Subgroups — #32976689

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618887	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556519	C₂	1	1	yes	maximal, characteristic
3	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608311	C₂	1	1	yes	maximal, characteristic
5	#22608212	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607767	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11924028	C₂²	1	1	yes	characteristic
9	#17988711	C₄	1	2	yes	complemented
10	#17988652	C₄	1	2	yes	complemented
11	#17988650	—	2	2	no	—
12	#17988638	—	2	1	no	—
13	C₂×#14556519	C₄	1	2	yes	—
14	C₄×#11924012	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10596831	—	2	2	no	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₇₃	—	2	1	no	—
19	#17988347	C₄	1	2	yes	—
20	#17988317	C₄	1	2	yes	complemented
21	#17988314	C₄	1	2	yes	complemented
22	#17988309	—	2	1	no	—
23	C₈×#10596831	—	2	1	no	—
24	#431×C₇₃	C₂²	1	1	yes	characteristic
25	#431×C₇₃	C₂²	1	1	yes	characteristic
26	C₄³×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924012	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	—	2	1	no	—
30	C₂²×#10596831	D₄	1	2	yes	—
31	C₂×C₈×D₇₃	—	2	2	no	—
32	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
34	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
35	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₇₃	—	2	1	no	—
37	C₄×C₂²⋊D₇₃	—	2	1	no	—
38	#14556550	C₂×C₄	1	1	yes	characteristic
39	#14556548	—	2	1	no	—
40	C₄²×D₇₃	—	2	1	no	—
41	#14556520	—	2	2	no	—
42	#14556519	C₈	1	8	yes	complemented
43	#14556503	—	2	2	no	—
44	#14556502	—	2	1	no	—
45	C₄×#10596831	—	2	8	no	—
46	C₄×#10596831	—	2	1	no	—
47	C₄×#10596831	—	2	1	no	—
48	C₄×#10596831	C₂×C₄	1	2	yes	—
49	C₄×#10596831	C₂×C₄	1	2	yes	—
50	C₄×#10596831	D₄	1	1	yes	characteristic
51	C₄×#10596831	D₄	1	1	yes	characteristic
52	C₈×Dic₇₃	—	2	2	no	—
53	C₂×C₄²×C₇₃	—	2	1	no	—
54	C₂×C₄²×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₇₃	—	2	1	no	—
60	C₂²⋊C₈×C₇₃	—	2	1	no	—
61	C₄×C₈×C₇₃	—	2	1	no	—
62	C₄×C₈×C₇₃	—	2	1	no	—
63	#11924028	—	2	2	no	—
64	#11924028	C₂×C₈	1	4	yes	—
65	C₂×Dic₁₄₆	C₄²	1	1	yes	characteristic
66	C₂×D₂₉₂	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₃	—	2	1	no	—
68	C₂×C₄×D₇₃	—	2	1	no	—
69	C₂×C₄×D₇₃	—	2	1	no	—
70	C₂×C₄×D₇₃	—	2	1	no	—
71	#11924012	—	2	2	no	—
72	#11924012	C₂×C₈	1	4	yes	complemented
73	C₂×#10596831	—	2	4	no	—
74	C₂×#10596831	—	2	2	no	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
78	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
79	C₈×D₇₃	—	4	4	no	—
80	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
81	C₄⋊Dic₇₃	C₄²	1	1	yes	characteristic
82	#11924007	—	2	1	no	—
83	#11924007	—	2	1	no	—
84	C₂²⋊Dic₇₃	—	2	1	no	—
85	#11924005	—	2	1	no	—
86	#11924005	—	2	1	no	—
87	C₄×Dic₇₃	—	2	1	no	—
88	C₄×Dic₇₃	—	2	1	no	—
89	C₄×Dic₇₃	—	2	1	no	—
90	C₂²×C₄×C₇₃	—	2	2	no	—
91	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₄²×C₇₃	—	2	4	no	—
97	C₄²×C₇₃	—	2	4	no	—
98	C₄²×C₇₃	—	2	2	no	—
99	C₄²×C₇₃	C₂×C₈	1	4	yes	complemented
