Subgroups — #34030268

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626693	C₂	1	2	yes	maximal, complemented
2	#22626329	C₂	1	2	yes	maximal, complemented
3	#22626328	C₂	1	2	yes	maximal, complemented
4	SD₃₂×#64408	C₂	1	4	yes	maximal, complemented
5	C₂×#17989514	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626239	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	#17989656	—	2	2	no	—
10	#17989643	C₂²	1	1	yes	characteristic
11	#17989626	—	2	4	no	—
12	Q₁₆×#64408	C₂²	1	4	yes	—
13	D₈×#64408	C₂²	1	4	yes	—
14	C₂×#14556660	C₂²	1	2	yes	—
15	#17989578	—	2	2	no	—
16	#17989550	C₂²	1	1	yes	characteristic
17	#17989514	C₂²	1	2	yes	—
18	#17989514	C₂²	1	2	yes	—
19	#17989500	C₂²	1	4	yes	—
20	#17989499	C₂²	1	4	yes	—
21	#17989474	C₂²	1	2	yes	—
22	C₁₆×#64408	C₂²	1	2	yes	—
23	#135×D₇₃	C₂²	1	2	yes	—
24	#17989020	C₄	1	1	yes	characteristic, complemented
25	#17989019	C₄	1	1	yes	characteristic, complemented
26	#17989018	C₄	1	2	yes	complemented
27	#17989017	C₄	1	2	yes	complemented
28	#17989016	C₄	1	1	yes	characteristic, complemented
29	Q₃₂×D₇₃	C₂²	1	1	yes	characteristic
30	SD₃₂×D₇₃	C₂²	1	2	yes	—
31	D₁₆×D₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
33	#508×C₇₃	C₄	1	1	yes	characteristic, complemented
34	D₇₃⋊M₁₆	D₄	1	1	yes	characteristic
35	Q₈×#64408	—	2	4	no	—
36	D₄×#64408	—	2	4	no	—
37	D₄×#64408	—	4	4	no	—
38	#14556670	—	4	2	no	—
39	#14556670	D₄	1	1	yes	characteristic
40	C₂×C₄⋊#64408	—	2	2	no	—
41	C₂×C₄⋊#64408	—	4	2	no	—
42	#14556666	—	2	1	no	—
43	#14556665	—	2	1	no	—
44	#14556664	C₂³	1	2	yes	—
45	#14556660	—	2	2	no	—
46	#14556660	C₂³	1	4	yes	—
47	#14556654	—	2	4	no	—
48	#14556653	—	2	4	no	—
49	#14556645	D₄	1	2	yes	—
50	C₈×#64408	C₂³	1	2	yes	—
51	#41×D₇₃	—	2	2	no	—
52	#14556614	C₂×C₄	1	2	yes	complemented
53	#14556613	C₂×C₄	1	2	yes	complemented
54	#14556612	—	2	2	no	—
55	#14556611	—	2	2	no	—
56	#14556610	C₂×C₄	1	2	yes	complemented
57	Q₁₆×D₇₃	C₂³	1	2	yes	—
58	SD₁₆×D₇₃	—	2	2	no	—
59	D₈×D₇₃	C₂³	1	2	yes	—
60	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
61	Dic₅₈₄	C₂×C₄	1	1	yes	characteristic, complemented
62	#14556537	C₂×C₄	1	2	yes	complemented
63	#14556536	C₂×C₄	1	2	yes	complemented
64	#14556535	C₂×C₄	1	2	yes	complemented
65	#14556534	C₂×C₄	1	2	yes	complemented
66	D₁₁₆₈	C₂×C₄	1	1	yes	characteristic, complemented
67	#14556532	C₂×C₄	1	2	yes	complemented
68	C₂×#11924003	C₂×C₄	1	1	yes	characteristic, complemented
69	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
70	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
71	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
72	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
73	#135×C₇₃	C₂×C₄	1	2	yes	complemented
74	Q₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
75	SD₃₂×C₇₃	C₂×C₄	1	2	yes	complemented
76	D₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
77	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂²×#64408	—	4	2	no	—
