Subgroups — #32991815

Order 18688.

774 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988351	C₂	1	4	yes	maximal
4	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂³×#10596831	C₂²	1	2	yes	—
7	#17988611	—	2	2	no	—
8	#17988611	C₄	1	4	yes	—
9	C₂×C₄⋊#10596831	—	2	2	no	—
10	C₂×#14556502	—	2	2	no	—
11	C₂×#14556502	—	2	2	no	—
12	C₂×#14556502	C₄	1	4	yes	—
13	#17988351	—	2	8	no	—
14	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	#423×C₇₃	—	2	2	no	—
17	#423×C₇₃	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	—	2	4	no	—
21	C₂×#11924012	—	2	4	no	—
22	C₂×#11924012	C₂×C₄	1	4	yes	—
23	C₂²×#10596831	—	2	2	no	—
24	C₂²×#10596831	—	2	2	no	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂²×#10596831	C₂×C₄	1	4	yes	—
29	C₂²×#10596831	D₄	1	4	yes	—
30	C₂²×#10596831	Q₈	1	4	yes	—
31	C₄⋊#10596831	—	2	8	no	—
32	#14556502	—	2	8	no	—
33	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₃	—	2	2	no	—
36	C₂×M₁₆×C₇₃	—	2	4	no	—
37	C₂×M₁₆×C₇₃	—	2	4	no	—
38	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	D₄	1	2	yes	—
44	C₂×C₄²×C₇₃	D₄	1	2	yes	—
45	C₂×C₄²×C₇₃	D₄	1	2	yes	—
46	C₄⋊C₈×C₇₃	—	2	8	no	—
47	#11924012	—	2	8	no	—
48	#11924012	—	4	4	no	—
49	C₂×#10596831	—	2	4	no	—
50	C₂×#10596831	—	2	4	no	—
51	C₂×#10596831	—	2	4	no	—
52	C₂×#10596831	—	2	4	no	—
53	C₂×#10596831	—	2	4	no	—
54	C₂×#10596831	—	2	4	no	—
55	C₂×#10596831	—	2	4	no	—
56	C₂×#10596831	—	4	2	no	—
57	C₂×#10596831	—	4	2	no	—
58	C₂×#10596831	C₄⋊C₄	1	8	yes	—
59	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	—	4	4	no	—
61	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₇₃	#41	1	2	yes	—
73	C₂²×C₄×C₇₃	#41	1	2	yes	—
74	M₁₆×C₇₃	—	2	8	no	—
75	M₁₆×C₇₃	—	4	4	no	—
76	C₂×C₈×C₇₃	—	2	4	no	—
77	C₂×C₈×C₇₃	—	4	2	no	—
78	C₂×C₈×C₇₃	—	4	2	no	—
79	C₄²×C₇₃	C₄²	1	4	yes	—
80	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
82	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
83	C₂³×C₇₃	—	2	4	no	—
84	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₇₃	#125	1	2	yes	—
88	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₇₃	—	2	8	no	—
90	C₂×C₄×C₇₃	—	4	4	no	—
91	C₂×C₄×C₇₃	—	4	4	no	—
92	C₂×C₄×C₇₃	#95	1	8	yes	—
93	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₃	#117	1	2	yes	—
99	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₇₃	#121	1	2	yes	—
104	C₂×C₄×C₇₃	#121	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₇₃	—	4	4	no	—
112	#10596831	—	2	8	no	—
113	#10596831	—	4	4	no	—
114	C₂²×C₇₃	—	2	4	no	—
115	C₂²×C₇₃	—	2	4	no	—
116	C₂²×C₇₃	—	2	2	no	—
117	C₂²×C₇₃	—	2	2	no	—
118	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
119	C₂²×C₇₃	#382	1	1	yes	characteristic
120	C₂²×C₇₃	#384	1	1	yes	characteristic
121	C₂²×C₇₃	#389	1	2	yes	—
122	C₂²×C₇₃	#389	1	2	yes	—
123	C₂²×C₇₃	#430	1	4	yes	—
124	C₄×C₇₃	—	2	8	no	—
125	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
126	C₄×C₇₃	#380	1	2	yes	—
127	C₄×C₇₃	#390	1	1	yes	characteristic
128	C₄×C₇₃	#390	1	1	yes	characteristic
129	C₄×C₇₃	#391	1	1	yes	characteristic
130	C₄×C₇₃	#391	1	1	yes	characteristic
131	#10845	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	2	no	—
133	C₂×C₇₃	—	2	2	no	—
134	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
135	C₂×C₇₃	#1738	1	2	yes	—
136	C₂×C₇₃	#1838	1	2	yes	—
137	C₂×C₇₃	#1849	1	2	yes	—
138	C₂×#423	—	73	1	no	—
139	C₂×#423	D₇₃	1	1	yes	characteristic
140	C₂²×#97	—	73	1	no	—
141	C₂×#341	—	73	4	no	—
142	C₇₃	#10845	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	73	2	no	—
144	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
145	C₂³×C₈	—	73	2	no	—
146	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
147	#423	—	2	2	no	—