100	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₃	#41	1	1	yes	characteristic
104	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₇₃	—	2	1	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₇₃	#131	1	2	yes	—
119	C₂²⋊D₇₃	—	2	4	no	—
120	C₂×Dic₇₃	—	2	1	no	—
121	C₂×Dic₇₃	—	2	1	no	—
122	C₂×Dic₇₃	—	2	1	no	—
123	D₂₉₂	—	2	1	no	—
124	D₂₉₂	C₄×C₈	1	2	yes	complemented
125	C₄×D₇₃	—	2	4	no	—
126	C₄×D₇₃	—	4	2	no	—
127	C₄×D₇₃	—	4	2	no	—
128	Dic₁₄₆	—	2	1	no	—
129	Dic₁₄₆	C₄×C₈	1	2	yes	complemented
130	C₈×C₇₃	—	4	2	no	—
131	C₈×C₇₃	—	4	2	no	—
132	#10596831	—	2	4	no	—
133	#10596831	—	2	2	no	—
134	#10596831	—	2	4	no	—
135	#10596831	#104	1	4	yes	complemented
136	C₂²×C₇₃	—	2	1	no	—
137	C₂²×C₇₃	—	2	1	no	—
138	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₇₃	C₈×D₄	1	2	yes	—
140	C₂×D₇₃	—	2	2	no	—
141	C₂×D₇₃	—	4	1	no	—
142	C₂×D₇₃	—	4	1	no	—
143	C₄×C₇₃	—	2	8	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₇₃	#405	1	1	yes	characteristic
148	C₄×C₇₃	C₂×#104	1	2	yes	—
149	C₄×C₇₃	C₈×D₄	1	2	yes	—
150	C₄×C₇₃	#443	1	2	yes	—
151	Dic₇₃	—	2	2	no	—
152	Dic₇₃	—	4	1	no	—
153	Dic₇₃	—	4	1	no	—
154	C₈×#104	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₇₃	C₄×#104	1	1	yes	characteristic
158	C₂×C₇₃	#1764	1	1	yes	characteristic
159	D₇₃	—	4	2	no	—
160	C₈×#41	—	73	1	no	—
161	C₄×#104	—	73	1	no	—
162	C₄²×C₈	—	73	1	no	—
163	#1476	—	73	1	no	—
164	C₈×M₁₆	—	73	1	no	—
165	#1278	—	73	1	no	—
166	#1276	D₇₃	1	1	yes	characteristic, complemented
167	C₇₃	C₈×#104	1	1	yes	characteristic, complemented, Sylow
168	C₄×#41	—	73	1	no	—
169	C₈×Q₈	—	73	2	no	—
170	C₈×D₄	—	73	2	no	—
171	C₈×D₄	—	146	2	no	—
172	#431	—	146	1	no	—
173	#431	C₂×D₇₃	1	1	yes	characteristic
174	#431	C₂×D₇₃	1	1	yes	characteristic
175	C₂×#104	—	73	2	no	—
176	C₄×M₁₆	—	73	1	no	—
177	C₂×C₄×C₈	—	73	1	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₂×C₄×C₈	—	146	1	no	—
180	C₂×C₄×C₈	—	146	2	no	—
181	C₄³	C₂×D₇₃	1	1	yes	characteristic
182	#339	—	73	2	no	—
183	Q₈⋊C₈	—	73	2	no	—
184	D₄⋊C₈	—	73	2	no	—
185	#322	—	146	1	no	—
186	C₈²	—	146	1	no	—
187	C₂×#41	—	73	1	no	—
188	C₂×M₁₆	—	73	2	no	—
189	C₂²×C₈	—	73	2	no	—
190	C₂²×C₈	—	146	2	no	—
191	C₂²×C₈	—	146	1	no	—
192	C₄×Q₈	—	73	1	no	—
193	C₄×Q₈	—	73	1	no	—
194	C₄×D₄	—	73	1	no	—
195	C₄×D₄	—	73	1	no	—
196	C₄×D₄	—	146	1	no	—
197	C₄×D₄	—	146	1	no	—
198	#117	—	73	1	no	—
199	#117	—	146	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	146	1	no	—
203	C₂×C₄²	C₄×D₇₃	1	2	yes	—
204	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
205	C₄⋊C₈	—	146	2	no	—
206	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
207	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
208	#104	—	73	8	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	146	2	no	—
212	#97	—	146	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	1	no	—
218	C₄×C₈	—	73	1	no	—
219	C₄×C₈	—	146	8	no	—
220	C₄×C₈	—	146	2	no	—
221	C₄×C₈	—	146	1	no	—
222	C₄×C₈	—	146	1	no	—
223	#41	—	73	4	no	—
224	#41	—	146	2	no	—
225	C₂×Q₈	—	73	1	no	—
226	C₂×D₄	—	73	1	no	—
227	C₂×D₄	—	146	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	146	1	no	—
