Subgroups — #32977220

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609800	C₂	1	2	yes	maximal
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	#22607772	C₂	1	2	yes	maximal
5	C₄×#107×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×#14556502	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	#17988342	—	2	2	no	—
11	#17988340	C₄	1	4	yes	complemented
12	C₈×#10596831	C₄	1	4	yes	complemented
13	C₂×#107×C₇₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	#340×C₇₃	C₄	1	2	yes	—
17	C₂²×#10596831	—	2	2	no	—
18	C₂²×#10596831	—	2	1	no	—
19	C₂²×#10596831	D₄	1	1	yes	characteristic
20	C₂²×#10596831	D₄	1	1	yes	characteristic
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	C₂×C₄	1	4	yes	—
23	#14556502	—	4	4	no	—
24	C₄×#10596831	—	2	1	no	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	Q₈	1	2	yes	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₃	—	2	2	no	—
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	#107×C₇₃	—	2	4	no	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	#95×C₇₃	—	2	2	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	—	2	4	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	—	4	2	no	—
44	C₂×#10596831	C₄⋊C₄	1	2	yes	—
45	C₂×#10596831	C₄⋊C₄	1	2	yes	—
46	C₂×#10596831	D₈	1	2	yes	—
47	C₂×#10596831	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₇₃	—	2	4	no	—
49	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	C₄²	1	2	yes	—
54	C₂×C₈×C₇₃	C₄²	1	2	yes	—
55	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
56	C₄⋊C₄×C₇₃	—	2	4	no	—
57	C₄⋊C₄×C₇₃	—	2	4	no	—
58	C₄²×C₇₃	—	4	2	no	—
59	C₄²×C₇₃	—	4	2	no	—
60	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₃	#41	1	1	yes	characteristic
64	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	4	4	no	—
67	C₂×C₄×C₇₃	—	4	4	no	—
68	C₂×C₄×C₇₃	—	4	4	no	—
69	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	#131	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
81	C₈×C₇₃	—	2	2	no	—
82	C₈×C₇₃	#97	1	4	yes	complemented
83	#10596831	—	2	4	no	—
84	#10596831	—	2	2	no	—
85	#10596831	—	4	4	no	—
86	#10596831	#107	1	4	yes	complemented
87	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₇₃	#436	1	2	yes	—
89	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
90	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
91	C₂²×C₇₃	#443	1	2	yes	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	8	4	no	—
95	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₇₃	#405	1	1	yes	characteristic
97	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
98	C₄×C₇₃	#427	1	1	yes	characteristic
99	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
100	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₇₃	#446	1	2	yes	—
102	#8282	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	#1589	1	2	yes	—
104	C₂×C₇₃	#1591	1	2	yes	—
105	C₂×C₇₃	#1770	1	1	yes	characteristic
106	C₂×C₇₃	#1772	1	1	yes	characteristic
107	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
108	C₄×#107	D₇₃	1	1	yes	characteristic
109	#1770	—	73	2	no	—
110	C₂×#335	—	73	1	no	—
111	C₂×C₈²	—	73	1	no	—
112	#1279	—	73	2	no	—
113	C₇₃	#8282	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	Dic₇₃	1	2	yes	—
115	C₂×C₄⋊C₈	—	73	2	no	—
116	C₂×#97	—	146	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
121	#340	Dic₇₃	1	2	yes	—
122	#336	—	146	2	no	—
123	#335	—	73	4	no	—
124	C₈²	—	73	4	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	—	146	2	no	—
128	C₂²×C₈	—	146	1	no	—
129	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	#107	—	2	4	no	—
135	C₄⋊C₈	—	73	4	no	—
136	C₄⋊C₈	—	146	2	no	—
137	#97	—	292	4	no	—
138	C₄×C₈	—	73	2	no	—
139	C₄×C₈	—	73	2	no	—
140	C₄×C₈	—	73	2	no	—
141	C₄×C₈	—	146	1	no	—
142	C₄×C₈	C₄×D₇₃	1	2	yes	—
143	C₄×C₈	C₄×D₇₃	1	2	yes	—
144	#95	—	2	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
148	C₂²×C₄	#11924027	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	292	2	no	—
151	C₂×C₈	—	292	2	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	2	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	146	4	no	—
157	C₂×C₈	—	146	4	no	—
158	C₂×C₈	—	146	2	no	—
159	C₂×C₈	—	146	1	no	—
160	C₂×C₈	—	146	1	no	—
161	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
162	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
163	C₂×C₈	#11924012	1	4	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
170	C₄²	Q₈×D₇₃	1	1	yes	characteristic
171	C₄²	#11924029	1	1	yes	characteristic
172	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
178	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
179	C₂×C₄	#14556549	1	1	yes	characteristic
180	C₂×C₄	#14556553	1	1	yes	characteristic
181	C₂×C₄	#14556556	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924012	1	2	yes	—
184	C₂×C₄	#14556598	1	2	yes	—
185	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
186	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
187	C₂×C₄	#14556610	1	1	yes	characteristic
188	C₂×C₄	#14556614	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	292	4	no	—
191	C₈	—	73	4	no	—
192	C₈	—	146	4	no	—
193	C₈	—	146	2	no	—
194	C₈	#14556502	1	4	yes	complemented
195	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
196	C₂²	#17988665	1	2	yes	—
197	C₂²	D₈×Dic₇₃	1	1	yes	characteristic
198	C₂²	Q₁₆×Dic₇₃	1	1	yes	characteristic
199	C₂²	#17988704	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	C₂×#14556502	1	1	yes	characteristic
204	C₄	#17988543	1	1	yes	characteristic
205	C₄	#107×D₇₃	1	1	yes	characteristic
206	C₄	#17988624	1	1	yes	characteristic
207	C₄	#17988669	1	1	yes	characteristic
208	C₄	#17988681	1	1	yes	characteristic
209	C₄	#17988718	1	2	yes	—
210	C₂	#22608816	1	2	yes	—
211	C₂	#22608828	1	2	yes	—
212	C₂	#22609800	1	1	yes	characteristic
213	C₂	#22609812	1	1	yes	characteristic
214	C₂	#107×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#107×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#107	D₇₃
hypocenter	C₇₃	#8282
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #17988340	C₄	semidirect
12. C₈×#10596831	C₄	semidirect
82. C₈×C₇₃	#97	semidirect
86. #10596831	#107	semidirect
113. C₇₃	#8282	semidirect
194. C₈	#14556502	semidirect

Maximal subgroups

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

subgroup	index
#22609800	2
C₂×#17988340	2
C₂×C₈×#10596831	2
#22607772	2
C₄×#107×C₇₃	2
#8282	73