Subgroups — #33055202

Order 18688.

572 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619048	C₂	1	2	yes	maximal
2	#22608720	C₂	1	2	yes	maximal, complemented
3	#22608716	C₂	1	2	yes	maximal, complemented
4	#22608711	C₂	1	2	yes	maximal, complemented
5	#22608706	C₂	1	2	yes	maximal, complemented
6	#22608698	C₂	1	2	yes	maximal, complemented
7	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608690	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2922	C₂	1	1	yes	maximal, characteristic
10	#17988718	—	2	2	no	—
11	#17988665	—	2	2	no	—
12	#17988659	—	2	2	no	—
13	#17988659	—	2	2	no	—
14	#17988644	C₂²	1	2	yes	—
15	#17988644	C₂²	1	2	yes	—
16	#17988640	C₂²	1	2	yes	—
17	#17988611	C₂²	1	2	yes	—
18	#17988611	C₂²	1	2	yes	—
19	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
20	#17988340	C₂²	1	2	yes	—
21	#17988340	C₂²	1	1	yes	characteristic
22	#17988340	C₂²	1	1	yes	characteristic
23	#17988340	C₂²	1	1	yes	characteristic
24	#17988340	C₂²	1	1	yes	characteristic
25	#17988340	C₂²	1	2	yes	—
26	#17988340	C₂²	1	2	yes	—
27	#17988338	C₂²	1	2	yes	—
28	#17988338	C₂²	1	2	yes	—
29	#17988338	C₂²	1	2	yes	—
30	C₄×#41×C₇₃	C₂²	1	1	yes	characteristic
31	#433×C₇₃	C₄	1	4	yes	—
32	#431×C₇₃	C₂²	1	2	yes	—
33	#431×C₇₃	C₂²	1	2	yes	—
34	#424×C₇₃	C₄	1	2	yes	—
35	#408×C₇₃	C₄	1	2	yes	—
36	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
38	C₂×#11924012	—	2	2	no	—
39	C₂×#11924012	—	2	2	no	—
40	C₂²×#10596831	—	2	2	no	—
41	C₄⋊#10596831	—	2	2	no	—
42	C₄⋊#10596831	—	2	2	no	—
43	C₄⋊#10596831	—	2	2	no	—
44	C₄⋊#10596831	C₂³	1	2	yes	—
45	C₄⋊#10596831	C₂³	1	2	yes	—
46	C₄⋊#10596831	C₂³	1	2	yes	—
47	C₄⋊#10596831	C₂³	1	2	yes	—
48	C₂²⋊#10596831	—	2	2	no	—
49	C₂²⋊#10596831	—	2	2	no	—
50	C₂²⋊#10596831	—	2	2	no	—
51	#14556502	—	2	2	no	—
52	#14556502	—	2	2	no	—
53	C₄×#10596831	—	2	2	no	—
54	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
58	C₄×Q₈×C₇₃	Q₈	1	1	yes	characteristic
59	C₄×D₄×C₇₃	D₄	1	1	yes	characteristic
60	C₄×D₄×C₇₃	D₄	1	2	yes	—
61	C₄×D₄×C₇₃	Q₈	1	2	yes	—
62	C₄×D₄×C₇₃	Q₈	1	1	yes	characteristic
63	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₇₃	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
71	#97×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	2	yes	—
76	#11924012	—	8	2	no	—
77	#11924012	—	8	2	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	2	2	no	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	—	8	2	no	—
83	#41×C₇₃	—	4	2	no	—
84	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×D₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₄×C₇₃	C₄⋊C₄	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
93	M₁₆×C₇₃	—	4	2	no	—
94	C₂×C₈×C₇₃	—	4	2	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₃	C₂×D₄	1	2	yes	—
108	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
110	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
112	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₇₃	—	4	1	no	—
115	D₄×C₇₃	—	4	1	no	—
116	D₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	4	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	4	1	no	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
134	C₈×C₇₃	—	2	2	no	—
135	C₈×C₇₃	—	2	2	no	—
136	C₈×C₇₃	—	2	4	no	—
137	#10596831	—	4	2	no	—
138	#10596831	—	4	2	no	—
139	#10596831	—	4	2	no	—
140	#10596831	—	4	2	no	—
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	2	no	—
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	2	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	2	2	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	#568	1	2	yes	—
156	C₄×C₇₃	#576	1	1	yes	characteristic
157	C₄×C₇₃	#578	1	1	yes	characteristic
158	#17825	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	4	1	no	—
160	C₂×C₇₃	—	4	2	no	—
161	C₂×C₇₃	#2905	1	1	yes	characteristic
162	C₂×C₇₃	#2913	1	1	yes	characteristic
163	C₂×C₇₃	#2916	1	1	yes	characteristic
164	#2989	—	73	2	no	—
165	#2922	D₇₃	1	1	yes	characteristic
166	#1574	—	73	2	no	—
167	#1573	—	73	2	no	—
168	#1572	—	73	2	no	—
169	#1571	—	73	2	no	—
170	#1569	—	73	2	no	—
171	#1568	—	73	1	no	—
172	#1567	—	73	1	no	—
173	C₇₃	#17825	1	1	yes	characteristic, complemented, Sylow
174	C₄×#41	C₂×D₇₃	1	1	yes	characteristic
175	#446	—	146	2	no	—
176	#436	—	146	2	no	—
177	#435	—	146	2	no	—
