Subgroups — #34030282

Order 18688.

642 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	#22626708	C₂	1	2	yes	maximal, complemented
2	#22626333	C₂	1	4	yes	maximal, complemented
3	#22626332	C₂	1	2	yes	maximal, complemented
4	#22626330	C₂	1	2	yes	maximal, complemented
5	C₂×#17989515	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626245	C₂	1	2	yes	maximal, complemented
7	#22626139	C₂	1	1	yes	maximal, characteristic, complemented
8	#508×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989666	—	2	2	no	—
10	#17989643	C₂²	1	1	yes	characteristic
11	#17989632	—	2	4	no	—
12	#17989631	C₂²	1	4	yes	complemented
13	#17989629	C₂²	1	4	yes	complemented
14	C₂×#14556662	C₂²	1	2	yes	—
15	#17989584	—	2	2	no	—
16	#17989550	C₂²	1	1	yes	characteristic
17	#17989515	C₂²	1	2	yes	complemented
18	#17989515	C₂²	1	2	yes	complemented
19	#17989502	C₂²	1	4	yes	complemented
20	#17989501	C₂²	1	4	yes	complemented
21	#17989474	C₂²	1	2	yes	complemented
22	C₁₆×#64408	C₂²	1	2	yes	complemented
23	#135×D₇₃	C₂²	1	2	yes	—
24	#17989020	C₂²	1	1	yes	characteristic
25	#17989019	C₄	1	1	yes	characteristic, complemented
26	#17989018	C₂²	1	2	yes	—
27	#17989017	C₄	1	2	yes	complemented
28	#17989016	C₂²	1	1	yes	characteristic
29	Q₃₂×D₇₃	C₄	1	1	yes	characteristic, complemented
30	SD₃₂×D₇₃	C₄	1	2	yes	complemented
31	D₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
32	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
33	#508×C₇₃	C₄	1	1	yes	characteristic, complemented
34	#14556679	—	2	4	no	—
35	#14556678	—	2	4	no	—
36	#14556678	—	4	4	no	—
37	C₂×#11924038	—	2	2	no	—
38	C₂×#11924038	—	4	2	no	—
39	D₇₃⋊M₁₆	—	4	2	no	—
40	D₇₃⋊M₁₆	D₄	1	1	yes	characteristic
41	#14556670	D₄	1	1	yes	characteristic
42	#14556666	—	2	1	no	—
43	#14556665	—	2	1	no	—
44	#14556664	C₂³	1	2	yes	—
45	#14556662	—	2	2	no	—
46	#14556662	C₂³	1	4	yes	—
47	#14556656	—	2	4	no	—
48	#14556655	—	2	4	no	—
49	#14556645	D₄	1	2	yes	—
50	C₈×#64408	C₂³	1	2	yes	—
51	#41×D₇₃	—	2	2	no	—
52	#14556614	C₂³	1	2	yes	—
53	#14556613	C₂×C₄	1	2	yes	—
54	#14556612	—	2	2	no	—
55	#14556611	—	2	2	no	—
56	#14556610	C₂³	1	2	yes	—
57	Q₁₆×D₇₃	C₂×C₄	1	2	yes	—
58	SD₁₆×D₇₃	—	2	2	no	—
59	D₈×D₇₃	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
61	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic
62	#14556537	C₂×C₄	1	2	yes	—
63	#14556536	C₂×C₄	1	2	yes	—
64	#14556535	C₂×C₄	1	2	yes	—
65	#14556534	C₂×C₄	1	2	yes	—
66	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic
67	#14556532	C₂×C₄	1	2	yes	—
68	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
69	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
70	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
71	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
72	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
73	#135×C₇₃	C₂×C₄	1	2	yes	—