79	C₂²×#64408	—	4	4	no	—
80	#11924038	—	2	1	no	—
81	#11924037	—	2	1	no	—
82	#11924036	C₂×D₄	1	2	yes	—
83	C₄⋊#64408	—	2	4	no	—
84	C₄⋊#64408	—	4	4	no	—
85	C₄⋊#64408	—	4	2	no	—
86	C₄⋊#64408	D₈	1	2	yes	—
87	C₂²⋊#64408	—	4	4	no	—
88	C₂²⋊#64408	D₈	1	2	yes	—
89	C₄×#64408	—	4	4	no	—
90	C₄×#64408	C₂×D₄	1	2	yes	—
91	#11924031	—	2	2	no	—
92	#11924029	—	2	2	no	—
93	#11924028	—	2	2	no	—
94	#11924028	—	4	2	no	—
95	#11924027	—	2	2	no	—
96	#11924027	—	4	2	no	—
97	Q₈×D₇₃	—	2	2	no	—
98	D₄×D₇₃	—	2	2	no	—
99	D₄×D₇₃	—	4	2	no	—
100	C₂×C₄×D₇₃	—	4	2	no	—
101	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
102	Dic₂₉₂	C₂²×C₄	1	2	yes	—
103	#11924018	—	2	2	no	—
104	#11924017	—	2	2	no	—
105	#11924016	—	2	2	no	—
106	#11924015	—	2	2	no	—
107	D₅₈₄	C₂²×C₄	1	2	yes	—
108	#11924013	—	2	2	no	—
109	C₂×#10596831	#41	1	1	yes	characteristic
110	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
112	C₈×D₇₃	#41	1	1	yes	characteristic
113	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
114	#11924003	C₂²×C₄	1	1	yes	characteristic
115	#11924003	C₂²×C₄	1	1	yes	characteristic
116	#41×C₇₃	—	2	2	no	—
117	Q₁₆×C₇₃	C₂²×C₄	1	2	yes	—
118	SD₁₆×C₇₃	—	2	2	no	—
119	D₈×C₇₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×D₇₃	—	4	2	no	—
124	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×#64408	—	4	4	no	—
126	C₂×#64408	—	16	4	no	—
127	C₂×#64408	—	16	2	no	—
128	C₂×#64408	C₂×D₈	1	4	yes	—
129	Q₈×C₇₃	—	2	2	no	—
130	D₄×C₇₃	—	2	2	no	—
131	D₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
134	C₂²⋊D₇₃	—	4	2	no	—
135	C₂²⋊D₇₃	—	4	2	no	—
136	C₂²⋊D₇₃	—	4	2	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	#135	1	1	yes	characteristic
139	D₂₉₂	—	2	2	no	—
140	D₂₉₂	—	4	2	no	—
141	C₄×D₇₃	—	4	2	no	—
142	C₄×D₇₃	—	4	2	no	—
143	C₄×D₇₃	—	4	2	no	—
144	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
145	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
146	C₄×D₇₃	#135	1	1	yes	characteristic
147	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
148	Dic₁₄₆	—	2	2	no	—
149	Dic₁₄₆	—	4	2	no	—
150	#10596833	—	2	2	no	—
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	—	2	2	no	—
158	C₂×D₇₃	—	4	2	no	—
159	C₂×D₇₃	—	8	2	no	—
160	C₂×D₇₃	—	8	2	no	—
161	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
162	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
163	#64408	—	2	4	no	—
164	#64408	—	8	4	no	—
165	C₄×C₇₃	—	4	2	no	—
166	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
167	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
168	Dic₇₃	—	4	2	no	—
169	Dic₇₃	C₄×D₈	1	1	yes	characteristic
170	Dic₇₃	C₂×#135	1	1	yes	characteristic
171	#33460	—	73	1	no	maximal, Sylow
172	C₂×C₇₃	—	2	1	no	—
173	C₂×C₇₃	—	8	2	no	—
174	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
175	D₇₃	—	2	1	no	—
176	D₇₃	—	8	2	no	—
177	D₇₃	#3416	1	2	yes	—
178	C₂×#508	—	73	1	no	—
179	#2951	—	73	2	no	—
180	#2177	—	73	2	no	—
