Subgroups — #33005602

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988415	C₂	1	2	yes	maximal
3	#22610664	C₂	1	1	yes	maximal, characteristic
4	#22608023	C₂	1	2	yes	maximal
5	C₇₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	—	2	1	no	—
7	#17988992	—	2	4	no	—
8	C₂×#14556525	—	2	1	no	—
9	C₂×#14556525	C₂²	1	1	yes	characteristic
10	C₂×#14556525	C₂²	1	2	yes	—
11	C₂×#14556525	C₂²	1	1	yes	characteristic
12	#17988485	—	2	1	no	—
13	#17988415	—	2	4	no	—
14	#17988350	—	2	1	no	—
15	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₆	—	4	1	no	—
18	#14556591	—	2	4	no	—
19	#14556591	—	4	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	2	1	no	—
22	C₂×C₄⋊Dic₇₃	—	2	1	no	—
23	C₂×C₄⋊Dic₇₃	—	4	1	no	—
24	C₂×#11924007	—	4	1	no	—
25	#14556525	—	2	4	no	—
26	#14556525	—	2	2	no	—
27	#14556525	—	2	2	no	—
28	#14556525	—	2	1	no	—
29	#14556525	D₄	1	4	yes	—
30	#14556525	Q₈	1	2	yes	complemented
31	#14556498	—	4	1	no	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂×C₁₆×C₇₃	Q₈	1	4	yes	complemented
38	C₄×C₈×C₇₃	—	2	2	no	—
39	C₂×Dic₁₄₆	—	4	4	no	—
40	C₂×Dic₁₄₆	—	8	2	no	—
41	C₂×Dic₁₄₆	—	8	2	no	—
42	C₂²×Dic₇₃	—	4	1	no	—
43	C₂²×Dic₇₃	—	4	2	no	—
44	C₂²×Dic₇₃	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	4	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	—	4	4	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₄⋊Dic₇₃	—	4	1	no	—
50	C₄⋊Dic₇₃	—	4	1	no	—
51	C₄⋊Dic₇₃	—	4	1	no	—
52	#11924007	—	8	2	no	—
53	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
59	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
63	C₄²×C₇₃	—	2	1	no	—
64	C₄²×C₇₃	—	2	1	no	—
65	C₁₆×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
72	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
73	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
79	C₂×Dic₇₃	—	4	4	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	—	8	1	no	—
82	C₂×Dic₇₃	—	8	2	no	—
83	C₂×Dic₇₃	—	8	1	no	—
84	C₂×Dic₇₃	—	8	2	no	—
85	C₂×Dic₇₃	—	8	1	no	—
86	C₂×Dic₇₃	—	8	1	no	—
87	C₂×Dic₇₃	—	8	1	no	—
88	Dic₁₄₆	—	8	8	no	—
89	C₈×C₇₃	—	2	2	no	—
90	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
92	C₂²×C₇₃	#447	1	1	yes	characteristic
93	C₂²×C₇₃	#474	1	1	yes	characteristic
94	C₂²×C₇₃	#500	1	1	yes	characteristic
95	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₇₃	#509	1	1	yes	characteristic
98	C₂²×C₇₃	#510	1	1	yes	characteristic
99	C₄×C₇₃	—	2	2	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	#395	1	1	yes	characteristic
102	C₄×C₇₃	#451	1	1	yes	characteristic
103	C₄×C₇₃	#477	1	1	yes	characteristic
104	C₄×C₇₃	#501	1	1	yes	characteristic
105	Dic₇₃	—	8	4	no	—
106	Dic₇₃	—	8	2	no	—
107	Dic₇₃	—	16	1	no	—
108	#13452	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#2057	1	1	yes	characteristic
110	C₂×C₇₃	#2185	1	1	yes	characteristic
111	C₂×C₇₃	#2187	1	1	yes	characteristic
112	C₂×C₇₃	#2225	1	1	yes	characteristic
113	C₂×C₇₃	#2226	1	1	yes	characteristic
114	C₂×C₇₃	#2254	1	1	yes	characteristic
115	C₂×C₇₃	#2255	1	1	yes	characteristic
116	C₂×#500	—	73	1	no	—
117	C₂×#367	—	73	2	no	—
118	#1942	—	73	1	no	—
119	#1380	—	73	2	no	—
120	#1374	D₇₃	1	1	yes	characteristic
121	C₇₃	#13452	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	—	146	1	no	—
123	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
124	#500	—	146	4	no	—
125	C₂×#107	—	73	2	no	—
126	C₂×#107	—	73	1	no	—
127	C₂×#107	—	73	1	no	—
