Subgroups — #32991902

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556526	C₂	1	2	yes	maximal
2	C₂×#17988543	C₂	1	1	yes	maximal, characteristic
3	#22609670	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988345	C₂	1	2	yes	maximal
5	C₇₃×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×#11924012	C₂²	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂×#14556526	—	2	4	no	—
10	C₂×#14556526	—	2	4	no	—
11	C₂×#14556526	C₄	1	8	yes	complemented
12	#17988543	—	2	4	no	—
13	C₂×#14556502	C₄	1	2	yes	—
14	C₂×C₄×#10596831	C₄	1	2	yes	—
15	#17988356	—	2	2	no	—
16	#17988345	—	2	4	no	—
17	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
19	#343×C₇₃	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	—	2	2	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	C₂×C₄	1	4	yes	—
28	C₂×#11924012	D₄	1	2	yes	—
29	C₂×#11924012	Q₈	1	2	yes	—
30	C₂²×#10596831	—	2	2	no	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	4	yes	—
33	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10596831	D₄	1	2	yes	—
36	C₂²×#10596831	D₄	1	4	yes	—
37	C₂²×#10596831	Q₈	1	2	yes	—
38	#14556526	—	2	16	no	—
39	#14556502	—	2	4	no	—
40	C₄×#10596831	—	2	4	no	—
41	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₇₃	—	2	2	no	—
43	C₂×M₁₆×C₇₃	—	2	2	no	—
44	C₂×M₁₆×C₇₃	—	2	2	no	—
45	C₂×M₁₆×C₇₃	—	2	2	no	—
46	C₂×M₁₆×C₇₃	—	2	2	no	—
47	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	#117×C₇₃	—	2	2	no	—
50	#117×C₇₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₇₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
54	#11924012	—	2	4	no	—
55	#11924012	—	2	4	no	—
56	#11924012	—	2	4	no	—
57	#11924012	—	2	4	no	—
58	#11924012	—	4	2	no	—
59	#11924012	—	4	2	no	—
60	C₂×#10596831	—	2	2	no	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	4	2	no	—
64	C₂×#10596831	—	4	2	no	—
65	C₂×#10596831	C₄²	1	2	yes	—
66	C₂×#10596831	C₄²	1	2	yes	—
67	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
68	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
69	C₂×#10596831	C₂²⋊C₄	1	8	yes	—
70	C₂×#10596831	C₄⋊C₄	1	4	yes	—
71	C₂×#10596831	C₄⋊C₄	1	2	yes	—
72	C₂×#10596831	C₄⋊C₄	1	2	yes	—
73	C₂×#10596831	C₄⋊C₄	1	8	yes	—
74	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	#41	1	2	yes	—
85	C₂²×C₄×C₇₃	#41	1	2	yes	—
86	M₁₆×C₇₃	—	2	4	no	—
87	M₁₆×C₇₃	—	2	4	no	—
88	M₁₆×C₇₃	—	4	2	no	—
89	M₁₆×C₇₃	—	4	2	no	—
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	—	4	2	no	—
92	C₂×C₈×C₇₃	—	4	2	no	—
93	C₄⋊C₄×C₇₃	—	2	4	no	—
94	C₂²⋊C₄×C₇₃	—	2	4	no	—
95	C₄²×C₇₃	—	2	2	no	—
96	C₄²×C₇₃	—	2	2	no	—
97	C₂³×C₇₃	—	4	2	no	—
98	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
100	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	4	4	no	—
105	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₇₃	#121	1	2	yes	—
117	C₂×C₄×C₇₃	#121	1	2	yes	—
118	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	#124	1	2	yes	—
120	C₂×C₄×C₇₃	#125	1	2	yes	—
121	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₇₃	—	4	4	no	—
124	#10596831	—	2	4	no	—
125	#10596831	—	4	4	no	—
126	#10596831	#95	1	4	yes	—
127	#10596831	#95	1	4	yes	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	—	4	2	no	—
131	C₂²×C₇₃	—	4	1	no	—
132	C₂²×C₇₃	—	4	1	no	—
133	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₇₃	#380	1	1	yes	characteristic
135	C₂²×C₇₃	#384	1	1	yes	characteristic
136	C₂²×C₇₃	#389	1	2	yes	—
137	C₂²×C₇₃	#391	1	2	yes	—
138	C₄×C₇₃	—	4	4	no	—
139	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
140	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₇₃	#380	1	1	yes	characteristic
142	C₄×C₇₃	#382	1	1	yes	characteristic
143	C₄×C₇₃	#389	1	2	yes	—
144	C₄×C₇₃	#390	1	2	yes	—
145	#10867	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	1	no	—
147	C₂×C₇₃	—	2	1	no	—
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
151	C₂×C₇₃	#1813	1	2	yes	—
152	C₂²×#108	—	73	2	no	—
153	C₂×#405	—	73	1	no	—
154	#1742	—	73	1	no	—
155	#1742	D₇₃	1	1	yes	characteristic
156	C₂×#338	—	73	2	no	—
