Subgroups — #34013315

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989508	C₂	1	4	yes	maximal
2	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	#17989508	—	2	8	no	—
6	C₄⋊#11924031	—	2	4	no	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
10	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	—	2	4	no	—
13	C₂×#11924031	—	2	8	no	—
14	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
15	C₂²×#10596831	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
17	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
20	C₄×#10596831	D₄	1	2	yes	—
21	C₄×#10596831	Q₈	1	2	yes	—
22	C₈×Dic₇₃	—	2	4	no	—
23	C₈×Dic₇₃	D₄	1	4	yes	—
24	C₈×Dic₇₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	—	2	2	no	—
28	#11924031	—	4	4	no	—
29	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
30	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
31	C₂×#10596831	—	2	2	no	—
32	C₂×#10596831	C₄⋊C₄	1	2	yes	—
33	C₂×#10596831	C₂×D₄	1	2	yes	—
34	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
35	C₂×#10596831	C₂×Q₈	1	2	yes	—
36	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
37	C₄×Dic₇₃	—	2	2	no	—
38	C₄×Dic₇₃	—	2	4	no	—
39	C₄×Dic₇₃	C₁₆	1	4	yes	complemented
40	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
41	C₄×Dic₇₃	M₁₆	1	4	yes	—
42	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	2	no	—
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
48	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₄²×C₇₃	M₁₆	1	2	yes	—
50	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
63	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
64	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
65	C₂×Dic₇₃	M₃₂	1	4	yes	—
66	C₈×C₇₃	—	2	4	no	—
67	#10596831	—	2	2	no	—
68	#10596831	#127	1	1	yes	characteristic
69	#10596831	C₄⋊Q₈	1	2	yes	—
70	#10596831	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
72	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
75	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	C₄⋊C₁₆	1	4	yes	complemented
78	C₄×C₇₃	#384	1	1	yes	characteristic
79	C₄×C₇₃	#423	1	2	yes	—
80	C₄×C₇₃	#423	1	1	yes	characteristic
81	Dic₇₃	—	2	4	no	—
82	Dic₇₃	C₄⋊C₁₆	1	8	yes	complemented
83	#12898	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	#1842	1	1	yes	characteristic
85	C₂×C₇₃	C₂×C₄⋊C₁₆	1	2	yes	—
86	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
87	C₂×C₇₃	#2150	1	2	yes	—
88	C₂×C₇₃	#2150	1	1	yes	characteristic
89	C₂×C₄⋊C₁₆	—	73	2	no	—
90	C₂×C₄⋊C₁₆	—	73	4	no	—
91	C₄²×C₈	—	73	1	no	—
92	C₇₃	#12898	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	73	4	no	—
94	C₂×C₄×C₈	—	73	2	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	C₂×C₄×C₈	—	73	2	no	—
97	C₂×C₄×C₈	#64408	1	1	yes	characteristic
98	C₄³	—	73	1	no	—
99	C₄⋊C₁₆	—	146	4	no	—
100	C₄⋊C₁₆	—	146	8	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	C₂×#64408	1	2	yes	—
104	C₂×C₄²	—	73	1	no	—
105	C₂×C₄²	—	73	2	no	—
106	C₂×C₄²	—	73	2	no	—
107	C₂×C₄²	—	73	1	no	—
108	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
109	C₂×C₁₆	—	146	4	no	—
110	C₂×C₁₆	—	146	8	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	73	4	no	—
113	C₄×C₈	—	73	4	no	—
114	C₄×C₈	—	73	2	no	—
115	C₄×C₈	—	73	2	no	—
116	C₄×C₈	—	146	4	no	—
117	C₂²×C₄	—	73	2	no	—
118	C₂²×C₄	—	73	2	no	—
119	C₂²×C₄	#11924036	1	1	yes	characteristic
120	C₂²×C₄	#11924036	1	1	yes	characteristic
121	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	C₄⋊#64408	1	4	yes	—
131	C₄²	—	73	4	no	—
132	C₄²	—	73	4	no	—
133	C₄²	—	73	4	no	—
134	C₄²	—	146	4	no	—
135	C₄²	—	146	2	no	—
136	C₄²	C₄⋊#64408	1	2	yes	—
137	C₄²	#11924037	1	2	yes	—
138	C₁₆	—	292	4	no	—
139	C₂³	C₂×#11924036	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	73	4	no	—
142	C₂×C₄	—	73	4	no	—
143	C₂×C₄	—	73	4	no	—
144	C₂×C₄	—	73	4	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	—	146	2	no	—
147	C₂×C₄	#14556658	1	2	yes	—
148	C₂×C₄	#14556658	1	2	yes	—
149	C₂×C₄	#14556664	1	2	yes	—
150	C₂×C₄	#14556666	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
153	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
154	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	73	1	no	—
157	C₈	—	73	2	no	—
158	C₈	—	73	1	no	—
159	C₈	—	146	2	no	—
160	C₂²	#17989588	1	2	yes	—
161	C₂²	C₂×#14556658	1	1	yes	characteristic
162	C₂²	C₂×#14556664	1	1	yes	characteristic
163	C₂²	#17989643	1	2	yes	—
164	C₂²	C₂×#14556666	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	73	8	no	—
167	C₄	—	146	4	no	—
168	C₄	#17989508	1	4	yes	complemented
169	C₄	#17989533	1	1	yes	characteristic
170	C₄	#17989590	1	2	yes	—
171	C₄	#17989592	1	1	yes	characteristic
172	C₂	#22625864	1	1	yes	characteristic
173	C₂	#22626261	1	2	yes	—
174	C₂	C₂×#17989508	1	1	yes	characteristic
175	C₂	#22626264	1	2	yes	—
176	C₂	#22626265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12898
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
39. C₄×Dic₇₃	C₁₆	semidirect
77. C₄×C₇₃	C₄⋊C₁₆	semidirect
82. Dic₇₃	C₄⋊C₁₆	semidirect
92. C₇₃	#12898	semidirect
168. C₄	#17989508	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989508	2
C₂×C₄⋊#11924031	2
C₄×C₈×Dic₇₃	2
#12898	73