Subgroups — #33086193

Order 18688.

894 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612554	C₂	1	2	yes	maximal
2	#22610065	C₂	1	4	yes	maximal, complemented
3	#22610062	C₂	1	2	yes	maximal, complemented
4	#22610059	C₂	1	2	yes	maximal, complemented
5	#22609651	C₂	1	2	yes	maximal
6	#22609648	C₂	1	1	yes	maximal, characteristic
7	C₂×#17988350	C₂	1	1	yes	maximal, characteristic
8	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556550	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	#17988486	—	2	2	no	—
12	#17988486	—	2	2	no	—
13	#17988462	—	2	4	no	—
14	#17988457	C₂²	1	4	yes	—
15	#17988453	C₂²	1	4	yes	—
16	#17988350	C₂²	1	1	yes	characteristic
17	#17988350	C₂²	1	2	yes	—
18	#17988350	C₂²	1	4	yes	—
19	#17988350	C₂²	1	1	yes	characteristic
20	#17988350	C₂²	1	4	yes	—
21	#17988350	C₂²	1	2	yes	—
22	#17988350	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
26	#405×C₇₃	C₄	1	8	yes	complemented
27	C₂×#97×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
29	C₂³×Dic₇₃	—	2	2	no	—
30	#14556550	—	2	4	no	—
31	#14556550	—	2	4	no	—
32	C₂×C₄⋊Dic₇₃	—	2	2	no	—
33	C₂×C₄⋊Dic₇₃	—	2	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	4	no	—
35	C₂×C₄⋊Dic₇₃	—	2	4	no	—
36	C₂×C₄⋊Dic₇₃	—	2	4	no	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
41	C₂×C₄×Dic₇₃	—	2	2	no	—
42	#14556498	—	2	4	no	—
43	#14556498	—	2	4	no	—
44	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₇₃	—	2	4	no	—
46	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₇₃	—	2	4	no	—
48	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	#117×C₇₃	C₂×C₄	1	4	yes	—
53	#117×C₇₃	Q₈	1	4	yes	—
54	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
57	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
58	#97×C₇₃	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
60	C₂²×Dic₇₃	—	2	2	no	—
61	C₂²×Dic₇₃	—	2	4	no	—
62	C₂²×Dic₇₃	—	2	2	no	—
63	C₂²×Dic₇₃	—	4	4	no	—
64	C₂²×Dic₇₃	—	8	2	no	—
65	C₂²×Dic₇₃	—	8	2	no	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	2	4	no	—
68	C₄⋊Dic₇₃	—	2	2	no	—
69	C₄⋊Dic₇₃	—	2	2	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	—	2	2	no	—
72	C₄⋊Dic₇₃	—	4	4	no	—
73	C₄⋊Dic₇₃	—	8	4	no	—
74	C₄⋊Dic₇₃	—	8	4	no	—
75	C₂²⋊Dic₇₃	—	4	4	no	—
76	C₄×Dic₇₃	—	4	4	no	—
77	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₃	#41	1	4	yes	—
84	C₂²×C₄×C₇₃	#41	1	4	yes	—
85	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₇₃	—	2	8	no	—
87	C₂×C₈×C₇₃	—	2	8	no	—
88	C₂×C₈×C₇₃	—	2	2	no	—
89	C₂×C₈×C₇₃	—	2	2	no	—
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
94	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
95	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
96	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
97	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
99	C₂³×C₇₃	—	2	1	no	—
100	C₂³×C₇₃	—	2	2	no	—
101	C₂³×C₇₃	—	2	1	no	—
102	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₇₃	#124	1	4	yes	—
104	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇₃	#117	1	4	yes	—
112	C₂×C₄×C₇₃	#117	1	4	yes	—
113	C₂×C₄×C₇₃	#124	1	4	yes	—
114	C₂×C₄×C₇₃	#125	1	8	yes	—
