Subgroups — #36370335

Order 22144.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₇₃⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19149766	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012538	C₂	1	2	yes	maximal, complemented
4	#26012536	C₂	1	2	yes	maximal, complemented
5	#26012534	C₂	1	2	yes	maximal, complemented
6	#26012533	C₂	1	2	yes	maximal, complemented
7	#26012506	C₂	1	1	yes	maximal, characteristic, complemented
8	#26012489	C₂	1	1	yes	maximal, characteristic, complemented
9	#26012488	C₂	1	1	yes	maximal, characteristic, complemented
10	#26012457	C₂	1	1	yes	maximal, characteristic, complemented
11	#26012287	C₂	1	1	yes	maximal, characteristic
12	#19149767	C₂²	1	1	yes	characteristic, complemented
13	#19149767	C₂²	1	1	yes	characteristic, complemented
14	#19149766	—	2	1	no	—
15	#19149766	—	2	1	no	—
16	#19149766	—	2	1	no	—
17	#19149766	—	2	1	no	—
18	#19149766	C₂²	1	1	yes	characteristic, complemented
19	#19149766	C₂²	1	2	yes	complemented
20	#19149766	C₂²	1	2	yes	complemented
21	#19149766	C₂²	1	1	yes	characteristic, complemented
22	C₂×#14583836	C₂²	1	1	yes	characteristic
23	D₁₇₃⋊M₁₆	—	2	1	no	—
24	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#14583835	—	2	1	no	—
26	C₂×D₁₇₃⋊C₈	—	2	1	no	—
27	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×#13063443	—	2	1	no	—
29	#19149758	C₂²	1	1	yes	characteristic
30	#19149750	C₂²	1	2	yes	complemented
31	#19149749	C₂²	1	2	yes	complemented
32	#19149744	C₂²	1	2	yes	complemented
33	#19149743	C₂²	1	2	yes	complemented
34	#19149743	C₂²	1	2	yes	complemented
35	#19149743	C₂²	1	2	yes	complemented
36	#19149733	C₂²	1	2	yes	complemented
37	C₈×#10621924	C₂²	1	2	yes	complemented
38	C₂×#14583825	C₂²	1	1	yes	characteristic
39	#41×D₁₇₃	C₂²	1	1	yes	characteristic
40	#19149712	C₄	1	1	yes	characteristic, complemented
41	#19149709	C₄	1	1	yes	characteristic, complemented
42	#19149708	C₄	1	2	yes	complemented
43	#19149705	C₄	1	1	yes	characteristic, complemented
44	#19149703	C₄	1	2	yes	complemented
45	#19149700	C₂²	1	2	yes	—
46	#19149698	C₂²	1	2	yes	—
47	M₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₁₇₃	C₄	1	1	yes	characteristic, complemented
49	#14583836	—	2	1	no	—
50	#14583836	—	2	1	no	—
51	#14583836	—	2	1	no	—
52	#14583836	C₂³	1	1	yes	characteristic
53	#14583836	C₂³	1	1	yes	characteristic
54	#14583836	C₂³	1	2	yes	—
55	#14583835	—	2	2	no	—
56	#14583835	—	2	2	no	—
57	#14583835	—	2	1	no	—
58	#14583835	—	2	1	no	—
59	#14583835	—	2	1	no	—
60	D₁₇₃⋊C₈	—	2	1	no	—
61	D₁₇₃⋊C₈	—	2	1	no	—
62	D₁₇₃⋊C₈	—	2	1	no	—
63	D₁₇₃⋊C₈	D₄	1	1	yes	characteristic
64	D₁₇₃⋊C₈	D₄	1	1	yes	characteristic
65	D₁₇₃⋊C₈	C₂³	1	1	yes	characteristic
66	D₁₇₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×#13063443	—	2	2	no	—
68	C₂×#13063443	—	2	2	no	—
69	C₂×#13063443	—	2	1	no	—
70	C₂×#13063443	—	2	1	no	—
71	C₂×#13063443	—	2	1	no	—
72	C₂×#13063443	D₄	1	1	yes	characteristic
73	C₂×#13063443	D₄	1	1	yes	characteristic
74	C₄⋊#10621924	—	2	1	no	—
75	C₂²⋊#10621924	—	2	1	no	—
