Subgroups — #33104817

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612888	C₂	1	2	yes	maximal
2	#22610701	C₂	1	4	yes	maximal
3	#22610690	C₂	1	2	yes	maximal
4	#22610095	C₂	1	2	yes	maximal
5	#22610088	C₂	1	1	yes	maximal, characteristic
6	C₄×#14556523	C₂	1	2	yes	maximal
7	C₄×#14556523	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556525	C₂²	1	4	yes	—
10	C₂×#14556523	C₂²	1	2	yes	—
11	C₂×#14556523	C₂²	1	1	yes	characteristic
12	C₂×#14556523	C₂²	1	2	yes	—
13	C₂×#14556523	C₂²	1	1	yes	characteristic
14	#17988494	—	2	2	no	—
15	#17988473	—	2	4	no	—
16	#17988457	C₂²	1	2	yes	—
17	#17988453	C₂²	1	4	yes	—
18	C₄⋊C₄×Dic₇₃	—	2	2	no	—
19	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
21	#17988350	C₂²	1	4	yes	—
22	#17988350	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₇₃	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
34	C₂×C₄×Dic₇₃	—	2	2	no	—
35	C₂×C₄×Dic₇₃	—	2	4	no	—
36	#14556525	—	4	4	no	—
37	#14556523	—	2	4	no	—
38	#14556523	—	2	2	no	—
39	#14556523	—	4	2	no	—
40	#14556523	Q₈	1	4	yes	complemented
41	#14556498	—	2	2	no	—
42	#14556498	—	2	4	no	—
43	#14556498	—	2	4	no	—
44	#14556498	—	2	4	no	—
45	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₇₃	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
56	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	—
59	C₂²×Dic₇₃	—	2	2	no	—
60	C₂²×Dic₇₃	—	2	2	no	—
61	C₂²×Dic₇₃	—	2	4	no	—
62	C₄⋊Dic₇₃	—	2	2	no	—
63	C₄⋊Dic₇₃	—	2	4	no	—
64	C₄⋊Dic₇₃	—	2	2	no	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	2	2	no	—
67	C₄⋊Dic₇₃	—	4	4	no	—
68	C₄⋊Dic₇₃	—	8	2	no	—
69	C₄⋊Dic₇₃	C₂×Q₈	1	4	yes	—
70	C₄×Dic₇₃	—	4	4	no	—
71	C₄×Dic₇₃	—	8	4	no	—
72	C₂×Q₈×C₇₃	—	2	4	no	—
73	C₂×Q₈×C₇₃	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	1	no	—
80	C₂×C₈×C₇₃	—	2	2	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₇₃	#41	1	2	yes	—
88	C₄⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
89	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
90	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
93	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
94	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₇₃	—	2	8	no	—
96	C₂×C₄×C₇₃	—	2	4	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₃	#135	1	4	yes	—
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×Dic₇₃	—	2	4	no	—
111	C₂×Dic₇₃	—	4	2	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	—	4	4	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	2	no	—
116	C₂×Dic₇₃	—	4	4	no	—
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	4	2	no	—
119	C₈×C₇₃	—	2	2	no	—
120	C₈×C₇₃	—	2	4	no	—
121	C₈×C₇₃	C₄×Q₈	1	4	yes	—
122	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
124	C₂²×C₇₃	#550	1	1	yes	characteristic
125	C₂²×C₇₃	C₂×#135	1	2	yes	—
126	C₂²×C₇₃	#577	1	2	yes	—
127	C₄×C₇₃	—	2	4	no	—
128	C₄×C₇₃	—	2	4	no	—
129	C₄×C₇₃	—	2	2	no	—
130	C₄×C₇₃	#427	1	4	yes	—
131	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₇₃	#520	1	1	yes	characteristic
133	C₄×C₇₃	#557	1	1	yes	characteristic
134	C₄×C₇₃	Q₈²	1	1	yes	characteristic
135	Dic₇₃	—	4	4	no	—
136	Dic₇₃	—	8	2	no	—
137	Dic₇₃	—	8	4	no	—
138	#22725	—	73	1	no	maximal, Sylow
139	C₂×C₇₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₇₃	C₂×#427	1	1	yes	characteristic
141	C₂×C₇₃	#2912	1	1	yes	characteristic
142	C₂×C₇₃	#3381	1	2	yes	—
143	C₂×C₇₃	#3394	1	2	yes	—
144	C₂×C₈×Q₈	D₇₃	1	1	yes	characteristic
145	#2355	—	73	2	no	—
146	#1949	—	73	4	no	—
147	#1947	—	73	2	no	—
148	#1846	—	73	2	no	—
149	#1844	—	73	1	no	—
150	C₄×#106	—	73	2	no	—
151	C₄×#106	—	73	1	no	—
152	C₇₃	#22725	1	1	yes	characteristic, complemented, Sylow
153	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
154	C₈×Q₈	Dic₇₃	1	8	yes	complemented
155	C₂×#107	—	73	4	no	—
156	C₂×#106	—	73	2	no	—
157	C₂×#106	—	73	1	no	—
158	C₂×#106	—	73	2	no	—
159	C₂×#106	—	73	1	no	—
160	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
162	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
