Subgroups — #33239364

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989016	C₂	1	1	yes	maximal, characteristic
2	#22618801	C₂	1	2	yes	maximal, complemented
3	#22611795	C₂	1	2	yes	maximal, complemented
4	#22611792	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611789	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611640	C₂	1	2	yes	maximal, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556610	C₂²	1	2	yes	—
13	#17989066	—	2	2	no	—
14	#17989016	C₄	1	8	yes	complemented
15	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
16	C₂×#14556535	C₂²	1	2	yes	—
17	C₂×#14556530	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
19	#17988687	—	2	2	no	—
20	#17988684	—	2	2	no	—
21	C₄×Dic₂₉₂	C₂²	1	2	yes	—
22	#17988681	C₂²	1	2	yes	—
23	D₈×Dic₇₃	C₂²	1	2	yes	—
24	#107×D₇₃	C₂²	1	2	yes	—
25	#17988585	—	2	2	no	—
26	#17988541	C₂²	1	1	yes	characteristic
27	#17988413	C₂²	1	1	yes	characteristic
28	#17988412	C₂²	1	2	yes	—
29	#17988392	C₂²	1	2	yes	—
30	#17988390	C₂²	1	2	yes	—
31	#17988388	C₂²	1	2	yes	—
32	#17988363	C₂²	1	1	yes	characteristic
33	#17988362	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
35	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₃	C₂²	1	2	yes	—
37	#366×C₇₃	C₂²	1	1	yes	characteristic
38	#357×C₇₃	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×#11924027	—	2	2	no	—
41	#14556610	—	2	4	no	—
42	#14556610	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₂₉₂	C₂³	1	2	yes	—
44	C₂×#11924016	—	2	2	no	—
45	C₂×#11924011	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₄₆	—	2	2	no	—
48	#14556556	—	2	2	no	—
49	D₄×Dic₇₃	—	2	2	no	—
50	C₄×C₂²⋊D₇₃	—	4	2	no	—
51	#14556551	—	4	2	no	—
52	#14556548	D₄	1	1	yes	characteristic
53	#14556547	—	4	2	no	—
54	C₂×#11924007	—	4	2	no	—
55	C₄⋊C₄×D₇₃	—	2	2	no	—
56	Dic₅₈₄	C₂×C₄	1	4	yes	complemented
57	#14556535	C₂×C₄	1	8	yes	complemented
58	#14556530	—	2	2	no	—
59	C₂×#11924003	C₂³	1	1	yes	characteristic
60	C₂×#11924003	C₂³	1	1	yes	characteristic
61	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
62	#14556525	C₂³	1	2	yes	—
63	#14556524	—	2	2	no	—
64	#14556516	—	2	2	no	—
65	#14556514	—	2	2	no	—
66	#14556512	—	2	2	no	—
67	#14556502	D₄	1	1	yes	characteristic
68	#14556501	D₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₇₃	C₂³	1	2	yes	—
71	C₄×D₄×C₇₃	—	2	2	no	—
72	D₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	C₂³	1	2	yes	—
76	#102×C₇₃	—	2	2	no	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	#11924027	—	2	8	no	—
79	#11924027	—	4	4	no	—
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×C₂²⋊D₇₃	—	4	2	no	—
82	C₂²×Dic₇₃	—	4	2	no	—
83	C₂²×Dic₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
86	Dic₂₉₂	—	2	2	no	—
87	Dic₂₉₂	C₂²×C₄	1	4	yes	—
88	#11924016	—	2	8	no	—
89	#11924011	—	2	2	no	—
90	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
91	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
92	C₈×D₇₃	C₂²×C₄	1	2	yes	—
93	C₈×D₇₃	#41	1	2	yes	—
94	C₄⋊Dic₇₃	—	2	2	no	—
95	#11924007	—	4	2	no	—
96	#11924007	—	4	2	no	—
97	#11924007	—	4	2	no	—
98	#11924007	D₈	1	1	yes	characteristic
99	#11924007	D₈	1	1	yes	characteristic
100	C₂²⋊Dic₇₃	—	4	2	no	—
101	#11924005	D₈	1	1	yes	characteristic
102	#11924005	D₈	1	1	yes	characteristic
103	C₄×Dic₇₃	—	4	2	no	—
104	C₄×Dic₇₃	—	4	2	no	—
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	#11924003	—	2	1	no	—