115	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
119	C₂×Dic₇₃	—	4	2	no	—
120	C₂×Dic₇₃	—	4	2	no	—
121	C₂×Dic₇₃	—	4	4	no	—
122	C₂×Dic₇₃	—	4	2	no	—
123	C₂×Dic₇₃	—	4	4	no	—
124	C₂×Dic₇₃	—	4	2	no	—
125	C₂×Dic₇₃	—	4	4	no	—
126	C₂×Dic₇₃	—	4	2	no	—
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	8	8	no	—
129	C₈×C₇₃	—	2	2	no	—
130	C₈×C₇₃	—	2	2	no	—
131	C₈×C₇₃	—	2	4	no	—
132	C₂²×C₇₃	—	2	4	no	—
133	C₂²×C₇₃	—	2	4	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	#382	1	4	yes	—
137	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₇₃	C₂×#124	1	2	yes	—
139	C₂²×C₇₃	#576	1	1	yes	characteristic
140	C₂²×C₇₃	#577	1	2	yes	—
141	C₂²×C₇₃	#578	1	1	yes	characteristic
142	C₄×C₇₃	—	2	4	no	—
143	C₄×C₇₃	—	2	4	no	—
144	C₄×C₇₃	#382	1	4	yes	—
145	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
146	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₇₃	C₂×#125	1	2	yes	—
148	Dic₇₃	—	8	2	no	—
149	Dic₇₃	—	8	4	no	—
150	Dic₇₃	—	8	2	no	—
151	#20763	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	—	2	4	no	—
153	C₂×C₇₃	C₂×#382	1	1	yes	characteristic
154	C₂×C₇₃	#2912	1	1	yes	characteristic
155	C₂×C₇₃	#2913	1	1	yes	characteristic
156	C₂×C₇₃	#3136	1	2	yes	—
157	C₂×C₇₃	#3137	1	2	yes	—
158	C₂×#405	D₇₃	1	1	yes	characteristic
159	#2317	—	73	2	no	—
160	#1836	—	73	4	no	—
161	#1835	—	73	2	no	—
162	#1834	—	73	2	no	—
163	#1736	—	73	2	no	—
164	#1735	—	73	1	no	—
165	C₂×#340	—	73	1	no	—
166	C₇₃	#20763	1	1	yes	characteristic, complemented, Sylow
167	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
168	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
169	C₂×#117	—	73	2	no	—
170	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
171	C₂²×C₄⋊C₄	—	73	2	no	—
172	#405	Dic₇₃	1	8	yes	complemented
173	C₂×#97	C₂×D₇₃	1	2	yes	—
174	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
175	#389	—	146	2	no	—
176	#389	—	146	2	no	—
177	#385	—	146	4	no	—
178	#384	—	73	4	no	—
179	#382	—	73	4	no	—
180	#340	—	73	1	no	—
181	#340	—	73	2	no	—
182	#340	—	73	4	no	—
183	#340	—	73	1	no	—
184	#340	—	73	4	no	—
185	#340	—	73	2	no	—
186	#340	—	73	2	no	—
187	C₂³×C₄	—	146	2	no	—
188	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
193	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
196	#117	—	146	4	no	—
197	#117	—	146	4	no	—
198	#117	Dic₁₄₆	1	4	yes	—
199	#117	C₂×Dic₇₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	73	4	no	—
201	C₂×C₄⋊C₄	—	73	2	no	—
202	C₂×C₄⋊C₄	—	73	2	no	—
203	C₂×C₄⋊C₄	—	146	2	no	—
204	C₂×C₄⋊C₄	—	146	4	no	—
205	C₂×C₄⋊C₄	—	146	4	no	—
206	C₂×C₄⋊C₄	—	146	4	no	—
207	C₂×C₄⋊C₄	—	146	4	no	—
208	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	146	2	no	—
211	C₂×C₂²⋊C₄	D₂₉₂	1	2	yes	—
212	C₂×C₄²	—	146	2	no	—
213	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
214	#97	C₂×Dic₇₃	1	8	yes	complemented
215	C₄×C₈	C₂×Dic₇₃	1	8	yes	complemented
216	#95	—	146	4	no	—
217	#95	—	146	4	no	—
218	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
219	C₂²×C₄	—	292	4	no	—
220	C₂²×C₄	—	584	2	no	—
221	C₂²×C₄	—	584	2	no	—
222	C₂²×C₄	—	146	2	no	—
223	C₂²×C₄	—	146	4	no	—
224	C₂²×C₄	—	146	2	no	—
