Subgroups — #34012633

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989509	C₂	1	2	yes	maximal
2	#22626137	C₂	1	2	yes	maximal
3	#22624975	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₂²	1	4	yes	—
6	#17989509	C₄	1	8	yes	complemented
7	#17989476	—	2	2	no	—
8	#17989473	—	2	2	no	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
16	#14556665	—	2	4	no	—
17	#14556665	C₂×C₄	1	8	yes	—
18	C₂×#11924031	—	2	4	no	—
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	—	2	2	no	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	—	2	2	no	—
25	C₈×Dic₇₃	—	2	2	no	—
26	C₈×Dic₇₃	C₈	1	4	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
28	C₈×Dic₇₃	D₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
31	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₈	1	4	yes	—
36	#11924031	—	2	8	no	—
37	C₂²×Dic₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	M₁₆	1	2	yes	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	C₂×C₈	1	2	yes	—
42	C₂×#10596831	C₂²×C₄	1	2	yes	—
43	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
44	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
45	C₂×#10596831	#41	1	2	yes	—
46	C₄×Dic₇₃	—	2	8	no	—
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	C₄²	1	2	yes	—
49	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₇₃	—	2	1	no	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄²	1	2	yes	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	C₂×C₈	1	2	yes	—
63	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	4	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	4	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	2	4	no	—
75	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
76	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
77	C₈×C₇₃	C₄×C₈	1	4	yes	—
78	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
80	#10596831	—	2	2	no	—
81	#10596831	C₄×D₄	1	2	yes	—
82	#10596831	C₄×Q₈	1	2	yes	—
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₇₃	#431	1	1	yes	characteristic
87	C₄×C₇₃	—	2	4	no	—
88	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₇₃	#431	1	1	yes	characteristic
92	Dic₇₃	—	2	4	no	—
93	Dic₇₃	—	2	2	no	—
94	Dic₇₃	#364	1	4	yes	complemented
95	C₄×#364	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	—	2	1	no	—
98	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₇₃	C₂×#364	1	2	yes	—
100	C₂×#364	—	73	2	no	—
101	C₄×M₃₂	—	73	2	no	—
102	C₄²×C₈	—	73	1	no	—
103	#1376	—	73	2	no	—
104	C₇₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	73	4	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	#64408	1	1	yes	characteristic
112	C₄³	—	73	1	no	—
113	#364	—	73	8	no	—
114	#346	—	146	2	no	—
115	C₄×C₁₆	—	146	2	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
120	C₂×C₄²	—	73	1	no	—
121	C₂×C₄²	—	73	1	no	—
122	C₂×C₄²	—	146	2	no	—
123	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
124	M₃₂	—	73	8	no	—
125	M₃₂	—	146	4	no	—
126	C₂×C₁₆	—	146	4	no	—
127	C₄×C₈	—	73	4	no	—
128	C₄×C₈	—	73	1	no	—
129	C₄×C₈	—	73	1	no	—
130	C₄×C₈	—	73	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₄×C₈	—	73	4	no	—
133	C₄×C₈	—	146	2	no	—
134	C₄×C₈	—	146	2	no	—
135	C₄×C₈	#10596833	1	4	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	73	2	no	—
138	C₂²×C₄	—	146	1	no	—
139	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
140	C₂×C₈	—	73	2	no	—
141	C₂×C₈	—	73	1	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	73	2	no	—
144	C₂×C₈	—	73	2	no	—
145	C₂×C₈	—	146	1	no	—
146	C₂×C₈	—	146	1	no	—
147	C₂×C₈	C₄×#64408	1	2	yes	—
148	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
149	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
150	C₂×C₈	C₂×#10596833	1	2	yes	—
151	C₂×C₈	C₂×#10596833	1	2	yes	—
152	C₂×C₈	#11924036	1	2	yes	—
153	C₂×C₈	#11924038	1	2	yes	—
154	C₄²	—	2	1	no	—
155	C₄²	—	73	2	no	—
156	C₄²	—	73	1	no	—
157	C₄²	—	73	1	no	—
158	C₄²	—	73	2	no	—
159	C₄²	—	146	8	no	—
160	C₄²	—	146	1	no	—
161	C₄²	C₂×#10596833	1	2	yes	—
162	C₁₆	—	146	8	no	—
163	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	73	2	no	—
167	C₂×C₄	—	73	2	no	—
168	C₂×C₄	—	146	4	no	—
169	C₂×C₄	—	146	4	no	—
170	C₂×C₄	—	146	2	no	—
171	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
173	C₂×C₄	#14556670	1	1	yes	characteristic
174	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
175	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
176	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
177	C₈	—	73	2	no	—
178	C₈	—	73	2	no	—
179	C₈	—	146	2	no	—
180	C₈	C₄×#10596833	1	4	yes	—
181	C₈	C₄⋊#10596833	1	2	yes	—
182	C₈	C₄⋊#10596833	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₄×#11924038	1	1	yes	characteristic
185	C₂²	#17989588	1	1	yes	characteristic
186	C₂²	#17989603	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	73	4	no	—
189	C₄	—	146	4	no	—
190	C₄	—	146	2	no	—
191	C₄	C₄×C₄⋊#64408	1	1	yes	characteristic
192	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
193	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
194	C₄	#17989606	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×C₄⋊#10596833	1	1	yes	characteristic
198	C₂	#22626270	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₄×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989509	C₄	semidirect
94. Dic₇₃	#364	semidirect
104. C₇₃	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989509	2
#22626137	2
#22624975	2
C₄×C₈×Dic₇₃	2
C₄×#364	73