Subgroups — #33090719

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610868	C₂	1	2	yes	maximal
2	#22610650	C₂	1	2	yes	maximal
3	#22610497	C₂	1	4	yes	maximal
4	#22609784	C₂	1	2	yes	maximal
5	#22609784	C₂	1	1	yes	maximal, characteristic
6	#22609680	C₂	1	2	yes	maximal
7	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	C₂×#14556502	C₂²	1	2	yes	—
12	C₂×#14556502	C₂²	1	2	yes	—
13	C₂×#14556502	C₂²	1	1	yes	characteristic
14	C₂×#14556502	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10596831	C₂²	1	2	yes	—
16	#17988342	C₂²	1	4	yes	—
17	#17988342	C₂²	1	4	yes	—
18	#17988342	C₂²	1	2	yes	—
19	#17988342	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
29	C₄³×C₇₃	C₂²	1	2	yes	—
30	C₄³×C₇₃	C₂²	1	1	yes	characteristic
31	C₂²×#10596831	C₂³	1	4	yes	—
32	C₂²×#10596831	C₂³	1	2	yes	—
33	C₂²×#10596831	C₂³	1	2	yes	—
34	C₄⋊#10596831	—	2	8	no	—
35	C₄⋊#10596831	—	2	4	no	—
36	#14556502	—	2	4	no	—
37	#14556502	—	2	2	no	—
38	#14556502	—	2	4	no	—
39	#14556502	Q₈	1	4	yes	complemented
40	C₄×#10596831	—	2	4	no	—
41	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₇₃	—	2	4	no	—
43	C₄×Q₈×C₇₃	—	2	4	no	—
44	C₄×Q₈×C₇₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
64	#95×C₇₃	C₂×C₄	1	2	yes	—
65	#95×C₇₃	C₂×C₄	1	2	yes	—
66	#95×C₇₃	C₂×C₄	1	2	yes	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	—	2	2	no	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	—	2	4	no	—
73	C₂×#10596831	—	2	4	no	—
74	C₂×#10596831	C₂×Q₈	1	4	yes	—
75	C₂×#10596831	#41	1	4	yes	—
76	C₂×Q₈×C₇₃	—	2	2	no	—
77	C₂×Q₈×C₇₃	—	2	2	no	—
78	C₂×Q₈×C₇₃	M₁₆	1	4	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂²×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	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₇₃	—	2	4	no	—
91	C₄⋊C₄×C₇₃	—	2	2	no	—
92	C₄⋊C₄×C₇₃	—	2	4	no	—
93	C₄⋊C₄×C₇₃	—	2	4	no	—
94	C₄⋊C₄×C₇₃	—	2	2	no	—
95	C₄⋊C₄×C₇₃	M₁₆	1	4	yes	—
96	C₄²×C₇₃	—	2	4	no	—
97	C₄²×C₇₃	—	2	4	no	—
98	C₄²×C₇₃	—	2	2	no	—
99	C₄²×C₇₃	—	2	2	no	—
100	C₄²×C₇₃	—	2	4	no	—
101	C₄²×C₇₃	—	2	2	no	—
102	C₄²×C₇₃	—	2	4	no	—
103	C₄²×C₇₃	—	4	1	no	—
104	C₄²×C₇₃	—	4	2	no	—
105	C₄²×C₇₃	—	4	1	no	—
106	C₄²×C₇₃	—	4	2	no	—
107	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
108	C₄²×C₇₃	#41	1	4	yes	—
109	C₄²×C₇₃	#41	1	2	yes	—
110	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₇₃	—	2	8	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	1	no	—
124	C₂×C₄×C₇₃	—	2	1	no	—
125	C₂×C₄×C₇₃	—	2	2	no	—
126	C₂×C₄×C₇₃	—	2	1	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	1	no	—
129	C₂×C₄×C₇₃	—	2	2	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	—	2	1	no	—
132	C₂×C₄×C₇₃	—	2	1	no	—
133	C₂×C₄×C₇₃	—	2	2	no	—
134	C₂×C₄×C₇₃	#117	1	2	yes	—
135	C₂×C₄×C₇₃	#117	1	2	yes	—
136	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
139	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
146	#10596831	—	2	4	no	—
147	#10596831	—	2	4	no	—
148	#10596831	—	4	4	no	—
149	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
150	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
152	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
153	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₇₃	#568	1	1	yes	characteristic
155	C₂²×C₇₃	#568	1	1	yes	characteristic
156	C₄×C₇₃	—	2	4	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	—	2	4	no	—
161	C₄×C₇₃	—	4	2	no	—
162	C₄×C₇₃	—	4	1	no	—
163	C₄×C₇₃	—	4	1	no	—
164	C₄×C₇₃	—	4	2	no	—
165	C₄×C₇₃	#432	1	4	yes	—
166	C₄×C₇₃	#529	1	2	yes	—
167	C₄×C₇₃	#533	1	2	yes	—
168	#21345	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	#2295	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#432	1	1	yes	characteristic
171	C₂×C₇₃	#2923	1	1	yes	characteristic
172	C₂×C₇₃	Q₈×M₁₆	1	1	yes	characteristic
173	C₂×C₇₃	Q₈×M₁₆	1	1	yes	characteristic
174	C₂×C₇₃	#2970	1	1	yes	characteristic
175	C₂×C₇₃	#2971	1	1	yes	characteristic
176	C₄²×Q₈	D₇₃	1	1	yes	characteristic
177	#1984	—	73	2	no	—
178	#1939	—	73	2	no	—
179	#1917	—	73	4	no	—
180	#1767	—	73	1	no	—
181	#1767	—	73	2	no	—