178	#435	—	146	2	no	—
179	#433	Dic₇₃	1	4	yes	—
180	#432	—	73	2	no	—
181	#432	—	73	2	no	—
182	#431	—	73	2	no	—
183	#431	C₂×D₇₃	1	2	yes	—
184	#431	C₂×D₇₃	1	2	yes	—
185	#424	Dic₇₃	1	2	yes	—
186	#423	—	73	2	no	—
187	#423	—	73	2	no	—
188	C₂×C₄⋊C₈	—	73	2	no	—
189	#408	Dic₇₃	1	2	yes	—
190	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
192	#335	—	73	2	no	—
193	#335	—	73	1	no	—
194	#335	—	73	1	no	—
195	#335	—	73	1	no	—
196	#335	—	73	1	no	—
197	#335	—	73	2	no	—
198	#335	—	73	2	no	—
199	#334	—	73	2	no	—
200	#334	—	73	2	no	—
201	#334	—	73	2	no	—
202	C₂×#41	C₂×Dic₇₃	1	1	yes	characteristic
203	C₂×M₁₆	—	146	2	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
206	C₂²×C₈	—	146	2	no	—
207	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
208	C₄×Q₈	Dic₁₄₆	1	1	yes	characteristic
209	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
210	C₄×D₄	Dic₁₄₆	1	1	yes	characteristic
211	C₄×D₄	Dic₁₄₆	1	2	yes	—
212	C₄×D₄	D₂₉₂	1	1	yes	characteristic
213	C₄×D₄	D₂₉₂	1	2	yes	—
214	#117	C₂×Dic₇₃	1	1	yes	characteristic
215	#117	C₂×Dic₇₃	1	2	yes	—
216	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
218	C₄⋊C₈	—	73	2	no	—
219	C₄⋊C₈	—	73	2	no	—
220	C₄⋊C₈	—	73	2	no	—
221	C₄⋊C₈	—	73	2	no	—
222	C₄⋊C₈	—	146	2	no	—
223	C₄⋊C₈	—	146	2	no	—
224	C₄⋊C₈	—	146	2	no	—
225	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
226	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
227	C₂²⋊C₈	—	146	2	no	—
228	C₂²⋊C₈	—	146	2	no	—
229	C₂²⋊C₈	—	146	2	no	—
230	C₂²⋊C₈	C₂×Dic₇₃	1	4	yes	—
231	C₂²⋊C₈	C₂×Dic₇₃	1	4	yes	—
232	#97	—	146	2	no	—
233	#97	—	146	2	no	—
234	#97	C₂×Dic₇₃	1	2	yes	—
235	C₄×C₈	—	146	2	no	—
236	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
237	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₇₃	1	2	yes	—
239	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
240	#41	—	4	2	no	—
241	C₂×Q₈	C₄⋊Dic₇₃	1	1	yes	characteristic
242	C₂×D₄	C₄⋊Dic₇₃	1	1	yes	characteristic
243	C₂×D₄	C₄⋊Dic₇₃	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
247	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
248	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	584	2	no	—
252	M₁₆	—	584	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	584	2	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	—	146	2	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
260	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
262	C₄⋊C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
264	C₄⋊C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
265	C₄⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
266	C₂²⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
267	C₂²⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
268	C₂²⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
269	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
270	C₄²	C₂²×Dic₇₃	1	2	yes	—
271	C₄²	C₂×D₂₉₂	1	2	yes	—
272	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
273	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
274	C₄²	C₂×Dic₁₄₆	1	2	yes	—
275	C₄²	C₂³×D₇₃	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊Dic₇₃	1	2	yes	—
277	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
297	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	2	4	no	—
301	C₈	—	292	2	no	—
302	C₈	—	292	2	no	—
303	C₈	—	292	2	no	—
304	C₈	—	292	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	#17989354	1	2	yes	—
320	C₄	#17989392	1	1	yes	characteristic
321	C₄	#17989404	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	4	2	no	—
324	C₂	#22618495	1	1	yes	characteristic
325	C₂	#22618531	1	1	yes	characteristic
326	C₂	#22618542	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#2922	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#2922	D₇₃
hypocenter	C₇₃	#17825
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #22608720	C₂	semidirect
3. #22608716	C₂	semidirect
4. #22608711	C₂	semidirect
5. #22608706	C₂	semidirect
6. #22608698	C₂	semidirect
7. #22608694	C₂	semidirect
8. #22608690	C₂	semidirect
173. C₇₃	#17825	semidirect

Maximal subgroups

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

subgroup	index
#22619048	2
#22608720	2
#22608716	2
#22608711	2
#22608706	2
#22608698	2
#22608694	2
#22608690	2
C₇₃×#2922	2
#17825	73