74	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
75	SD₃₂×C₇₃	C₂×C₄	1	2	yes	—
76	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
78	#11924038	—	2	4	no	—
79	#11924038	—	4	2	no	—
80	#11924038	—	4	4	no	—
81	#11924038	D₈	1	1	yes	characteristic
82	#11924038	D₈	1	1	yes	characteristic
83	#11924037	—	4	4	no	—
84	#11924037	D₈	1	2	yes	—
85	#11924036	—	4	4	no	—
86	#11924036	C₂×D₄	1	2	yes	—
87	C₂×#10596833	—	4	2	no	—
88	C₂×#10596833	—	4	4	no	—
89	C₄⋊#64408	—	2	1	no	—
90	C₂²⋊#64408	—	2	1	no	—
91	C₄×#64408	C₂×D₄	1	2	yes	—
92	#11924031	—	2	2	no	—
93	#11924029	—	2	2	no	—
94	#11924028	—	2	2	no	—
95	#11924028	—	4	2	no	—
96	#11924027	—	2	2	no	—
97	#11924027	—	4	2	no	—
98	Q₈×D₇₃	—	2	2	no	—
99	D₄×D₇₃	—	2	2	no	—
100	D₄×D₇₃	—	4	2	no	—
101	C₂×C₄×D₇₃	—	4	2	no	—
102	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
103	Dic₂₉₂	C₂²×C₄	1	2	yes	—
104	#11924018	—	2	2	no	—
105	#11924017	—	2	2	no	—
106	#11924016	—	2	2	no	—
107	#11924015	—	2	2	no	—
108	D₅₈₄	C₂²×C₄	1	2	yes	—
109	#11924013	—	2	2	no	—
110	C₂×#10596831	#41	1	1	yes	characteristic
111	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
113	C₈×D₇₃	#41	1	1	yes	characteristic
114	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
115	#11924003	C₂²×C₄	1	1	yes	characteristic
116	#11924003	C₂²×C₄	1	1	yes	characteristic
117	#41×C₇₃	—	2	2	no	—
118	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
119	SD₁₆×C₇₃	—	2	2	no	—
120	D₈×C₇₃	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×D₇₃	—	4	2	no	—
125	C₂²×D₇₃	#135	1	1	yes	characteristic
126	C₂×#64408	—	2	2	no	—
127	Q₈×C₇₃	—	2	2	no	—
128	D₄×C₇₃	—	2	2	no	—
129	D₄×C₇₃	—	4	2	no	—
130	C₂×C₄×C₇₃	—	4	2	no	—
131	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
132	C₂²⋊D₇₃	—	4	2	no	—
133	C₂²⋊D₇₃	—	4	2	no	—
134	C₂²⋊D₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
137	D₂₉₂	—	2	2	no	—
138	D₂₉₂	—	4	2	no	—
139	C₄×D₇₃	—	4	2	no	—
140	C₄×D₇₃	—	4	2	no	—
141	C₄×D₇₃	—	4	2	no	—
142	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
143	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
144	C₄×D₇₃	#135	1	1	yes	characteristic
145	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
146	Dic₁₄₆	—	2	2	no	—
147	Dic₁₄₆	—	4	2	no	—
148	#10596833	—	4	4	no	—
149	#10596833	C₂×D₈	1	2	yes	—
150	#10596833	C₂×D₈	1	2	yes	—
151	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
153	#10596831	C₄×D₄	1	1	yes	characteristic
154	#10596831	C₂×#41	1	1	yes	characteristic
155	C₂²×C₇₃	—	4	2	no	—
156	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
157	C₂×D₇₃	—	4	1	no	—
158	C₂×D₇₃	—	4	2	no	—
159	C₂×D₇₃	—	16	2	no	—
160	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
161	C₂×D₇₃	C₂×#135	1	1	yes	characteristic
162	#64408	—	4	2	no	—
163	C₄×C₇₃	—	4	2	no	—
164	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