182	#1745	—	73	2	no	—
183	C₄×#97	—	73	1	no	—
184	C₇₃	#21345	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	Dic₇₃	1	1	yes	characteristic
186	C₂×C₄×Q₈	Dic₇₃	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
188	C₂×C₄⋊C₈	—	73	4	no	—
189	C₂×C₄⋊C₈	—	73	2	no	—
190	C₂×#97	—	73	1	no	—
191	C₂×#97	—	73	2	no	—
192	C₂×#97	—	73	1	no	—
193	C₂×#97	—	73	2	no	—
194	C₂×C₄×C₈	—	73	2	no	—
195	C₄×C₄⋊C₄	Dic₇₃	1	1	yes	characteristic
196	C₄×C₄⋊C₄	Dic₇₃	1	2	yes	—
197	C₄×C₄⋊C₄	Dic₇₃	1	2	yes	—
198	C₄×C₄⋊C₄	Dic₇₃	1	1	yes	characteristic
199	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
200	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
201	C₄³	C₂×D₇₃	1	1	yes	characteristic
202	C₄³	C₂×D₇₃	1	2	yes	—
203	#336	—	73	2	no	—
204	#336	—	73	4	no	—
205	#336	—	73	4	no	—
206	#336	—	73	4	no	—
207	C₂²×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	73	2	no	—
210	C₂²×C₈	—	73	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	4	no	—
214	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
221	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
224	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
225	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
226	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
228	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
229	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
231	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
233	C₄⋊C₈	—	146	8	no	—
234	C₄⋊C₈	—	146	4	no	—
235	#97	—	73	4	no	—
236	#97	—	146	4	no	—
237	#97	—	146	4	no	—
238	#97	—	146	2	no	—
239	C₄×C₈	—	146	4	no	—
240	#95	C₂×Dic₇₃	1	2	yes	—
241	#95	C₂×Dic₇₃	1	2	yes	—
242	#95	C₂×Dic₇₃	1	2	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	#11924012	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
257	C₂×C₈	—	73	4	no	—
258	C₂×C₈	—	73	4	no	—
259	C₂×C₈	—	146	4	no	—
260	C₂×C₈	—	146	4	no	—
261	C₂×C₈	—	146	4	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	—	146	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	#11924012	1	4	yes	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	2	no	—
283	C₄²	Q₈×D₇₃	1	2	yes	—
284	C₄²	#11924028	1	4	yes	—
285	C₄²	#11924029	1	2	yes	—
286	C₂³	C₂³×Dic₇₃	1	1	yes	characteristic
287	Q₈	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	#14556550	1	2	yes	—
311	C₂×C₄	#14556550	1	2	yes	—
312	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
313	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
314	C₂×C₄	C₂×#11924012	1	4	yes	—
315	C₂×C₄	C₂×#11924012	1	4	yes	—
316	C₂×C₄	C₂×#11924012	1	2	yes	—
317	C₂×C₄	C₂×#11924012	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
319	C₂×C₄	#41×D₇₃	1	2	yes	—
320	C₂×C₄	C₂×#11924028	1	2	yes	—
321	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
322	C₈	—	292	4	no	—
323	C₈	—	146	4	no	—
324	C₈	—	146	4	no	—
325	C₂²	C₂×#14556550	1	1	yes	characteristic
326	C₂²	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
327	C₂²	#41×Dic₇₃	1	1	yes	characteristic
328	C₂²	C₂²×#11924012	1	1	yes	characteristic
329	C₂²	C₂²×#11924012	1	1	yes	characteristic
330	C₂²	#17989354	1	1	yes	characteristic
331	C₂²	#17989354	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	#17988644	1	4	yes	—
342	C₄	#17989124	1	2	yes	—
343	C₄	#17989141	1	2	yes	—
344	C₂	#22612409	1	1	yes	characteristic
345	C₂	C₂×#17988644	1	1	yes	characteristic
346	C₂	#22618582	1	1	yes	characteristic
347	C₂	Q₈×#11924012	1	1	yes	characteristic
348	C₂	Q₈×#11924012	1	1	yes	characteristic
349	C₂	#22618945	1	1	yes	characteristic
350	C₂	#22618951	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂³×C₄
Fitting	C₄²×Q₈×C₇₃	C₂
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄²×Q₈	D₇₃
hypocenter	C₇₃	#21345
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
39. #14556502	Q₈	semidirect
184. C₇₃	#21345	semidirect

Maximal subgroups

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

subgroup	index
#22610868	2
#22610650	2
#22610497	2
#22609784	2
#22609680	2
C₄×#14556502	2
C₄²×Q₈×C₇₃	2
#21345	73