181	#2176	—	73	2	no	—
182	C₄×SD₃₂	—	73	4	no	—
183	C₂×#367	—	73	1	no	—
184	#2139	—	73	2	no	—
185	#2108	—	73	1	no	—
186	C₇₃	#33460	1	1	yes	characteristic, complemented, Sylow
187	C₂×#135	—	73	2	no	—
188	#520	—	146	2	no	—
189	#508	—	73	2	no	—
190	#508	—	73	1	no	—
191	#508	—	73	2	no	—
192	#508	—	73	1	no	—
193	#508	—	73	1	no	—
194	#508	#64408	1	1	yes	characteristic, complemented
195	C₂×Q₃₂	—	73	1	no	—
196	C₂×SD₃₂	—	73	2	no	—
197	C₂×D₁₆	—	73	1	no	—
198	C₂×M₃₂	—	73	1	no	—
199	C₂²×C₁₆	—	73	1	no	—
200	#440	—	146	4	no	—
201	C₄×Q₁₆	—	73	4	no	—
202	C₄×D₈	—	73	4	no	—
203	C₂×#107	—	73	2	no	—
204	#417	—	146	2	no	—
205	#405	—	73	1	no	—
206	#367	—	73	2	no	—
207	#367	—	73	2	no	—
208	#358	—	73	4	no	—
209	#357	—	73	4	no	—
210	#346	—	73	2	no	—
211	C₄×C₁₆	—	73	2	no	—
212	C₂×#41	—	146	2	no	—
213	#135	—	73	2	no	—
214	#135	—	73	2	no	—
215	#135	—	73	2	no	—
216	#135	—	146	2	no	—
217	#135	—	146	2	no	—
218	#135	C₂×#64408	1	2	yes	complemented
219	C₂×Q₁₆	—	73	2	no	—
220	C₂×SD₁₆	—	146	2	no	—
221	C₂×D₈	—	73	2	no	—
222	C₂×M₁₆	—	73	1	no	—
223	C₂²×C₈	—	73	1	no	—
224	C₄×Q₈	—	146	4	no	—
225	C₄×D₄	—	292	4	no	—
226	C₄×D₄	—	146	4	no	—
227	#117	—	292	2	no	—
228	#117	—	73	1	no	—
229	C₂×C₄⋊C₄	—	292	2	no	—
230	C₂×C₄⋊C₄	—	146	2	no	—
231	Q₃₂	—	73	2	no	—
232	Q₃₂	—	73	1	no	—
233	Q₃₂	C₂×#64408	1	1	yes	characteristic, complemented
234	SD₃₂	—	73	2	no	—
235	SD₃₂	—	73	2	no	—
236	SD₃₂	—	73	2	no	—
237	SD₃₂	C₂×#64408	1	2	yes	complemented
238	D₁₆	—	73	2	no	—
239	D₁₆	—	73	1	no	—
240	D₁₆	C₂×#64408	1	1	yes	characteristic, complemented
241	M₃₂	—	146	1	no	—
242	M₃₂	—	146	1	no	—
243	C₂×C₁₆	—	73	1	no	—
244	C₂×C₁₆	—	73	1	no	—
245	C₂×C₁₆	—	73	1	no	—
246	C₂×C₁₆	—	73	1	no	—
247	C₂×C₁₆	—	73	1	no	—
248	C₂×C₁₆	—	73	2	no	—
249	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic, complemented
250	#107	—	73	4	no	—
251	#107	—	146	2	no	—
252	#103	—	146	4	no	—
253	#102	—	146	4	no	—
254	#97	—	73	2	no	—
255	C₄×C₈	—	73	2	no	—
256	#41	—	2	2	no	—
257	#41	—	292	2	no	—
258	#41	—	292	2	no	—
259	#41	—	146	2	no	—
260	#41	—	146	2	no	—
261	#41	—	146	2	no	—
262	C₂×Q₈	—	146	2	no	—
263	C₂×D₄	—	292	2	no	—
264	C₂×D₄	—	146	2	no	—
265	C₂²×C₄	—	292	2	no	—
266	C₂²×C₄	—	292	2	no	—
267	C₂²×C₄	—	292	4	no	—
268	C₂²×C₄	—	73	1	no	—
269	Q₁₆	—	73	2	no	—
270	Q₁₆	—	146	2	no	—
271	Q₁₆	C₂²×#64408	1	2	yes	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	146	2	no	—
274	SD₁₆	—	146	2	no	—
275	SD₁₆	—	146	2	no	—
276	D₈	—	73	2	no	—
277	D₈	—	146	2	no	—
278	D₈	C₂²×#64408	1	2	yes	—
279	M₁₆	—	146	1	no	—
280	M₁₆	—	146	1	no	—
281	C₂×C₈	—	73	1	no	—
282	C₂×C₈	—	73	1	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	1	no	—
285	C₂×C₈	—	73	1	no	—
286	C₂×C₈	—	73	2	no	—
287	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
288	C₄⋊C₄	—	292	2	no	—
289	C₄⋊C₄	—	292	4	no	—
290	C₄⋊C₄	—	73	2	no	—
291	C₄⋊C₄	—	146	4	no	—
292	C₂²⋊C₄	—	292	4	no	—
293	C₂²⋊C₄	—	73	2	no	—
294	C₄²	—	292	4	no	—