107	#11924003	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₇₃	—	2	2	no	—
109	C₂²×C₄×C₇₃	—	4	2	no	—
110	D₈×C₇₃	—	2	2	no	—
111	D₈×C₇₃	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	—	2	2	no	—
115	C₂²⋊C₄×C₇₃	—	4	2	no	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₃	—	2	1	no	—
118	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
119	C₂³×C₇₃	—	4	2	no	—
120	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
121	D₄×C₇₃	—	2	4	no	—
122	D₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	16	2	no	—
125	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	4	8	no	—
129	C₂×Dic₇₃	—	4	2	no	—
130	C₂×Dic₇₃	—	4	8	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	16	2	no	—
133	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
136	C₄×D₇₃	—	16	2	no	—
137	C₄×D₇₃	C₄×D₄	1	2	yes	—
138	C₄×D₇₃	#135	1	2	yes	—
139	Dic₁₄₆	—	2	4	no	—
140	Dic₁₄₆	—	4	2	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
144	#10596831	—	2	1	no	—
145	#10596831	C₄×D₄	1	2	yes	—
146	C₂²×C₇₃	—	4	4	no	—
147	C₂²×C₇₃	—	8	2	no	—
148	C₂²×C₇₃	—	8	2	no	—
149	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	2	1	no	—
152	C₂×D₇₃	C₄×D₈	1	2	yes	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	—	8	2	no	—
156	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
158	Dic₇₃	—	2	1	no	—
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	4	4	no	—
161	Dic₇₃	—	8	2	no	—
162	Dic₇₃	C₄×D₈	1	2	yes	—
163	#33459	—	73	1	no	maximal, Sylow
164	C₂×C₇₃	—	8	4	no	—
165	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₇₃	#3415	1	1	yes	characteristic
167	C₂×C₇₃	#3417	1	1	yes	characteristic
168	D₇₃	—	2	2	no	—
169	C₂×#508	—	73	1	no	—
170	#2951	—	73	2	no	—
171	#2175	—	73	2	no	—
172	#2175	—	73	2	no	—
173	C₄×Q₃₂	—	73	1	no	—
174	C₄×Q₃₂	—	73	1	no	—
175	C₄×D₁₆	—	73	1	no	—
176	C₄×D₁₆	D₇₃	1	1	yes	characteristic, complemented
177	C₂×#366	—	73	1	no	—
178	#2139	—	73	2	no	—
179	#2108	—	73	1	no	—
180	C₇₃	#33459	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	73	2	no	—
182	#520	—	146	2	no	—
183	#508	—	73	8	no	—
184	C₂×Q₃₂	—	73	1	no	—
185	C₂×SD₃₂	—	73	2	no	—
186	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
187	C₂×M₃₂	—	73	1	no	—
188	C₂²×C₁₆	—	73	1	no	—
189	#440	—	146	2	no	—
190	#440	—	146	2	no	—
191	C₄×Q₁₆	—	73	2	no	—
192	C₄×Q₁₆	—	73	2	no	—
193	C₄×D₈	—	73	2	no	—
194	C₄×D₈	C₂×D₇₃	1	2	yes	—
195	C₂×#107	—	73	2	no	—
196	#417	—	146	2	no	—
197	#405	—	73	1	no	—
198	#366	—	73	1	no	—
199	#366	—	73	2	no	—
200	#366	C₂×D₇₃	1	1	yes	characteristic
201	#358	—	73	2	no	—
202	#358	—	73	2	no	—
203	#357	—	73	2	no	—
204	#357	C₂×D₇₃	1	2	yes	—
205	#346	—	73	1	no	—
206	#346	—	73	1	no	—
207	C₄×C₁₆	—	73	1	no	—
208	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
209	C₂×#41	—	146	2	no	—
210	#135	—	73	8	no	—
211	#135	—	146	4	no	—
212	C₂×Q₁₆	—	73	2	no	—
213	C₂×SD₁₆	—	146	2	no	—
214	C₂×D₈	C₂²×D₇₃	1	2	yes	—
215	C₂×M₁₆	—	73	1	no	—
216	C₂²×C₈	—	73	1	no	—
217	C₄×Q₈	—	146	2	no	—
218	C₄×Q₈	—	146	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	292	2	no	—
221	C₄×D₄	—	292	2	no	—
222	C₄×D₄	—	146	2	no	—
223	#117	—	292	2	no	—
224	#117	—	73	1	no	—
225	C₂×C₄⋊C₄	—	292	2	no	—
226	C₂×C₄⋊C₄	—	146	2	no	—
227	Q₃₂	—	73	4	no	—
228	SD₃₂	—	73	8	no	—
229	D₁₆	C₄×D₇₃	1	4	yes	complemented
230	M₃₂	—	146	2	no	—
231	C₂×C₁₆	—	73	1	no	—
232	C₂×C₁₆	—	73	4	no	—
233	C₂×C₁₆	—	73	1	no	—
234	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	#107	—	73	2	no	—
237	#107	—	146	2	no	—