164	#391	—	146	2	no	—
165	#387	—	146	4	no	—
166	#384	—	73	2	no	—
167	#382	—	73	4	no	—
168	C₄×C₄⋊C₄	—	73	2	no	—
169	C₄×C₄⋊C₄	—	73	4	no	—
170	C₄×C₄⋊C₄	—	146	2	no	—
171	#340	—	73	4	no	—
172	#340	—	73	2	no	—
173	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
176	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
177	C₄×Q₈	Dic₁₄₆	1	4	yes	—
178	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
179	C₂×C₄⋊C₄	—	73	2	no	—
180	C₂×C₄⋊C₄	—	73	2	no	—
181	C₂×C₄⋊C₄	—	73	4	no	—
182	C₂×C₄⋊C₄	—	146	2	no	—
183	C₂×C₄⋊C₄	—	146	2	no	—
184	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
185	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
186	C₂×C₄²	—	146	2	no	—
187	C₂×C₄²	—	146	4	no	—
188	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
189	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
190	#107	—	292	4	no	—
191	#106	—	292	2	no	—
192	#106	—	73	4	no	—
193	#106	—	146	4	no	—
194	#106	—	146	2	no	—
195	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
196	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
197	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
198	C₄×C₈	C₂×Dic₇₃	1	8	yes	—
199	#95	—	146	2	no	—
200	#95	—	146	4	no	—
201	#95	—	146	4	no	—
202	#95	—	146	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	C₄⋊Dic₇₃	1	4	yes	—
205	C₂²×C₄	—	146	2	no	—
206	C₂²×C₄	—	146	4	no	—
207	C₂²×C₄	—	146	2	no	—
208	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
209	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
210	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
218	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
219	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
220	C₂×C₈	Q₈×D₇₃	1	2	yes	—
221	C₂×C₈	#11924029	1	2	yes	—
222	C₄⋊C₄	—	292	4	no	—
223	C₄⋊C₄	—	584	2	no	—
224	C₄⋊C₄	—	73	4	no	—
225	C₄⋊C₄	—	146	2	no	—
226	C₄⋊C₄	—	146	4	no	—
227	C₄⋊C₄	—	146	4	no	—
228	C₄⋊C₄	—	146	2	no	—
229	C₄⋊C₄	—	146	2	no	—
230	C₄⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
231	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
232	C₄²	—	292	4	no	—
233	C₄²	—	584	4	no	—
234	C₄²	C₂²×Dic₇₃	1	4	yes	—
235	C₄²	C₂²×Dic₇₃	1	2	yes	—
236	C₄²	C₂×Dic₁₄₆	1	4	yes	—
237	C₄²	C₂×Dic₁₄₆	1	2	yes	—
238	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	292	2	no	—
244	C₂×C₄	—	292	2	no	—
245	C₂×C₄	—	292	4	no	—
246	C₂×C₄	—	292	2	no	—
247	C₂×C₄	—	292	4	no	—
248	C₂×C₄	—	292	2	no	—
249	C₂×C₄	—	292	2	no	—
250	C₂×C₄	—	292	4	no	—
251	C₂×C₄	—	146	4	no	—
252	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
256	C₂×C₄	#14556613	1	4	yes	—
257	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
260	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
261	C₂×C₄	#14556637	1	1	yes	characteristic
262	C₂×C₄	#14556638	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	Q₈×Dic₇₃	1	4	yes	—
266	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
267	C₂²	Q₈×D₂₉₂	1	1	yes	characteristic
268	C₂²	#17989259	1	1	yes	characteristic
269	C₂²	C₂×#14556613	1	2	yes	—
270	C₂²	#17989399	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	292	4	no	—
275	C₄	—	584	4	no	—
276	C₄	—	584	2	no	—
277	C₄	#17988625	1	4	yes	—
278	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
279	C₄	#17989070	1	1	yes	characteristic
280	C₄	#17989316	1	1	yes	characteristic
281	C₄	Q₈×Dic₁₄₆	1	1	yes	characteristic
282	C₂	Q₈×C₄⋊Dic₇₃	1	1	yes	characteristic
283	C₂	C₂×#17988625	1	1	yes	characteristic
284	C₂	#22618526	1	1	yes	characteristic
285	C₂	#22622808	1	2	yes	—
286	C₂	#22622929	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₈×Q₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₃
hypocenter	C₇₃	#22725
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. C₈×Q₈×C₇₃	C₄	semidirect
40. #14556523	Q₈	semidirect
152. C₇₃	#22725	semidirect
154. C₈×Q₈	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612888	2
#22610701	2
#22610690	2
#22610095	2
#22610088	2
C₄×#14556523	2
C₂×C₈×Q₈×C₇₃	2
#22725	73