295	C₄²	—	73	2	no	—
296	C₁₆	—	73	1	no	—
297	C₁₆	—	73	1	no	—
298	C₁₆	—	146	2	no	—
299	C₁₆	C₂²×#64408	1	1	yes	characteristic
300	C₁₆	C₂²×#64408	1	1	yes	characteristic
301	C₂³	—	292	2	no	—
302	C₂³	—	73	1	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	292	2	no	—
305	Q₈	—	146	2	no	—
306	D₄	—	2	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	292	2	no	—
309	D₄	—	292	2	no	—
310	D₄	—	292	2	no	—
311	D₄	—	292	2	no	—
312	D₄	—	146	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	292	2	no	—
315	C₂×C₄	—	292	2	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	292	2	no	—
318	C₂×C₄	—	292	4	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	1	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	4	no	—
325	C₂×C₄	—	1168	2	no	—
326	C₂×C₄	—	1168	4	no	—
327	C₂×C₄	D₄×#64408	1	1	yes	characteristic
328	C₈	—	73	1	no	—
329	C₈	—	73	1	no	—
330	C₈	—	146	2	no	—
331	C₈	D₄×#64408	1	1	yes	characteristic
332	C₈	C₂³×#64408	1	1	yes	characteristic
333	C₂²	—	4	2	no	—
334	C₂²	—	292	2	no	—
335	C₂²	—	584	2	no	—
336	C₂²	—	584	2	no	—
337	C₂²	—	73	1	no	—
338	C₂²	—	73	1	no	—
339	C₂²	—	146	2	no	—
340	C₂²	D₈×#64408	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	292	2	no	—
343	C₄	—	584	4	no	—
344	C₄	—	73	1	no	—
345	C₄	—	73	1	no	—
346	C₄	—	146	4	no	—
347	C₄	D₈×#64408	1	1	yes	characteristic
348	C₄	C₂×D₄×#64408	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	8	2	no	—
351	C₂	—	584	2	no	—
352	C₂	—	73	2	no	—
353	C₂	—	146	1	no	—
354	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	#33460
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. #22626693	C₂	semidirect
2. #22626329	C₂	semidirect
3. #22626328	C₂	semidirect
4. SD₃₂×#64408	C₂	semidirect
5. C₂×#17989514	C₂	semidirect
6. #22626239	C₂	semidirect
7. #22626139	C₂	semidirect
24. #17989020	C₄	semidirect
25. #17989019	C₄	semidirect
26. #17989018	C₄	semidirect
27. #17989017	C₄	semidirect
28. #17989016	C₄	semidirect
33. #508×C₇₃	C₄	semidirect
52. #14556614	C₂×C₄	semidirect
53. #14556613	C₂×C₄	semidirect
56. #14556610	C₂×C₄	semidirect
61. Dic₅₈₄	C₂×C₄	semidirect
62. #14556537	C₂×C₄	semidirect
63. #14556536	C₂×C₄	semidirect
64. #14556535	C₂×C₄	semidirect
65. #14556534	C₂×C₄	semidirect
66. D₁₁₆₈	C₂×C₄	semidirect
67. #14556532	C₂×C₄	semidirect
68. C₂×#11924003	C₂×C₄	semidirect
69. C₁₆×D₇₃	C₂×C₄	semidirect
70. C₁₆×D₇₃	C₂×C₄	semidirect
73. #135×C₇₃	C₂×C₄	semidirect
74. Q₃₂×C₇₃	C₂×C₄	semidirect
75. SD₃₂×C₇₃	C₂×C₄	semidirect
76. D₁₆×C₇₃	C₂×C₄	semidirect
77. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
186. C₇₃	#33460	semidirect
194. #508	#64408	semidirect
218. #135	C₂×#64408	semidirect
233. Q₃₂	C₂×#64408	semidirect
237. SD₃₂	C₂×#64408	semidirect
240. D₁₆	C₂×#64408	semidirect
249. C₂×C₁₆	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626693	2
#22626329	2
#22626328	2
SD₃₂×#64408	2
C₂×#17989514	2
#22626239	2
#22626139	2
#508×D₇₃	2
#33460	73