238	#107	C₂²×D₇₃	1	2	yes	—
239	#103	—	146	2	no	—
240	#103	—	146	2	no	—
241	#102	—	2	2	no	—
242	#102	—	146	2	no	—
243	#97	—	73	1	no	—
244	#97	—	73	1	no	—
245	C₄×C₈	—	73	1	no	—
246	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
247	#41	—	292	4	no	—
248	#41	—	146	8	no	—
249	C₂×Q₈	—	146	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	292	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	292	2	no	—
254	C₂²×C₄	—	292	2	no	—
255	C₂²×C₄	—	292	2	no	—
256	C₂²×C₄	—	73	1	no	—
257	Q₁₆	—	73	4	no	—
258	Q₁₆	—	146	2	no	—
259	SD₁₆	—	146	8	no	—
260	D₈	—	2	2	no	—
261	D₈	C₂×C₄×D₇₃	1	4	yes	—
262	M₁₆	—	146	2	no	—
263	C₂×C₈	—	73	1	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	73	2	no	—
266	C₂×C₈	—	73	1	no	—
267	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	292	2	no	—
271	C₄⋊C₄	—	292	2	no	—
272	C₄⋊C₄	—	292	2	no	—
273	C₄⋊C₄	—	73	1	no	—
274	C₄⋊C₄	—	73	1	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	292	2	no	—
278	C₂²⋊C₄	—	73	1	no	—
279	C₂²⋊C₄	—	73	1	no	—
280	C₄²	—	292	2	no	—
281	C₄²	—	292	2	no	—
282	C₄²	—	73	1	no	—
283	C₄²	D₄×D₇₃	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	73	2	no	—
286	C₁₆	—	146	1	no	—
287	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
288	C₂³	—	4	2	no	—
289	C₂³	—	73	1	no	—
290	Q₈	—	292	2	no	—
291	Q₈	—	146	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	292	8	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	16	2	no	—
297	C₂×C₄	—	292	2	no	—
298	C₂×C₄	—	292	8	no	—
299	C₂×C₄	—	292	2	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	2	no	—
302	C₂×C₄	—	73	2	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	1168	2	no	—
306	C₂×C₄	—	1168	2	no	—
307	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
308	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	73	2	no	—
312	C₈	—	146	1	no	—
313	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
314	C₈	#41×D₇₃	1	1	yes	characteristic
315	C₂²	—	4	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	73	2	no	—
319	C₂²	—	146	1	no	—
320	C₂²	—	146	1	no	—
321	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	292	4	no	—
326	C₄	—	584	2	no	—
327	C₄	—	73	2	no	—
328	C₄	—	146	1	no	—
329	C₄	—	146	1	no	—
330	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
331	C₄	#135×D₇₃	1	1	yes	characteristic
332	C₂	—	8	4	no	—
333	C₂	—	146	2	no	—
334	C₂	C₄×D₈×D₇₃	1	1	yes	characteristic
335	C₂	#22623068	1	1	yes	characteristic
336	C₂	#22623081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×D₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989016	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₄×D₁₆	D₇₃
hypocenter	C₇₃	#33459
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #22618801	C₂	semidirect
3. #22611795	C₂	semidirect
4. #22611792	C₂	semidirect
5. C₄×Dic₅₈₄	C₂	semidirect
6. #22611789	C₂	semidirect
7. D₁₆×Dic₇₃	C₂	semidirect
8. #366×D₇₃	C₂	semidirect
9. #22611640	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₄×D₁₆×C₇₃	C₂	semidirect
14. #17989016	C₄	semidirect
42. #14556610	C₂×C₄	semidirect
56. Dic₅₈₄	C₂×C₄	semidirect
57. #14556535	C₂×C₄	semidirect
61. C₁₆×D₇₃	C₂×C₄	semidirect
72. D₁₆×C₇₃	C₂×C₄	semidirect
176. C₄×D₁₆	D₇₃	semidirect
180. C₇₃	#33459	semidirect
229. D₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989016	2
#22618801	2
#22611795	2
#22611792	2
C₄×Dic₅₈₄	2
#22611789	2
D₁₆×Dic₇₃	2
#366×D₇₃	2
#22611640	2
#22611526	2
C₄×D₁₆×C₇₃	2
#33459	73