128	C₂×#107	—	146	1	no	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	#389	—	146	1	no	—
131	#367	—	146	4	no	—
132	#340	—	146	1	no	—
133	C₂²×Q₈	—	292	1	no	—
134	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	292	2	no	—
137	C₄⋊Q₈	—	146	4	no	—
138	C₂×C₄⋊C₄	—	292	1	no	—
139	C₂×C₄⋊C₄	—	292	1	no	—
140	C₂×C₄⋊C₄	—	146	1	no	—
141	C₂×C₄⋊C₄	—	146	2	no	—
142	C₂×C₄⋊C₄	—	146	1	no	—
143	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	Dic₁₄₆	1	4	yes	complemented
147	#107	—	73	4	no	—
148	#107	—	73	2	no	—
149	#107	—	146	2	no	—
150	#107	—	146	4	no	—
151	#107	—	146	2	no	—
152	#107	—	146	1	no	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	292	1	no	—
155	C₂×Q₈	—	292	4	no	—
156	C₂×Q₈	—	584	2	no	—
157	C₂×Q₈	—	584	2	no	—
158	C₂²×C₄	—	292	1	no	—
159	C₂²×C₄	—	292	2	no	—
160	C₂²×C₄	—	292	1	no	—
161	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
162	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	Dic₂₉₂	1	2	yes	—
167	C₂×C₈	D₄×D₇₃	1	2	yes	—
168	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
169	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
170	C₂×C₈	#11924027	1	1	yes	characteristic
171	C₄⋊C₄	—	292	4	no	—
172	C₄⋊C₄	—	292	2	no	—
173	C₄⋊C₄	—	292	1	no	—
174	C₄⋊C₄	—	292	1	no	—
175	C₄⋊C₄	—	292	1	no	—
176	C₄⋊C₄	—	584	2	no	—
177	C₄⋊C₄	—	146	4	no	—
178	C₄⋊C₄	—	146	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	4	no	—
182	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
183	Q₈	—	584	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	292	2	no	—
189	C₂×C₄	—	292	4	no	—
190	C₂×C₄	—	584	1	no	—
191	C₂×C₄	—	584	2	no	—
192	C₂×C₄	—	584	2	no	—
193	C₂×C₄	—	584	1	no	—
194	C₂×C₄	—	584	1	no	—
195	C₂×C₄	—	584	1	no	—
196	C₂×C₄	—	584	1	no	—
197	C₂×C₄	#14556536	1	2	yes	—
198	C₂×C₄	#14556536	1	2	yes	—
199	C₂×C₄	#14556561	1	1	yes	characteristic
200	C₂×C₄	#14556571	1	1	yes	characteristic
201	C₂×C₄	#14556591	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
203	C₂×C₄	#14556619	1	1	yes	characteristic
204	C₂×C₄	#14556624	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#14556568	1	2	yes	—
207	C₈	#14556590	1	2	yes	—
208	C₂²	#17988723	1	1	yes	characteristic
209	C₂²	#17988868	1	1	yes	characteristic
210	C₂²	#17988992	1	1	yes	characteristic
211	C₂²	C₂×#14556536	1	1	yes	characteristic
212	C₂²	C₂×#14556536	1	1	yes	characteristic
213	C₂²	#17989025	1	1	yes	characteristic
214	C₂²	#17989030	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	584	2	no	—
218	C₄	—	584	4	no	—
219	C₄	—	1168	1	no	—
220	C₄	#17988508	1	1	yes	characteristic
221	C₄	#17988745	1	1	yes	characteristic
222	C₄	#17988883	1	1	yes	characteristic
223	C₄	#17988996	1	1	yes	characteristic
224	C₂	#22611260	1	1	yes	characteristic
225	C₂	#22611850	1	1	yes	characteristic
226	C₂	#22611862	1	1	yes	characteristic
227	C₂	#22612074	1	1	yes	characteristic
228	C₂	#22612079	1	1	yes	characteristic
229	C₂	#22612213	1	1	yes	characteristic
230	C₂	#22612217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1374	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#1374	D₇₃
hypocenter	C₇₃	#13452
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
30. #14556525	Q₈	semidirect
37. C₂×C₁₆×C₇₃	Q₈	semidirect
121. C₇₃	#13452	semidirect
146. C₂×C₁₆	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988992	2
C₂×#17988415	2
#22610664	2
#22608023	2
C₇₃×#1374	2
#13452	73