157	C₇₃	#10867	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	73	1	no	—
159	C₂²×M₁₆	—	73	2	no	—
160	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
161	C₂³×C₈	—	73	1	no	—
162	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
163	C₂×#108	—	73	8	no	—
164	C₂×#108	—	146	4	no	—
165	C₂×#108	—	146	4	no	—
166	#405	—	146	4	no	—
167	C₂×#97	—	73	2	no	—
168	C₂×C₄×C₈	—	73	2	no	—
169	#343	—	2	2	no	—
170	#343	—	146	2	no	—
171	#338	—	146	4	no	—
172	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	73	2	no	—
179	C₂×M₁₆	—	73	2	no	—
180	C₂×M₁₆	—	73	4	no	—
181	C₂×M₁₆	—	146	2	no	—
182	C₂×M₁₆	—	146	2	no	—
183	C₂×M₁₆	—	146	2	no	—
184	C₂×M₁₆	—	146	2	no	—
185	C₂×M₁₆	—	146	2	no	—
186	C₂×M₁₆	—	146	2	no	—
187	C₂×M₁₆	—	146	2	no	—
188	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	73	1	no	—
191	C₂²×C₈	—	73	1	no	—
192	C₂²×C₈	—	73	4	no	—
193	C₂²×C₈	—	73	4	no	—
194	C₂²×C₈	—	73	2	no	—
195	C₂²×C₈	—	73	2	no	—
196	C₂²×C₈	—	73	2	no	—
197	C₂²×C₈	—	146	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	2	yes	—
201	C₂×C₂²⋊C₄	Dic₁₄₆	1	2	yes	—
202	C₂×C₄²	C₄×D₇₃	1	2	yes	—
203	#108	—	146	16	no	—
204	#97	—	146	4	no	—
205	C₄×C₈	—	146	4	no	—
206	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
207	C₂²×C₄	#11924007	1	4	yes	—
208	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
210	C₂²×C₄	D₄×D₇₃	1	2	yes	—
211	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
215	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	#11924027	1	2	yes	—
217	C₂²×C₄	#11924028	1	2	yes	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	292	2	no	—
223	M₁₆	—	292	2	no	—
224	M₁₆	—	146	4	no	—
225	M₁₆	—	146	4	no	—
226	M₁₆	—	146	4	no	—
227	M₁₆	—	146	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	292	2	no	—
232	C₂×C₈	—	292	2	no	—
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	73	2	no	—
235	C₂×C₈	—	73	2	no	—
236	C₂×C₈	—	73	2	no	—
237	C₂×C₈	—	73	8	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	8	no	—
241	C₂×C₈	—	73	4	no	—
242	C₂×C₈	—	146	4	no	—
243	C₂×C₈	—	146	2	no	—
244	C₂×C₈	—	146	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×#11924007	1	1	yes	characteristic
251	C₂³	C₄×Dic₁₄₆	1	2	yes	—
252	C₂³	#14556568	1	2	yes	—
253	C₂³	#14556590	1	1	yes	characteristic
254	C₂³	#14556590	1	1	yes	characteristic
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
259	C₂×C₄	C₂²⋊C₄×D₇₃	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
262	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
263	C₂×C₄	#14556547	1	1	yes	characteristic
264	C₂×C₄	#14556548	1	1	yes	characteristic
265	C₂×C₄	#14556551	1	2	yes	—
266	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
267	C₂×C₄	#14556556	1	2	yes	—
268	C₂×C₄	#14556563	1	2	yes	—
269	C₂×C₄	#14556564	1	2	yes	—
270	C₂×C₄	#14556569	1	2	yes	—
271	C₂×C₄	#14556577	1	2	yes	—
272	C₂×C₄	#14556581	1	2	yes	—
273	C₂×C₄	#14556587	1	1	yes	characteristic
274	C₂×C₄	#14556589	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₈	—	292	4	no	—
277	C₈	—	73	4	no	—
278	C₈	—	73	4	no	—
279	C₈	—	146	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₄×#11924007	1	1	yes	characteristic
286	C₂²	#17988445	1	1	yes	characteristic
287	C₂²	#17988456	1	1	yes	characteristic
288	C₂²	#17988484	1	2	yes	—
289	C₂²	#17988491	1	2	yes	—
290	C₄	—	4	4	no	—
291	C₄	#95×D₇₃	1	1	yes	characteristic
292	C₄	#17988438	1	1	yes	characteristic
293	C₄	#17988446	1	1	yes	characteristic
294	C₄	#17988451	1	1	yes	characteristic
295	C₄	#17988482	1	2	yes	—
296	C₄	#17988487	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#22608159	1	1	yes	characteristic
302	C₂	#22609979	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556547
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1742	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1742	D₇₃
hypocenter	C₇₃	#10867
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
11. C₂×#14556526	C₄	semidirect
157. C₇₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556526	2
C₂×#17988543	2
#22609670	2
C₂×#17988345	2
C₇₃×#1742	2
#10867	73