₃	C₄×D₄	1	2	yes	—
127	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₇₃	—	2	2	no	—
129	C₂×D₁₇₃	—	2	4	no	—
130	C₂×D₁₇₃	—	4	8	no	—
131	C₂×D₁₇₃	—	4	2	no	—
132	C₂×D₁₇₃	—	4	1	no	—
133	C₂×D₁₇₃	—	4	1	no	—
134	C₂×D₁₇₃	—	4	2	no	—
135	C₄×C₁₇₃	—	4	2	no	—
136	C₄×C₁₇₃	—	4	1	no	—
137	C₄×C₁₇₃	—	4	1	no	—
138	C₄×C₁₇₃	C₄×D₄	1	2	yes	—
139	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
141	Dic₁₇₃	—	2	2	no	—
142	Dic₁₇₃	—	2	1	no	—
143	Dic₁₇₃	—	2	1	no	—
144	Dic₁₇₃	—	2	1	no	—
145	Dic₁₇₃	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₂×C₁₇₃	—	2	2	no	—
147	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₁₇₃	#575	1	1	yes	characteristic
150	D₁₇₃	—	4	2	no	—
151	D₁₇₃	—	4	4	no	—
152	C₁₇₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	173	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	173	1	no	—
155	C₄×#41	—	173	1	no	—
156	C₂×C₄×D₄	—	173	1	no	—
157	#442	—	173	2	no	—
158	#440	—	173	2	no	—
159	C₄×SD₁₆	—	173	2	no	—
160	C₄×D₈	—	173	2	no	—
161	#428	D₁₇₃	1	1	yes	characteristic, complemented
162	#418	—	173	1	no	—
163	#417	—	173	1	no	—
164	C₄×M₁₆	—	173	1	no	—
165	C₂×#41	—	173	1	no	—
166	C₂²×D₄	—	173	1	no	—
167	C₂³×C₄	—	346	1	no	—
168	AGL(1,ℤ/8)	—	173	8	no	—
169	C₂×SD₁₆	—	173	2	no	—
170	C₂×D₈	—	173	2	no	—
171	C₂×M₁₆	—	173	1	no	—
172	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
173	C₄×Q₈	—	173	1	no	—
174	C₄×Q₈	—	173	1	no	—
175	C₄×D₄	—	346	1	no	—
176	C₄×D₄	—	346	1	no	—
177	C₄×D₄	—	346	1	no	—
178	C₄×D₄	—	346	1	no	—
179	C₄×D₄	—	173	2	no	—
180	C₄×D₄	—	173	2	no	—
181	C₄×D₄	—	173	1	no	—
182	C₄×D₄	—	173	1	no	—
183	#117	—	346	1	no	—
184	#117	C₂×D₁₇₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	346	1	no	—
187	C₂×C₄²	—	346	1	no	—
188	C₂×C₄²	—	173	1	no	—
189	#107	C₂×D₁₇₃	1	2	yes	complemented
190	#106	C₂×D₁₇₃	1	2	yes	complemented
191	#103	—	173	2	no	—
192	#102	—	173	2	no	—
193	#102	—	173	2	no	—
194	#102	—	173	2	no	—
195	#97	—	173	2	no	—
196	C₄×C₈	—	173	2	no	—
197	C₂⁴	—	346	1	no	—
198	#41	—	346	2	no	—
199	#41	—	173	4	no	—
200	C₂×Q₈	—	173	1	no	—
201	C₂×D₄	—	346	1	no	—
202	C₂×D₄	—	346	1	no	—
203	C₂×D₄	—	346	2	no	—
204	C₂×D₄	—	173	4	no	—
205	C₂×D₄	—	173	1	no	—
206	C₂×D₄	—	173	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	346	1	no	—
209	C₂²×C₄	—	346	1	no	—
210	C₂²×C₄	—	346	2	no	—
211	C₂²×C₄	—	346	2	no	—
212	C₂²×C₄	—	346	1	no	—
213	C₂²×C₄	—	173	1	no	—
214	C₂²×C₄	—	173	1	no	—
215	C₂²×C₄	—	692	4	no	—
216	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
217	SD₁₆	—	173	8	no	—
218	D₈	—	173	8	no	—
219	M₁₆	—	346	2	no	—
220	M₁₆	C₄×D₁₇₃	1	4	yes	complemented
221	C₂×C₈	—	173	2	no	—
222	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	346	1	no	—
225	C₄⋊C₄	—	346	1	no	—
226	C₄⋊C₄	C₂²×D₁₇₃	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₇₃	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₁₇₃	1	2	yes	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	346	2	no	—
231	C₂²⋊C₄	—	346	2	no	—
232	C₂²⋊C₄	—	346	1	no	—
233	C₂²⋊C₄	—	346	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	346	1	no	—
236	C₄²	—	346	1	no	—
237	C₄²	—	173	1	no	—
238	C₄²	—	173	1	no	—
239	C₄²	—	173	1	no	—
240	C₄²	—	173	1	no	—
241	C₂³	—	346	4	no	—