165	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
166	Dic₇₃	—	4	2	no	—
167	Dic₇₃	C₄×D₈	1	1	yes	characteristic
168	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
169	#33465	—	73	1	no	maximal, Sylow
170	C₂×C₇₃	—	2	1	no	—
171	C₂×C₇₃	—	8	2	no	—
172	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
173	D₇₃	—	2	1	no	—
174	D₇₃	—	2	1	no	—
175	D₇₃	—	8	2	no	—
176	C₂×#508	—	73	1	no	—
177	#2957	—	73	2	no	—
178	#2179	—	73	4	no	—
179	#2178	—	73	2	no	—
180	#2178	—	73	2	no	—
181	C₂×#368	—	73	1	no	—
182	#2143	—	73	2	no	—
183	#2108	—	73	1	no	—
184	C₇₃	#33465	1	1	yes	characteristic, complemented, Sylow
185	C₂×#135	—	73	2	no	—
186	#568	—	146	2	no	—
187	#508	—	73	2	no	—
188	#508	—	73	1	no	—
189	#508	—	73	2	no	—
190	#508	—	73	1	no	—
191	#508	—	73	1	no	—
192	#508	#64408	1	1	yes	characteristic, complemented
193	C₂×Q₃₂	—	73	1	no	—
194	C₂×SD₃₂	—	73	2	no	—
195	C₂×D₁₆	—	73	1	no	—
196	C₂×M₃₂	—	73	1	no	—
197	C₂²×C₁₆	—	73	1	no	—
198	#444	—	146	4	no	—
199	#443	—	73	4	no	—
200	#443	—	73	4	no	—
201	C₂×#108	—	73	2	no	—
202	#421	—	146	2	no	—
203	#405	—	73	1	no	—
204	#368	—	73	2	no	—
205	#368	—	73	2	no	—
206	#359	—	73	4	no	—
207	#359	—	73	4	no	—
208	#346	—	73	2	no	—
209	C₄×C₁₆	—	73	2	no	—
210	C₂×#41	—	146	2	no	—
211	#135	—	73	2	no	—
212	#135	—	73	2	no	—
213	#135	—	73	2	no	—
214	#135	—	146	2	no	—
215	#135	—	146	2	no	—
216	#135	C₂×#64408	1	2	yes	—
217	C₂×Q₁₆	—	73	2	no	—
218	C₂×SD₁₆	—	146	2	no	—
219	C₂×D₈	—	73	2	no	—
220	#131	—	292	4	no	—
221	#131	—	146	4	no	—
222	#131	—	146	4	no	—
223	C₂×M₁₆	—	292	2	no	—
224	C₂×M₁₆	—	292	2	no	—
225	C₂×M₁₆	—	73	1	no	—
226	C₂×M₁₆	—	146	2	no	—
227	C₂²×C₈	—	73	1	no	—
228	#117	—	73	1	no	—
229	Q₃₂	—	73	2	no	—
230	Q₃₂	—	73	1	no	—
231	Q₃₂	C₂×#64408	1	1	yes	characteristic
232	SD₃₂	—	73	2	no	—
233	SD₃₂	—	73	2	no	—
234	SD₃₂	—	73	2	no	—
235	SD₃₂	C₂×#64408	1	2	yes	—
236	D₁₆	—	73	2	no	—
237	D₁₆	—	73	1	no	—
238	D₁₆	C₂×#64408	1	1	yes	characteristic
239	M₃₂	—	146	1	no	—
240	M₃₂	—	146	1	no	—
241	C₂×C₁₆	—	73	1	no	—
242	C₂×C₁₆	—	73	1	no	—
243	C₂×C₁₆	—	73	1	no	—
244	C₂×C₁₆	—	73	1	no	—
245	C₂×C₁₆	—	73	1	no	—
246	C₂×C₁₆	—	73	2	no	—
247	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
248	#108	—	73	4	no	—
249	#108	—	146	2	no	—
250	#104	—	146	4	no	—
251	#104	—	146	4	no	—
252	#97	—	73	2	no	—
253	C₄×C₈	—	73	2	no	—
254	#41	—	2	2	no	—
255	#41	—	292	2	no	—
256	#41	—	292	2	no	—
257	#41	—	146	2	no	—
258	#41	—	146	2	no	—
259	#41	—	146	2	no	—
260	C₂×Q₈	—	146	2	no	—
261	C₂×D₄	—	292	2	no	—
262	C₂×D₄	—	146	2	no	—
263	C₂²×C₄	—	292	2	no	—
264	C₂²×C₄	—	73	1	no	—
265	Q₁₆	—	73	2	no	—
266	Q₁₆	—	146	2	no	—
267	Q₁₆	C₂²×#64408	1	2	yes	—
268	SD₁₆	—	2	2	no	—
269	SD₁₆	—	146	2	no	—
270	SD₁₆	—	146	2	no	—
271	SD₁₆	—	146	2	no	—
272	D₈	—	73	2	no	—
273	D₈	—	146	2	no	—
274	D₈	C₂²×#64408	1	2	yes	—
275	M₁₆	—	292	2	no	—
276	M₁₆	—	292	4	no	—
