Subgroups — #34030266

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626692	C₂	1	2	yes	maximal
2	#22626328	C₂	1	4	yes	maximal, complemented
3	Q₃₂×#64408	C₂	1	4	yes	maximal, complemented
4	#22626282	C₂	1	1	yes	maximal, characteristic
5	#22626241	C₂	1	2	yes	maximal
6	#22626139	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989655	—	2	2	no	—
9	#17989643	C₂²	1	1	yes	characteristic
10	#17989627	—	2	4	no	—
11	Q₁₆×#64408	C₂²	1	8	yes	—
12	#17989597	C₂²	1	2	yes	—
13	#17989580	—	2	2	no	—
14	#17989550	C₂²	1	1	yes	characteristic
15	#17989514	C₂²	1	2	yes	—
16	#17989513	C₂²	1	2	yes	—
17	#17989500	C₂²	1	8	yes	—
18	#17989474	C₂²	1	2	yes	—
19	C₁₆×#64408	C₂²	1	2	yes	—
20	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
21	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic, complemented
22	C₂×#14556537	C₄	1	2	yes	complemented
23	Q₃₂×D₇₃	C₄	1	2	yes	complemented
24	Q₃₂×D₇₃	C₄	1	2	yes	complemented
25	Q₃₂×D₇₃	C₂²	1	2	yes	—
26	Q₃₂×D₇₃	C₂²	1	2	yes	—
27	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
28	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
29	D₇₃⋊M₁₆	D₄	1	1	yes	characteristic
30	Q₈×#64408	—	2	8	no	—
31	Q₈×#64408	—	4	4	no	—
32	#14556670	—	2	2	no	—
33	#14556670	—	4	4	no	—
34	#14556670	D₄	1	1	yes	characteristic
35	#14556666	—	2	1	no	—
36	#14556665	—	2	1	no	—
37	#14556664	C₂³	1	2	yes	—
38	#14556660	C₂³	1	4	yes	—
39	#14556659	—	2	2	no	—
40	#14556654	—	2	8	no	—
41	#14556645	D₄	1	2	yes	—
42	C₈×#64408	C₂³	1	2	yes	—
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
45	C₂×#11924018	—	2	2	no	—
46	Q₁₆×D₇₃	—	2	2	no	—
47	Q₁₆×D₇₃	—	2	2	no	—
48	Q₁₆×D₇₃	C₂×C₄	1	4	yes	—
49	Q₁₆×D₇₃	C₂³	1	4	yes	—
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	Dic₅₈₄	C₂×C₄	1	2	yes	—
52	Dic₅₈₄	C₂×C₄	1	2	yes	—
53	#14556537	C₂×C₄	1	4	yes	—
54	#14556537	C₂×C₄	1	4	yes	—
55	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
59	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
61	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
62	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#11924038	—	2	1	no	—
65	#11924037	—	2	1	no	—
66	#11924036	C₂×D₄	1	2	yes	—
67	C₄⋊#64408	—	2	4	no	—
68	C₄⋊#64408	—	4	8	no	—
69	C₄⋊#64408	D₈	1	2	yes	—
70	C₂²⋊#64408	—	4	2	no	—
71	C₂²⋊#64408	D₈	1	2	yes	—
72	C₄×#64408	—	4	8	no	—
73	C₄×#64408	—	4	2	no	—
74	C₄×#64408	C₂×D₄	1	2	yes	—
75	#11924031	—	2	2	no	—
76	C₂×Dic₁₄₆	—	2	2	no	—
77	C₂×Dic₁₄₆	—	4	2	no	—
78	Q₈×D₇₃	—	2	4	no	—
79	Q₈×D₇₃	—	2	4	no	—
80	Q₈×D₇₃	—	4	2	no	—
81	Q₈×D₇₃	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
84	Dic₂₉₂	—	2	2	no	—
85	Dic₂₉₂	C₂²×C₄	1	4	yes	—
86	#11924018	—	2	4	no	—
87	#11924018	—	2	4	no	—
88	C₂×#10596831	#41	1	1	yes	characteristic
89	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
91	C₈×D₇₃	#41	1	1	yes	characteristic
92	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
