Subgroups — #32991813

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 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	2	yes	maximal
2	C₂×#17988352	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924012	C₂²	1	4	yes	—
5	#17988611	—	2	4	no	—
6	#17988611	C₄	1	8	yes	—
7	C₂×C₄⋊#10596831	—	2	4	no	—
8	#17988352	—	2	8	no	—
9	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	—	2	4	no	—
12	C₂×C₄×C₈×C₇₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	2	4	no	—
15	C₂×#11924012	—	2	8	no	—
16	C₂×#11924012	—	2	8	no	—
17	C₂×#11924012	C₂×C₄	1	8	yes	—
18	C₂²×#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	2	16	no	—
20	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₇₃	—	2	4	no	—
23	C₂²×C₈×C₇₃	—	2	4	no	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₃	D₄	1	4	yes	—
26	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	D₄	1	4	yes	—
32	C₄×C₈×C₇₃	—	2	4	no	—
33	C₄×C₈×C₇₃	—	2	4	no	—
34	#11924012	—	2	16	no	—
35	#11924012	—	4	8	no	—
36	C₂×#10596831	—	2	8	no	—
37	C₂×#10596831	—	4	4	no	—
38	C₂×#10596831	—	4	4	no	—
39	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	—	2	8	no	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	8	yes	—
42	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₇₃	#41	1	4	yes	—
51	C₂×C₈×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	—	2	4	no	—
53	C₂×C₈×C₇₃	—	2	8	no	—
54	C₂×C₈×C₇₃	—	2	8	no	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
56	C₄²×C₇₃	C₄²	1	2	yes	—
57	C₄²×C₇₃	C₄²	1	2	yes	—
58	C₄²×C₇₃	C₂²⋊C₄	1	8	yes	—
59	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
61	C₂³×C₇₃	—	2	4	no	—
62	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₇₃	#125	1	2	yes	—
65	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	2	16	no	—
68	C₂×C₄×C₇₃	#95	1	4	yes	—
69	C₂×C₄×C₇₃	#95	1	4	yes	—
70	C₂×C₄×C₇₃	#108	1	8	yes	—
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₇₃	#117	1	2	yes	—
76	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
78	C₂×C₄×C₇₃	#121	1	4	yes	—
79	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₇₃	#124	1	2	yes	—
81	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₇₃	—	2	8	no	—
84	#10596831	—	4	8	no	—
85	C₂²×C₇₃	—	2	4	no	—
86	C₂²×C₇₃	—	2	4	no	—
87	C₂²×C₇₃	—	2	4	no	—
88	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
89	C₂²×C₇₃	#382	1	1	yes	characteristic
90	C₂²×C₇₃	#384	1	1	yes	characteristic
91	C₂²×C₇₃	#389	1	4	yes	—
92	C₂²×C₇₃	C₂×#108	1	4	yes	—
93	C₄×C₇₃	—	2	4	no	—
94	C₄×C₇₃	#341	1	8	yes	complemented
95	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
96	C₄×C₇₃	#380	1	2	yes	—
97	C₄×C₇₃	#390	1	2	yes	—
98	C₄×C₇₃	#391	1	2	yes	—
99	#10844	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	4	no	—
101	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
102	C₂×C₇₃	C₂×#341	1	2	yes	—
103	C₂×C₇₃	#1837	1	2	yes	—
104	C₂×C₇₃	#1848	1	2	yes	—
105	C₂×#423	—	73	2	no	—
106	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic
107	C₂×#341	—	73	4	no	—
108	C₇₃	#10844	1	1	yes	characteristic, complemented, Sylow
109	C₂²×M₁₆	—	73	4	no	—
110	C₂³×C₈	C₂×D₇₃	1	2	yes	—
111	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
112	#423	—	73	8	no	—
113	#423	—	146	4	no	—
114	C₂×C₄⋊C₈	—	146	4	no	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	Dic₇₃	1	2	yes	—
117	C₂×C₄×C₈	Dic₇₃	1	2	yes	—
118	#341	—	146	8	no	—
119	C₂³×C₄	Dic₁₄₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
121	C₂×M₁₆	—	73	8	no	—
122	C₂×M₁₆	—	146	8	no	—
123	C₂×M₁₆	—	146	8	no	—
124	C₂×M₁₆	—	146	4	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	146	4	no	—
128	C₂²×C₈	Dic₁₄₆	1	4	yes	—
129	C₂²×C₈	D₂₉₂	1	4	yes	—
130	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
131	C₂×C₄²	C₄×D₇₃	1	4	yes	—
132	C₂×C₄²	D₂₉₂	1	2	yes	—
133	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₇₃	1	4	yes	—
136	C₄⋊C₈	—	146	16	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	#11924007	1	8	yes	—
142	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
147	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
148	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
150	C₂²×C₄	#11924028	1	4	yes	—
151	M₁₆	—	292	8	no	—
152	M₁₆	—	146	16	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	292	4	no	—
158	C₂×C₈	—	292	4	no	—
159	C₂×C₈	—	146	8	no	—
160	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	—
161	C₄²	C₄×Dic₇₃	1	2	yes	—
162	C₄²	C₄×Dic₇₃	1	2	yes	—
163	C₄²	#11924005	1	8	yes	—
164	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
165	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
166	C₂³	—	2	4	no	—
167	C₂³	C₂×#11924007	1	2	yes	—
168	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
169	C₂³	#14556583	1	2	yes	—
170	C₂³	#14556591	1	2	yes	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	#14556498	1	4	yes	—
174	C₂×C₄	#14556498	1	4	yes	—
175	C₂×C₄	#14556527	1	8	yes	—
176	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
177	C₂×C₄	C₂×#11924005	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
180	C₂×C₄	#14556550	1	2	yes	—
181	C₂×C₄	C₄×D₂₉₂	1	4	yes	—
182	C₂×C₄	C₄×Dic₁₄₆	1	4	yes	—
183	C₂×C₄	#14556565	1	4	yes	—
184	C₂×C₄	#14556570	1	4	yes	—
185	C₂×C₄	#14556579	1	2	yes	—
186	C₂×C₄	#14556588	1	1	yes	characteristic
187	C₂×C₄	#14556591	1	1	yes	characteristic
188	C₈	—	2	8	no	—
189	C₈	—	292	8	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×#14556498	1	1	yes	characteristic
194	C₂²	#17988453	1	1	yes	characteristic
195	C₂²	#17988457	1	1	yes	characteristic
196	C₂²	#17988485	1	4	yes	—
197	C₂²	C₂×#14556527	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	#17988352	1	8	yes	complemented
200	C₄	C₄×C₄⋊Dic₇₃	1	2	yes	—
201	C₄	#17988447	1	2	yes	—
202	C₄	#17988489	1	2	yes	—
203	C₄	#17988493	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	#22608161	1	1	yes	characteristic
206	C₂	C₂×#17988352	1	2	yes	—
207	C₂	#22610069	1	2	yes	—
208	C₂	#22610103	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	#10844
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
94. C₄×C₇₃	#341	semidirect
108. C₇₃	#10844	semidirect
199. C₄	#17988352	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
C₂×#17988352	2
C₂²×C₄×C₈×C₇₃	2
#10844	73