225	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
226	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
227	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
228	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
230	C₂²×C₄	#11924028	1	4	yes	—
231	C₂²×C₄	#11924028	1	4	yes	—
232	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
239	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
240	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
241	C₄⋊C₄	—	292	4	no	—
242	C₄⋊C₄	—	584	4	no	—
243	C₄⋊C₄	—	584	4	no	—
244	C₄⋊C₄	—	146	4	no	—
245	C₄⋊C₄	—	146	4	no	—
246	C₄⋊C₄	—	146	2	no	—
247	C₄⋊C₄	—	146	2	no	—
248	C₄⋊C₄	—	146	2	no	—
249	C₄⋊C₄	—	146	2	no	—
250	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
251	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
252	C₂²⋊C₄	—	292	4	no	—
253	C₂²⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
254	C₄²	—	292	4	no	—
255	C₄²	C₂²×Dic₇₃	1	4	yes	—
256	C₄²	C₂×Dic₁₄₆	1	4	yes	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
261	C₂³	#14556579	1	4	yes	—
262	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	292	2	no	—
267	C₂×C₄	—	292	4	no	—
268	C₂×C₄	—	292	2	no	—
269	C₂×C₄	—	292	4	no	—
270	C₂×C₄	—	292	2	no	—
271	C₂×C₄	—	292	2	no	—
272	C₂×C₄	—	292	2	no	—
273	C₂×C₄	—	292	2	no	—
274	C₂×C₄	—	292	4	no	—
275	C₂×C₄	—	584	8	no	—
276	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
277	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
278	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
279	C₂×C₄	#14556550	1	4	yes	—
280	C₂×C₄	#14556550	1	4	yes	—
281	C₂×C₄	#14556579	1	4	yes	—
282	C₂×C₄	#14556583	1	8	yes	—
283	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#11924028	1	2	yes	—
286	C₂×C₄	C₂×#11924028	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	#17988453	1	4	yes	—
295	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
296	C₂²	C₂×#14556579	1	2	yes	—
297	C₂²	#17989392	1	1	yes	characteristic
298	C₂²	#17989399	1	2	yes	—
299	C₂²	#17989404	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	2	4	no	—
302	C₄	—	584	4	no	—
303	C₄	—	584	2	no	—
304	C₄	—	584	2	no	—
305	C₄	#17988453	1	4	yes	—
306	C₄	C₂×#14556550	1	1	yes	characteristic
307	C₄	C₂×#14556550	1	1	yes	characteristic
308	C₄	C₂×#14556583	1	2	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₂×#17988453	1	1	yes	characteristic
311	C₂	#22618526	1	1	yes	characteristic
312	C₂	#22618531	1	1	yes	characteristic
313	C₂	#22620287	1	2	yes	—
314	C₂	#22620298	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#405×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#20763
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22610065	C₂	semidirect
3. #22610062	C₂	semidirect
4. #22610059	C₂	semidirect
26. #405×C₇₃	C₄	semidirect
58. #97×C₇₃	C₂×C₄	semidirect
59. C₄×C₈×C₇₃	C₂×C₄	semidirect
166. C₇₃	#20763	semidirect
172. #405	Dic₇₃	semidirect
214. #97	C₂×Dic₇₃	semidirect
215. C₄×C₈	C₂×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612554	2
#22610065	2
#22610062	2
#22610059	2
#22609651	2
#22609648	2
C₂×#17988350	2
C₂×#405×C₇₃	2
#20763	73