277	M₁₆	—	292	4	no	—
278	M₁₆	—	73	2	no	—
279	M₁₆	—	73	1	no	—
280	M₁₆	—	73	1	no	—
281	M₁₆	—	146	4	no	—
282	C₂×C₈	—	292	2	no	—
283	C₂×C₈	—	292	4	no	—
284	C₂×C₈	—	292	4	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	73	1	no	—
287	C₂×C₈	—	73	1	no	—
288	C₂×C₈	—	73	1	no	—
289	C₂×C₈	—	73	1	no	—
290	C₂×C₈	—	73	1	no	—
291	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
292	C₄⋊C₄	—	146	1	no	—
293	C₂²⋊C₄	—	146	1	no	—
294	C₄²	—	73	2	no	—
295	C₁₆	—	73	1	no	—
296	C₁₆	—	73	1	no	—
297	C₁₆	—	146	2	no	—
298	C₁₆	C₂²×#64408	1	1	yes	characteristic
299	C₁₆	C₂²×#64408	1	1	yes	characteristic
300	C₂³	—	292	2	no	—
301	C₂³	—	73	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	292	2	no	—
304	Q₈	—	146	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	292	2	no	—
308	D₄	—	292	2	no	—
309	D₄	—	292	2	no	—
310	D₄	—	292	2	no	—
311	D₄	—	146	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	292	2	no	—
314	C₂×C₄	—	292	2	no	—
315	C₂×C₄	—	292	2	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	73	1	no	—
318	C₂×C₄	—	73	1	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	D₄×#64408	1	1	yes	characteristic
324	C₈	—	292	4	no	—
325	C₈	—	73	2	no	—
326	C₈	—	73	2	no	—
327	C₈	—	73	1	no	—
328	C₈	—	73	1	no	—
329	C₈	D₄×#64408	1	1	yes	characteristic
330	C₈	C₂³×#64408	1	1	yes	characteristic
331	C₂²	—	4	2	no	—
332	C₂²	—	292	1	no	—
333	C₂²	—	292	2	no	—
334	C₂²	—	73	1	no	—
335	C₂²	—	73	1	no	—
336	C₂²	—	1168	2	no	—
337	C₂²	D₈×#64408	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	292	2	no	—
340	C₄	—	292	2	no	—
341	C₄	—	73	1	no	—
342	C₄	—	73	1	no	—
343	C₄	D₈×#64408	1	1	yes	characteristic
344	C₄	C₂×D₄×#64408	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	8	2	no	—
347	C₂	—	584	2	no	—
348	C₂	—	146	1	no	—
349	C₂	—	146	1	no	—
350	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	#508×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	#508×D₇₃	C₂
cosocle	C₈×D₇₃	C₂⁴
hypercenter	#508	#64408
hypocenter	C₇₃	#33465
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #22626708	C₂	semidirect
2. #22626333	C₂	semidirect
3. #22626332	C₂	semidirect
4. #22626330	C₂	semidirect
5. C₂×#17989515	C₂	semidirect
6. #22626245	C₂	semidirect
7. #22626139	C₂	semidirect
12. #17989631	C₂²	semidirect
13. #17989629	C₂²	semidirect
17. #17989515	C₂²	semidirect
18. #17989515	C₂²	semidirect
19. #17989502	C₂²	semidirect
20. #17989501	C₂²	semidirect
21. #17989474	C₂²	semidirect
22. C₁₆×#64408	C₂²	semidirect
25. #17989019	C₄	semidirect
27. #17989017	C₄	semidirect
29. Q₃₂×D₇₃	C₄	semidirect
30. SD₃₂×D₇₃	C₄	semidirect
31. D₁₆×D₇₃	C₄	semidirect
33. #508×C₇₃	C₄	semidirect
184. C₇₃	#33465	semidirect
192. #508	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626708	2
#22626333	2
#22626332	2
#22626330	2
C₂×#17989515	2
#22626245	2
#22626139	2
#508×D₇₃	2
#33465	73