242	C₂³	—	346	1	no	—
243	C₂³	—	346	2	no	—
244	C₂³	—	692	2	no	—
245	C₂³	—	692	2	no	—
246	C₂³	D₄×D₁₇₃	1	1	yes	characteristic
247	Q₈	—	346	1	no	—
248	Q₈	—	173	2	no	—
249	D₄	—	346	4	no	—
250	D₄	—	346	4	no	—
251	D₄	—	346	2	no	—
252	D₄	—	346	1	no	—
253	D₄	—	173	4	no	—
254	D₄	—	173	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	346	1	no	—
260	C₂×C₄	—	346	2	no	—
261	C₂×C₄	—	346	4	no	—
262	C₂×C₄	—	173	1	no	—
263	C₂×C₄	—	173	1	no	—
264	C₂×C₄	—	173	1	no	—
265	C₂×C₄	—	173	1	no	—
266	C₂×C₄	—	692	2	no	—
267	C₂×C₄	—	692	4	no	—
268	C₂×C₄	—	692	2	no	—
269	C₂×C₄	—	692	4	no	—
270	C₂×C₄	—	692	1	no	—
271	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
272	C₂×C₄	D₄×D₁₇₃	1	1	yes	characteristic
273	C₂×C₄	#14583827	1	2	yes	—
274	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
275	C₈	—	346	2	no	—
276	C₈	C₂×C₄×D₁₇₃	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	346	2	no	—
280	C₂²	—	346	4	no	—
281	C₂²	—	692	8	no	—
282	C₂²	—	692	2	no	—
283	C₂²	—	692	1	no	—
284	C₂²	—	692	1	no	—
285	C₂²	—	692	2	no	—
286	C₂²	#19149641	1	2	yes	—
287	C₂²	C₂×D₄×D₁₇₃	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	346	2	no	—
292	C₄	—	346	1	no	—
293	C₄	—	346	1	no	—
294	C₄	—	346	1	no	—
295	C₄	—	173	2	no	—
296	C₄	#19149641	1	2	yes	—
297	C₄	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
298	C₄	C₂×#14583827	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	692	2	no	—
301	C₂	—	692	4	no	—
302	C₂	C₂×#19149641	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₁₇₃	1	1	yes	characteristic
304	C₂	#26012290	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇₃
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	#428×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
prime radical	C₂×#19149707	C₂
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	#428	D₁₇₃
hypocenter	C₁₇₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #26011959	C₂	semidirect
3. C₂×#19149641	C₂	semidirect
4. #26011602	C₂	semidirect
5. #26011590	C₂	semidirect
6. #26011578	C₂	semidirect
7. #26011573	C₂	semidirect
8. #26011495	C₂	semidirect
9. #26011490	C₂	semidirect
10. M₁₆×Dic₁₇₃	C₂	semidirect
11. #428×C₁₇₃	C₂	semidirect
15. #19149707	C₄	semidirect
19. #19149644	C₂²	semidirect
20. #19149644	C₂²	semidirect
25. #19149641	C₂²	semidirect
26. #19149641	C₂²	semidirect
33. #19149602	C₂²	semidirect
36. #19149599	C₂²	semidirect
37. #19149589	C₂²	semidirect
38. C₈×Dic₁₇₃	C₂²	semidirect
42. #107×C₁₇₃	C₂²	semidirect
43. #106×C₁₇₃	C₂²	semidirect
46. #14583826	C₂×C₄	semidirect
50. C₂×D₆₉₂	C₂×C₄	semidirect
62. #14583815	C₂×C₄	semidirect
63. D₁₃₈₄	C₂×C₄	semidirect
80. M₁₆×C₁₇₃	C₂×C₄	semidirect
94. Dic₃₄₆	C₂²×C₄	semidirect
99. D₆₉₂	C₂²×C₄	semidirect
123. C₈×C₁₇₃	C₂²×C₄	semidirect
145. Dic₁₇₃	AGL(1,ℤ/8)	semidirect
152. C₁₇₃	C₄×AGL(1,ℤ/8)	semidirect
161. #428	D₁₇₃	semidirect
189. #107	C₂×D₁₇₃	semidirect
190. #106	C₂×D₁₇₃	semidirect
220. M₁₆	C₄×D₁₇₃	semidirect
276. C₈	C₂×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×#19149707	2
#26011959	2
C₂×#19149641	2
#26011602	2
#26011590	2
#26011578	2
#26011573	2
#26011495	2
#26011490	2
M₁₆×Dic₁₇₃	2
#428×C₁₇₃	2
C₄×AGL(1,ℤ/8)	173