93	#11924003	C₂²×C₄	1	1	yes	characteristic
94	#11924003	C₂²×C₄	1	1	yes	characteristic
95	C₂×Q₈×C₇₃	—	2	2	no	—
96	Q₁₆×C₇₃	—	2	2	no	—
97	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
102	C₂×#64408	—	4	4	no	—
103	C₂×#64408	C₂×D₈	1	4	yes	—
104	Q₈×C₇₃	—	2	4	no	—
105	Q₈×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂×Dic₇₃	—	4	2	no	—
109	C₂×Dic₇₃	#135	1	1	yes	characteristic
110	C₄×D₇₃	—	4	4	no	—
111	C₄×D₇₃	—	4	4	no	—
112	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
113	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
114	C₄×D₇₃	#135	1	1	yes	characteristic
115	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
116	Dic₁₄₆	—	2	4	no	—
117	Dic₁₄₆	—	4	2	no	—
118	Dic₁₄₆	—	4	4	no	—
119	Dic₁₄₆	—	4	4	no	—
120	#10596833	—	2	2	no	—
121	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	#10596831	C₄×D₄	1	1	yes	characteristic
124	#10596831	C₂×#41	1	1	yes	characteristic
125	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
126	C₂×D₇₃	—	2	2	no	—
127	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
128	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
129	#64408	—	2	4	no	—
130	#64408	—	8	4	no	—
131	C₄×C₇₃	—	4	4	no	—
132	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
133	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
134	Dic₇₃	—	4	4	no	—
135	Dic₇₃	C₄×D₈	1	1	yes	characteristic
136	Dic₇₃	C₂×#135	1	1	yes	characteristic
137	#33458	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	1	no	—
139	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
140	D₇₃	—	2	1	no	—
141	D₇₃	#3417	1	2	yes	—
142	C₂²×Q₃₂	—	73	1	no	—
143	#2950	—	73	2	no	—
144	#2176	—	73	4	no	—
145	C₄×Q₃₂	—	73	4	no	—
146	#2158	—	73	1	no	—
147	#2141	—	73	2	no	—
148	#2108	—	73	1	no	—
149	C₇₃	#33458	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₁₆	—	73	2	no	—
151	#519	—	146	2	no	—
152	C₂×Q₃₂	—	73	2	no	—
153	C₂×Q₃₂	—	73	2	no	—
154	C₂×Q₃₂	—	73	2	no	—
155	C₂×Q₃₂	—	73	2	no	—
156	C₂×Q₃₂	—	73	2	no	—
157	C₂×Q₃₂	—	73	1	no	—
158	C₂×Q₃₂	#64408	1	1	yes	characteristic, complemented
159	C₂×M₃₂	—	73	1	no	—
160	C₂²×C₁₆	—	73	1	no	—
161	#441	—	146	4	no	—
162	C₄×Q₁₆	—	73	8	no	—
163	#427	—	73	2	no	—
164	#419	—	146	2	no	—
165	#405	—	73	1	no	—
166	#367	—	73	2	no	—
167	#366	—	73	2	no	—
168	#358	—	73	8	no	—
169	#346	—	73	2	no	—
170	C₄×C₁₆	—	73	2	no	—
171	C₂²×Q₈	—	146	2	no	—
172	C₂×Q₁₆	—	73	4	no	—
173	C₂×Q₁₆	—	73	4	no	—
174	C₂×Q₁₆	—	73	2	no	—
175	C₂×Q₁₆	—	146	2	no	—
176	C₂×Q₁₆	—	146	2	no	—
177	C₂×Q₁₆	—	146	2	no	—
178	C₂×Q₁₆	C₂×#64408	1	2	yes	—
179	C₂×M₁₆	—	73	1	no	—
180	C₂²×C₈	—	73	1	no	—
181	C₄×Q₈	—	292	4	no	—
182	C₄×Q₈	—	146	8	no	—
183	#117	—	292	4	no	—
184	#117	—	73	1	no	—
185	#117	—	146	2	no	—
186	Q₃₂	—	73	4	no	—
187	Q₃₂	—	73	4	no	—
188	Q₃₂	—	73	2	no	—
189	Q₃₂	—	73	2	no	—
190	Q₃₂	C₂×#64408	1	2	yes	—
191	Q₃₂	C₂×#64408	1	2	yes	—
192	M₃₂	—	146	1	no	—