76	C₄×#10621924	C₂³	1	2	yes	—
77	#14583827	C₂³	1	1	yes	characteristic
78	#14583826	—	2	1	no	—
79	#14583826	C₂×C₄	1	1	yes	characteristic
80	#14583825	—	2	1	no	—
81	#14583825	—	2	1	no	—
82	#14583825	—	2	1	no	—
83	#14583825	C₂×C₄	1	2	yes	—
84	#14583825	C₂³	1	1	yes	characteristic
85	#14583825	C₂³	1	2	yes	—
86	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
87	Q₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
89	D₄×D₁₇₃	—	2	1	no	—
90	D₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₇₃	—	2	1	no	—
92	C₂×C₄×D₁₇₃	—	2	1	no	—
93	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
94	Dic₆₉₂	C₂×C₄	1	2	yes	—
95	#14583816	C₂×C₄	1	2	yes	—
96	#14583815	C₂×C₄	1	2	yes	—
97	#14583814	C₂×C₄	1	2	yes	—
98	#14583814	C₂×C₄	1	2	yes	—
99	#14583811	C₂×C₄	1	2	yes	—
100	#14583810	C₂×C₄	1	1	yes	characteristic
101	#14583809	C₂×C₄	1	2	yes	—
102	C₈×D₁₇₃	C₂³	1	2	yes	—
103	#41×C₁₇₃	C₂×C₄	1	1	yes	characteristic
104	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
105	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
106	D₈×C₁₇₃	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
108	C₂×#10621924	—	2	2	no	—
109	#13063443	—	2	2	no	—
110	#13063443	—	2	1	no	—
111	#13063443	—	2	1	no	—
112	#13063443	C₂×D₄	1	1	yes	characteristic
113	#13063443	C₂×D₄	1	1	yes	characteristic
114	#13063443	C₂×D₄	1	1	yes	characteristic
115	#13063443	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₇₃	—	2	1	no	—
117	C₂²×D₁₇₃	#41	1	1	yes	characteristic
118	Dic₃₄₆	—	2	1	no	—
119	Dic₃₄₆	—	2	1	no	—
120	Dic₃₄₆	C₂²×C₄	1	1	yes	characteristic
121	Dic₃₄₆	C₂²×C₄	1	2	yes	—
122	D₆₉₂	—	2	1	no	—
123	D₆₉₂	C₂²×C₄	1	1	yes	characteristic
124	C₂²⋊D₁₇₃	—	2	1	no	—
125	C₂²⋊D₁₇₃	—	2	2	no	—
126	C₂²⋊D₁₇₃	—	2	1	no	—
127	C₂²⋊D₁₇₃	—	2	2	no	—
128	C₂²⋊D₁₇₃	—	2	1	no	—
129	C₂×Dic₁₇₃	—	2	1	no	—
130	C₂×Dic₁₇₃	—	2	2	no	—
131	C₂×Dic₁₇₃	—	2	1	no	—
132	C₂×Dic₁₇₃	—	2	2	no	—
133	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄×D₁₇₃	—	2	1	no	—
135	C₄×D₁₇₃	—	2	1	no	—
136	C₄×D₁₇₃	—	2	1	no	—
137	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
139	C₄×D₁₇₃	#41	1	1	yes	characteristic
140	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
141	#13063436	C₂²×C₄	1	2	yes	—
142	C₂³×C₁₇₃	—	2	1	no	—
143	Q₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
144	D₄×C₁₇₃	—	2	1	no	—
145	D₄×C₁₇₃	—	2	1	no	—
146	D₄×C₁₇₃	C₂²×C₄	1	2	yes	—
147	D₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₇₃	—	2	1	no	—
149	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
150	C₈×C₁₇₃	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₇₃	—	2	1	no	—
152	C₂²×C₁₇₃	—	2	2	no	—
153	C₂²×C₁₇₃	—	4	2	no	—
154	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₇₃	—	2	1	no	—
156	C₂×D₁₇₃	—	4	1	no	—
157	C₂×D₁₇₃	—	4	2	no	—
158	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₇₃	C₂×#41	1	1	yes	characteristic
160	#10621924	—	4	2	no	—
161	C₄×C₁₇₃	—	2	1	no	—
162	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
164	Dic₁₇₃	—	2	1	no	—
165	Dic₁₇₃	—	2	2	no	—