230	C₂²×C₄	—	146	1	no	—
231	C₂²×C₄	—	146	1	no	—
232	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
234	C₂²×C₄	#11924027	1	1	yes	characteristic
235	M₁₆	—	73	4	no	—
236	M₁₆	—	146	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	292	4	no	—
240	C₂×C₈	—	73	2	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	146	4	no	—
243	C₂×C₈	—	146	4	no	—
244	C₂×C₈	—	146	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₄⋊C₄	—	73	1	no	—
247	C₄⋊C₄	—	73	1	no	—
248	C₄⋊C₄	—	146	1	no	—
249	C₄⋊C₄	—	146	1	no	—
250	C₂²⋊C₄	—	146	1	no	—
251	C₂²⋊C₄	—	146	1	no	—
252	C₂²⋊C₄	—	146	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	146	1	no	—
257	C₄²	—	146	1	no	—
258	C₄²	—	146	1	no	—
259	C₄²	C₈×D₇₃	1	4	yes	complemented
260	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
262	C₄²	D₄×D₇₃	1	1	yes	characteristic
263	C₄²	#11924029	1	1	yes	characteristic
264	C₂³	—	146	1	no	—
265	C₂³	#14556551	1	1	yes	characteristic
266	Q₈	—	73	2	no	—
267	Q₈	—	146	1	no	—
268	D₄	—	73	2	no	—
269	D₄	—	146	4	no	—
270	D₄	—	146	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	292	2	no	—
277	C₂×C₄	—	292	2	no	—
278	C₂×C₄	—	146	1	no	—
279	C₂×C₄	—	146	1	no	—
280	C₂×C₄	—	146	1	no	—
281	C₂×C₄	—	146	4	no	—
282	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
284	C₂×C₄	#14556549	1	1	yes	characteristic
285	C₂×C₄	#14556551	1	1	yes	characteristic
286	C₂×C₄	#14556553	1	1	yes	characteristic
287	C₂×C₄	#14556553	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
289	C₂×C₄	#14556599	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	73	4	no	—
293	C₈	—	146	4	no	—
294	C₈	—	146	4	no	—
295	C₈	—	146	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	292	1	no	—
299	C₂²	—	292	1	no	—
300	C₂²	—	146	2	no	—
301	C₂²	#17988434	1	1	yes	characteristic
302	C₂²	#17988650	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	292	1	no	—
307	C₄	—	292	1	no	—
308	C₄	—	146	2	no	—
309	C₄	C₄×C₈×D₇₃	1	1	yes	characteristic
310	C₄	#17988544	1	1	yes	characteristic
311	C₄	#104×D₇₃	1	2	yes	—
312	C₄	#17988652	1	2	yes	—
313	C₄	#17988701	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	292	2	no	—
316	C₂	#22609701	1	1	yes	characteristic
317	C₂	#22609735	1	1	yes	characteristic
318	C₂	#22609765	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1276	C₂
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1276	D₇₃
hypocenter	C₇₃	C₈×#104
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. C₄²×#10596831	C₂	semidirect
5. #22608212	C₂	semidirect
6. #22607767	C₂	semidirect
7. C₇₃×#1276	C₂	semidirect
9. #17988711	C₄	semidirect
10. #17988652	C₄	semidirect
20. #17988317	C₄	semidirect
21. #17988314	C₄	semidirect
42. #14556519	C₈	semidirect
72. #11924012	C₂×C₈	semidirect
99. C₄²×C₇₃	C₂×C₈	semidirect
124. D₂₉₂	C₄×C₈	semidirect
129. Dic₁₄₆	C₄×C₈	semidirect
135. #10596831	#104	semidirect
166. #1276	D₇₃	semidirect
167. C₇₃	C₈×#104	semidirect
259. C₄²	C₈×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618887	2
C₄×#14556519	2
C₄²×#10596831	2
#22608311	2
#22608212	2
#22607767	2
C₇₃×#1276	2
C₈×#104	73