193	M₃₂	—	146	1	no	—
194	C₂×C₁₆	—	73	1	no	—
195	C₂×C₁₆	—	73	1	no	—
196	C₂×C₁₆	—	73	1	no	—
197	C₂×C₁₆	—	73	1	no	—
198	C₂×C₁₆	—	73	1	no	—
199	C₂×C₁₆	—	73	2	no	—
200	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
201	#107	—	73	4	no	—
202	#106	—	146	2	no	—
203	#103	—	146	8	no	—
204	#97	—	73	2	no	—
205	C₄×C₈	—	73	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	292	2	no	—
208	C₂×Q₈	—	292	2	no	—
209	C₂×Q₈	—	292	2	no	—
210	C₂×Q₈	—	146	4	no	—
211	C₂×Q₈	—	146	4	no	—
212	C₂×Q₈	—	146	2	no	—
213	C₂²×C₄	—	292	2	no	—
214	C₂²×C₄	—	73	1	no	—
215	Q₁₆	—	2	2	no	—
216	Q₁₆	—	73	4	no	—
217	Q₁₆	—	146	4	no	—
218	Q₁₆	—	146	4	no	—
219	Q₁₆	—	146	2	no	—
220	Q₁₆	C₂²×#64408	1	4	yes	—
221	M₁₆	—	146	1	no	—
222	M₁₆	—	146	1	no	—
223	C₂×C₈	—	73	1	no	—
224	C₂×C₈	—	73	1	no	—
225	C₂×C₈	—	73	1	no	—
226	C₂×C₈	—	73	1	no	—
227	C₂×C₈	—	73	1	no	—
228	C₂×C₈	—	73	2	no	—
229	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
230	C₄⋊C₄	—	292	8	no	—
231	C₄⋊C₄	—	73	2	no	—
232	C₄⋊C₄	—	146	4	no	—
233	C₂²⋊C₄	—	292	2	no	—
234	C₂²⋊C₄	—	73	2	no	—
235	C₄²	—	292	8	no	—
236	C₄²	—	292	2	no	—
237	C₄²	—	73	2	no	—
238	C₁₆	—	73	1	no	—
239	C₁₆	—	73	1	no	—
240	C₁₆	—	146	2	no	—
241	C₁₆	C₂²×#64408	1	1	yes	characteristic
242	C₁₆	C₂²×#64408	1	1	yes	characteristic
243	C₂³	—	73	1	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	4	2	no	—
246	Q₈	—	292	4	no	—
247	Q₈	—	292	4	no	—
248	Q₈	—	292	2	no	—
249	Q₈	—	146	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	292	4	no	—
252	C₂×C₄	—	292	4	no	—
253	C₂×C₄	—	292	2	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	73	1	no	—
256	C₂×C₄	—	73	1	no	—
257	C₂×C₄	—	73	1	no	—
258	C₂×C₄	—	73	1	no	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	73	4	no	—
261	C₂×C₄	D₄×#64408	1	1	yes	characteristic
262	C₈	—	73	1	no	—
263	C₈	—	73	1	no	—
264	C₈	—	146	2	no	—
265	C₈	D₄×#64408	1	1	yes	characteristic
266	C₈	C₂³×#64408	1	1	yes	characteristic
267	C₂²	—	73	1	no	—
268	C₂²	—	73	1	no	—
269	C₂²	—	146	2	no	—
270	C₂²	D₈×#64408	1	1	yes	characteristic
271	C₄	—	4	4	no	—
272	C₄	—	292	4	no	—
273	C₄	—	584	4	no	—
274	C₄	—	73	1	no	—
275	C₄	—	73	1	no	—
276	C₄	—	146	4	no	—
277	C₄	D₈×#64408	1	1	yes	characteristic
278	C₄	C₂×D₄×#64408	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	73	2	no	—
281	C₂	—	146	1	no	—
282	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	C₂²×D₇₃	C₂×D₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	#33458
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22626328	C₂	semidirect
3. Q₃₂×#64408	C₂	semidirect
21. C₂×Dic₅₈₄	C₄	semidirect
22. C₂×#14556537	C₄	semidirect
23. Q₃₂×D₇₃	C₄	semidirect
24. Q₃₂×D₇₃	C₄	semidirect
28. C₂×Q₃₂×C₇₃	C₄	semidirect
149. C₇₃	#33458	semidirect
158. C₂×Q₃₂	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626692	2
#22626328	2
Q₃₂×#64408	2
#22626282	2
#22626241	2
#22626139	2
C₂×Q₃₂×D₇₃	2
#33458	73