Subgroups — #32991812

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₄×#10596831	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×C₄×#10596831	—	2	2	no	—
11	C₂×C₄×#10596831	—	2	2	no	—
12	C₂×C₄×#10596831	C₄	1	2	yes	—
13	C₂×C₄×#10596831	C₄	1	2	yes	—
14	#17988351	—	2	8	no	—
15	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#423×C₇₃	—	2	2	no	—
18	#423×C₇₃	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	4	no	—
22	C₂×#11924012	—	2	4	no	—
23	C₂×#11924012	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
29	C₂²×#10596831	C₂×C₄	1	4	yes	—
30	C₂²×#10596831	D₄	1	4	yes	—
31	C₂²×#10596831	Q₈	1	4	yes	—
32	C₄⋊#10596831	—	2	8	no	—
33	C₄×#10596831	—	2	4	no	—
34	C₄×#10596831	—	2	4	no	—
35	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₇₃	—	2	2	no	—
38	C₂×M₁₆×C₇₃	—	2	4	no	—
39	C₂×M₁₆×C₇₃	—	2	4	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₇₃	—	2	2	no	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	D₄	1	2	yes	—
48	C₂×C₄²×C₇₃	D₄	1	2	yes	—
49	C₄⋊C₈×C₇₃	—	2	8	no	—
50	#11924012	—	2	8	no	—
51	#11924012	—	4	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	—	2	4	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₂×#10596831	C₄⋊C₄	1	8	yes	—
62	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	8	no	—
64	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇₃	#41	1	2	yes	—
77	C₂²×C₄×C₇₃	#41	1	2	yes	—
78	M₁₆×C₇₃	—	2	8	no	—
79	M₁₆×C₇₃	—	4	4	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	4	2	no	—
82	C₂×C₈×C₇₃	—	4	2	no	—
83	C₄²×C₇₃	C₄²	1	2	yes	—
84	C₄²×C₇₃	C₄²	1	2	yes	—
85	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
89	C₂³×C₇₃	—	2	4	no	—
90	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₇₃	#125	1	2	yes	—
94	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	2	8	no	—
97	C₂×C₄×C₇₃	—	2	8	no	—
98	C₂×C₄×C₇₃	#95	1	4	yes	—
99	C₂×C₄×C₇₃	#95	1	4	yes	—
100	C₂×C₄×C₇₃	#108	1	8	yes	—
101	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₃	#117	1	2	yes	—
107	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₇₃	#121	1	2	yes	—
112	C₂×C₄×C₇₃	#121	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₇₃	—	4	4	no	—
120	#10596831	—	2	8	no	—
121	#10596831	—	4	4	no	—
122	C₂²×C₇₃	—	2	4	no	—
123	C₂²×C₇₃	—	2	4	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
127	C₂²×C₇₃	#382	1	1	yes	characteristic
128	C₂²×C₇₃	#384	1	1	yes	characteristic
129	C₂²×C₇₃	#389	1	2	yes	—
130	C₂²×C₇₃	#389	1	2	yes	—
131	C₂²×C₇₃	C₂×#108	1	4	yes	—
132	C₄×C₇₃	—	2	4	no	—
133	C₄×C₇₃	#341	1	8	yes	complemented
134	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
135	C₄×C₇₃	#380	1	2	yes	—
136	C₄×C₇₃	#390	1	1	yes	characteristic
137	C₄×C₇₃	#390	1	1	yes	characteristic
138	C₄×C₇₃	#391	1	1	yes	characteristic
139	C₄×C₇₃	#391	1	1	yes	characteristic
140	#10844	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	—	2	2	no	—
143	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
144	C₂×C₇₃	C₂×#341	1	2	yes	—
145	C₂×C₇₃	#1837	1	2	yes	—
146	C₂×C₇₃	#1848	1	2	yes	—
147	C₂×#423	—	73	1	no	—
148	C₂×#423	D₇₃	1	1	yes	characteristic
149	C₂²×C₄×C₈	—	73	1	no	—
150	C₂×#341	—	73	4	no	—
151	C₇₃	#10844	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	73	2	no	—
153	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
154	C₂³×C₈	—	73	2	no	—
155	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
156	#423	—	2	2	no	—
157	#423	—	73	4	no	—
158	#423	—	146	2	no	—
159	#423	Dic₇₃	1	4	yes	—
160	C₂×C₄⋊C₈	—	2	2	no	—
161	C₂×C₄⋊C₈	—	146	2	no	—
162	C₂×C₄×C₈	—	73	2	no	—
163	C₂×C₄×C₈	—	73	2	no	—
164	C₂×C₄×C₈	—	146	2	no	—
165	C₂×C₄×C₈	—	146	2	no	—
166	#341	—	146	8	no	—