166	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
167	Dic₁₇₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₇₃	—	2	1	no	—
169	C₂×C₁₇₃	—	4	1	no	—
170	C₂×C₁₇₃	—	4	2	no	—
171	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
172	D₁₇₃	—	2	1	no	—
173	D₁₇₃	—	2	1	no	—
174	D₁₇₃	—	4	1	no	—
175	C₁₇₃	#2955	1	1	yes	characteristic, complemented, Sylow
176	#2955	—	173	1	no	maximal, Sylow
177	#575	—	173	1	no	—
178	C₂×#131	—	173	1	no	—
179	C₂×#131	—	173	1	no	—
180	#444	—	173	2	no	—
181	#444	—	173	2	no	—
182	#443	—	173	2	no	—
183	#443	—	173	2	no	—
184	#430	—	173	1	no	—
185	#421	—	173	1	no	—
186	#421	—	173	1	no	—
187	#405	—	173	1	no	—
188	C₂×#41	—	173	1	no	—
189	C₂×#41	—	173	1	no	—
190	#137	—	173	1	no	—
191	#137	—	173	2	no	—
192	#137	—	173	1	no	—
193	AGL(1,ℤ/8)	—	173	1	no	—
194	AGL(1,ℤ/8)	—	173	2	no	—
195	AGL(1,ℤ/8)	#10621924	1	1	yes	characteristic, complemented
196	#135	—	173	2	no	—
197	#135	—	173	2	no	—
198	#131	—	346	1	no	—
199	#131	—	346	1	no	—
200	#131	—	346	1	no	—
201	#131	—	346	1	no	—
202	#131	—	173	2	no	—
203	#131	—	173	1	no	—
204	#131	—	173	1	no	—
205	#131	—	173	2	no	—
206	#131	—	173	1	no	—
207	#131	—	173	1	no	—
208	C₂×M₁₆	—	346	1	no	—
209	C₂×M₁₆	—	346	1	no	—
210	C₂×M₁₆	—	173	1	no	—
211	C₂×M₁₆	—	173	1	no	—
212	C₂×M₁₆	—	173	1	no	—
213	C₂²×C₈	—	346	1	no	—
214	C₂²×C₈	—	346	1	no	—
215	C₂²×C₈	—	173	1	no	—
216	#117	—	173	1	no	—
217	#108	—	173	2	no	—
218	#108	—	173	2	no	—
219	#104	—	173	2	no	—
220	#104	—	173	2	no	—
221	#104	—	173	2	no	—
222	#104	—	173	2	no	—
223	#97	—	173	2	no	—
224	C₄×C₈	—	173	2	no	—
225	#41	—	346	1	no	—
226	#41	—	346	1	no	—
227	#41	—	346	1	no	—
228	#41	—	346	1	no	—
229	#41	—	173	1	no	—
230	#41	—	173	2	no	—
231	#41	—	173	2	no	—
232	#41	—	173	1	no	—
233	#41	—	173	1	no	—
234	#41	C₂×#10621924	1	1	yes	characteristic
235	C₂×Q₈	—	173	1	no	—
236	C₂×Q₈	—	173	1	no	—
237	C₂×D₄	—	346	1	no	—
238	C₂×D₄	—	346	1	no	—
239	C₂×D₄	—	173	1	no	—
240	C₂×D₄	C₂×#10621924	1	1	yes	characteristic
241	C₂²×C₄	—	346	1	no	—
242	C₂²×C₄	—	346	1	no	—
243	C₂²×C₄	—	173	1	no	—
244	Q₁₆	—	173	2	no	—
245	Q₁₆	—	173	2	no	—
246	SD₁₆	—	173	2	no	—
247	SD₁₆	—	173	2	no	—
248	SD₁₆	—	173	2	no	—
249	SD₁₆	C₂×#10621924	1	2	yes	—
250	D₈	—	173	2	no	—
251	D₈	C₂×#10621924	1	2	yes	—
252	M₁₆	—	346	1	no	—
253	M₁₆	—	346	1	no	—
254	M₁₆	—	346	2	no	—
255	M₁₆	—	346	1	no	—
256	M₁₆	—	346	2	no	—
257	M₁₆	—	346	1	no	—
258	M₁₆	—	346	1	no	—
259	M₁₆	—	346	1	no	—
260	M₁₆	—	173	1	no	—
261	M₁₆	—	173	1	no	—
262	M₁₆	—	173	2	no	—
263	M₁₆	—	173	1	no	—
264	M₁₆	—	173	2	no	—
265	M₁₆	C₂×#10621924	1	1	yes	characteristic
266	C₂×C₈	—	346	1	no	—
267	C₂×C₈	—	346	1	no	—
268	C₂×C₈	—	346	2	no	—
269	C₂×C₈	—	346	1	no	—
270	C₂×C₈	—	346	2	no	—
271	C₂×C₈	—	346	1	no	—
272	C₂×C₈	—	346	1	no	—
273	C₂×C₈	—	346	1	no	—
274	C₂×C₈	—	173	1	no	—
275	C₂×C₈	—	173	1	no	—
276	C₂×C₈	—	173	1	no	—
277	C₂×C₈	—	173	1	no	—
278	C₂×C₈	—	173	1	no	—
279	C₂×C₈	—	173	1	no	—
280	C₂×C₈	—	173	2	no	—
281	C₄⋊C₄	—	346	1	no	—
282	C₂²⋊C₄	—	346	1	no	—