148	#423	—	73	4	no	—
149	#423	—	146	2	no	—
150	#423	Dic₇₃	1	4	yes	—
151	C₂×C₄⋊C₈	—	2	2	no	—
152	C₂×C₄⋊C₈	—	146	2	no	—
153	C₂×#97	—	73	4	no	—
154	C₂×#97	—	146	2	no	—
155	C₂×#97	—	146	2	no	—
156	#341	—	146	8	no	—
157	C₂³×C₄	Dic₁₄₆	1	2	yes	—
158	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	73	4	no	—
163	C₂×M₁₆	—	146	4	no	—
164	C₂×M₁₆	—	146	4	no	—
165	C₂×M₁₆	—	146	2	no	—
166	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	73	4	no	—
169	C₂²×C₈	—	73	4	no	—
170	C₂²×C₈	—	73	4	no	—
171	C₂²×C₈	—	146	2	no	—
172	C₂²×C₈	—	146	2	no	—
173	C₂²×C₈	—	146	2	no	—
174	C₂²×C₈	—	146	2	no	—
175	C₂²×C₈	—	146	2	no	—
176	C₂×C₄²	C₄×D₇₃	1	2	yes	—
177	C₂×C₄²	C₄×D₇₃	1	2	yes	—
178	C₂×C₄²	D₂₉₂	1	2	yes	—
179	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
182	C₄⋊C₈	—	2	8	no	—
183	C₄⋊C₈	—	146	8	no	—
184	#97	—	146	8	no	—
185	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	#11924007	1	4	yes	—
188	C₂²×C₄	#11924007	1	4	yes	—
189	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
190	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₇₃	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
197	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
198	C₂²×C₄	#11924027	1	2	yes	—
199	C₂²×C₄	#11924029	1	2	yes	—
200	M₁₆	—	2	8	no	—
201	M₁₆	—	4	4	no	—
202	M₁₆	—	292	4	no	—
203	M₁₆	—	146	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	292	2	no	—
208	C₂×C₈	—	292	2	no	—
209	C₂×C₈	—	73	8	no	—
210	C₂×C₈	—	146	4	no	—
211	C₂×C₈	—	146	4	no	—
212	C₂×C₈	—	146	4	no	—
213	C₂×C₈	—	146	4	no	—
214	C₂×C₈	—	146	4	no	—
215	C₂×C₈	—	146	4	no	—
216	C₂×C₈	—	146	4	no	—
217	C₄²	C₄×Dic₇₃	1	4	yes	—
218	C₄²	#11924005	1	4	yes	—
219	C₄²	#11924005	1	4	yes	—
220	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×#11924007	1	1	yes	characteristic
223	C₂³	C₂×#11924007	1	1	yes	characteristic
224	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
225	C₂³	#14556582	1	2	yes	—
226	C₂³	#14556590	1	2	yes	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	#14556498	1	8	yes	—
231	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
233	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
235	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
236	C₂×C₄	#14556549	1	2	yes	—
237	C₂×C₄	#14556553	1	2	yes	—
238	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
239	C₂×C₄	#14556556	1	2	yes	—
240	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
241	C₂×C₄	#14556562	1	2	yes	—
242	C₂×C₄	#14556566	1	2	yes	—
243	C₂×C₄	#14556567	1	2	yes	—
244	C₂×C₄	#14556571	1	2	yes	—
245	C₂×C₄	#14556578	1	1	yes	characteristic
246	C₂×C₄	#14556580	1	1	yes	characteristic
247	C₂×C₄	#14556587	1	1	yes	characteristic
248	C₂×C₄	#14556589	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	292	4	no	—
251	C₈	—	146	8	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×#14556498	1	1	yes	characteristic
257	C₂²	#17988452	1	1	yes	characteristic
258	C₂²	#17988456	1	1	yes	characteristic
259	C₂²	#17988483	1	2	yes	—
260	C₂²	#17988486	1	2	yes	—
261	C₂²	#17988635	1	4	yes	—
262	C₄	—	2	8	no	—
263	C₄	C₄⋊C₄×Dic₇₃	1	2	yes	—
264	C₄	#17988444	1	2	yes	—
265	C₄	#17988488	1	1	yes	characteristic
266	C₄	#17988490	1	1	yes	characteristic
267	C₄	#17988492	1	1	yes	characteristic
268	C₄	#17988494	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	#22608160	1	1	yes	characteristic
272	C₂	#22609657	1	2	yes	—
273	C₂	#22610071	1	2	yes	—
274	C₂	#22610105	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#10845
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
142. C₇₃	#10845	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
C₂²×#14556502	2
C₂×#17988351	2
C₂×#423×C₇₃	2
#10845	73