167	C₂³×C₄	Dic₁₄₆	1	2	yes	—
168	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	73	4	no	—
173	C₂×M₁₆	—	146	4	no	—
174	C₂×M₁₆	—	146	4	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	73	4	no	—
179	C₂²×C₈	—	73	4	no	—
180	C₂²×C₈	—	73	4	no	—
181	C₂²×C₈	—	146	2	no	—
182	C₂²×C₈	—	146	2	no	—
183	C₂²×C₈	—	146	2	no	—
184	C₂²×C₈	—	146	2	no	—
185	C₂²×C₈	—	146	2	no	—
186	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₇₃	1	2	yes	—
189	C₂×C₄²	D₂₉₂	1	2	yes	—
190	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	146	8	no	—
195	C₄×C₈	—	146	4	no	—
196	C₄×C₈	—	146	4	no	—
197	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	#11924007	1	2	yes	—
200	C₂²×C₄	#11924007	1	4	yes	—
201	C₂²×C₄	#11924007	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₇₃	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
210	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
211	C₂²×C₄	#11924027	1	2	yes	—
212	C₂²×C₄	#11924029	1	2	yes	—
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	292	4	no	—
216	M₁₆	—	146	8	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	292	2	no	—
221	C₂×C₈	—	292	2	no	—
222	C₂×C₈	—	73	8	no	—
223	C₂×C₈	—	146	4	no	—
224	C₂×C₈	—	146	4	no	—
225	C₂×C₈	—	146	4	no	—
226	C₂×C₈	—	146	4	no	—
227	C₂×C₈	—	146	4	no	—
228	C₂×C₈	—	146	4	no	—
229	C₂×C₈	—	146	4	no	—
230	C₄²	C₄×Dic₇₃	1	2	yes	—
231	C₄²	C₄×Dic₇₃	1	2	yes	—
232	C₄²	#11924005	1	2	yes	—
233	C₄²	#11924005	1	2	yes	—
234	C₄²	#11924005	1	4	yes	—
235	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂×#11924007	1	1	yes	characteristic
238	C₂³	C₂×#11924007	1	1	yes	characteristic
239	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
240	C₂³	#14556582	1	2	yes	—
241	C₂³	#14556590	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	#14556498	1	4	yes	—
246	C₂×C₄	#14556498	1	4	yes	—
247	C₂×C₄	#14556526	1	8	yes	—
248	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
250	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
252	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
253	C₂×C₄	#14556549	1	2	yes	—
254	C₂×C₄	#14556553	1	2	yes	—
255	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
256	C₂×C₄	#14556556	1	2	yes	—
257	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
258	C₂×C₄	#14556562	1	2	yes	—
259	C₂×C₄	#14556566	1	2	yes	—
260	C₂×C₄	#14556567	1	2	yes	—
261	C₂×C₄	#14556571	1	2	yes	—
262	C₂×C₄	#14556578	1	1	yes	characteristic
263	C₂×C₄	#14556580	1	1	yes	characteristic
264	C₂×C₄	#14556587	1	1	yes	characteristic
265	C₂×C₄	#14556589	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	292	4	no	—
268	C₈	—	146	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₂×#14556498	1	1	yes	characteristic
274	C₂²	#17988452	1	1	yes	characteristic
275	C₂²	#17988456	1	1	yes	characteristic
276	C₂²	#17988483	1	2	yes	—
277	C₂²	#17988486	1	2	yes	—
278	C₂²	C₂×#14556526	1	4	yes	—
279	C₄	—	2	4	no	—
280	C₄	#17988351	1	8	yes	complemented
281	C₄	C₄⋊C₄×Dic₇₃	1	2	yes	—
282	C₄	#17988444	1	2	yes	—
283	C₄	#17988488	1	1	yes	characteristic
284	C₄	#17988490	1	1	yes	characteristic
285	C₄	#17988492	1	1	yes	characteristic
286	C₄	#17988494	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#22608160	1	1	yes	characteristic
290	C₂	C₂×#17988351	1	2	yes	—
291	C₂	#22610068	1	2	yes	—
292	C₂	#22610102	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	#10844
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
133. C₄×C₇₃	#341	semidirect
151. C₇₃	#10844	semidirect
280. C₄	#17988351	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×C₄×#10596831	2
C₂×#17988351	2
C₂×#423×C₇₃	2
#10844	73