283	C₄²	—	173	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	346	1	no	—
286	C₂³	—	173	1	no	—
287	Q₈	—	346	1	no	—
288	Q₈	—	346	1	no	—
289	Q₈	—	173	1	no	—
290	Q₈	—	173	2	no	—
291	Q₈	C₂²×#10621924	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	346	1	no	—
295	D₄	—	346	1	no	—
296	D₄	—	346	2	no	—
297	D₄	—	346	2	no	—
298	D₄	—	346	1	no	—
299	D₄	—	346	1	no	—
300	D₄	—	173	1	no	—
301	D₄	C₂²×#10621924	1	2	yes	—
302	D₄	C₂²×#10621924	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	346	1	no	—
305	C₂×C₄	—	346	1	no	—
306	C₂×C₄	—	346	1	no	—
307	C₂×C₄	—	346	1	no	—
308	C₂×C₄	—	346	2	no	—
309	C₂×C₄	—	346	2	no	—
310	C₂×C₄	—	346	1	no	—
311	C₂×C₄	—	346	2	no	—
312	C₂×C₄	—	173	1	no	—
313	C₂×C₄	—	173	1	no	—
314	C₂×C₄	—	173	1	no	—
315	C₂×C₄	—	173	1	no	—
316	C₂×C₄	—	173	1	no	—
317	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
318	C₈	—	346	2	no	—
319	C₈	—	346	1	no	—
320	C₈	—	346	1	no	—
321	C₈	—	173	1	no	—
322	C₈	—	173	1	no	—
323	C₈	—	173	1	no	—
324	C₈	—	173	1	no	—
325	C₈	—	173	2	no	—
326	C₈	C₂²×#10621924	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	346	1	no	—
331	C₂²	—	173	1	no	—
332	C₂²	—	173	1	no	—
333	C₂²	—	692	1	no	—
334	C₂²	—	692	2	no	—
335	C₂²	D₄×#10621924	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	346	1	no	—
338	C₄	—	346	2	no	—
339	C₄	—	173	1	no	—
340	C₄	—	173	1	no	—
341	C₄	—	692	2	no	—
342	C₄	D₄×#10621924	1	1	yes	characteristic
343	C₄	C₂³×#10621924	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	346	1	no	—
348	C₂	—	346	1	no	—
349	C₂	—	692	1	no	—
350	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	AGL(1,ℤ/8)×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	#26012287	C₂
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	AGL(1,ℤ/8)	#10621924
hypocenter	C₁₇₃	#2955
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₁₇₃⋊#131	C₂	semidirect
2. C₂×#19149766	C₂	semidirect
3. #26012538	C₂	semidirect
4. #26012536	C₂	semidirect
5. #26012534	C₂	semidirect
6. #26012533	C₂	semidirect
7. #26012506	C₂	semidirect
8. #26012489	C₂	semidirect
9. #26012488	C₂	semidirect
10. #26012457	C₂	semidirect
12. #19149767	C₂²	semidirect
13. #19149767	C₂²	semidirect
18. #19149766	C₂²	semidirect
19. #19149766	C₂²	semidirect
20. #19149766	C₂²	semidirect
21. #19149766	C₂²	semidirect
30. #19149750	C₂²	semidirect
31. #19149749	C₂²	semidirect
32. #19149744	C₂²	semidirect
33. #19149743	C₂²	semidirect
34. #19149743	C₂²	semidirect
35. #19149743	C₂²	semidirect
36. #19149733	C₂²	semidirect
37. C₈×#10621924	C₂²	semidirect
40. #19149712	C₄	semidirect
41. #19149709	C₄	semidirect
42. #19149708	C₄	semidirect
43. #19149705	C₄	semidirect
44. #19149703	C₄	semidirect
48. AGL(1,ℤ/8)×C₁₇₃	C₄	semidirect
175. C₁₇₃	#2955	semidirect
195. AGL(1,ℤ/8)	#10621924	semidirect

Maximal subgroups

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

subgroup	index
D₁₇₃⋊#131	2
C₂×#19149766	2
#26012538	2
#26012536	2
#26012534	2
#26012533	2
#26012506	2
#26012489	2
#26012488	2
#26012457	2
#26012287	2
#2955	173