Subgroups — #34013267

Order 18688.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
2	#22624969	C₂	1	4	yes	maximal
3	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	C₄⋊#11924031	—	2	4	no	—
6	C₂×C₄×#10596831	C₄	1	2	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄³×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	4	no	—
12	C₂×#11924031	—	2	8	no	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂×C₄	1	2	yes	—
15	C₂²×#10596831	C₂³	1	1	yes	characteristic
16	C₄×#10596831	—	2	4	no	—
17	C₄×#10596831	—	2	4	no	—
18	C₄×#10596831	—	2	2	no	—
19	C₄×#10596831	D₄	1	2	yes	—
20	C₄×#10596831	Q₈	1	2	yes	—
21	C₂×C₄²×C₇₃	C₈	1	2	yes	—
22	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	#11924031	—	4	4	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	—	2	2	no	—
30	C₂×#10596831	C₄⋊C₄	1	2	yes	—
31	C₂×#10596831	M₁₆	1	4	yes	—
32	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
33	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
34	C₂×#10596831	#41	1	4	yes	—
35	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	—	2	4	no	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	—	2	2	no	—
44	C₄²×C₇₃	C₁₆	1	4	yes	complemented
45	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₇₃	M₁₆	1	2	yes	—
47	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
58	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#117	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
63	#10596831	—	2	2	no	—
64	#10596831	—	2	4	no	—
65	#10596831	#124	1	2	yes	—
66	#10596831	#125	1	2	yes	—
67	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₇₃	#431	1	2	yes	—
69	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
71	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	—	2	4	no	—
74	C₄×C₇₃	—	2	4	no	—
75	C₄×C₇₃	C₄⋊C₁₆	1	4	yes	—
76	C₄×C₇₃	#382	1	1	yes	characteristic
77	C₄×C₇₃	#423	1	1	yes	characteristic
78	C₄×C₇₃	#432	1	2	yes	—
79	#12879	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1831	1	1	yes	characteristic
81	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
82	C₂×C₇₃	#2150	1	1	yes	characteristic
83	C₂×C₇₃	#2162	1	2	yes	—
84	C₂×C₇₃	#2163	1	2	yes	—
85	C₂×C₄⋊C₁₆	—	73	2	no	—
86	C₄²×C₈	—	73	1	no	—
87	#1374	—	73	4	no	—
88	C₇₃	#12879	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	73	4	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	—	73	2	no	—
92	C₂×C₄×C₈	—	73	2	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	C₄³	#64408	1	1	yes	characteristic
95	C₄⋊C₁₆	—	146	4	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂×C₄²	#10596833	1	2	yes	—
100	C₂×C₄²	#10596833	1	1	yes	characteristic
101	C₂×C₄²	#10596833	1	1	yes	characteristic
102	C₂×C₄²	C₂×#64408	1	2	yes	—
103	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
104	C₂×C₁₆	—	146	4	no	—
105	C₂×C₁₆	—	146	8	no	—
106	C₄×C₈	—	73	2	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	146	4	no	—
109	C₄×C₈	—	146	4	no	—
110	C₄×C₈	—	146	2	no	—
111	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
112	C₂²×C₄	C₂×#10596833	1	2	yes	—
113	C₂²×C₄	C₂×#10596833	1	2	yes	—
114	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
115	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	4	no	—
119	C₂×C₈	—	73	4	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	#11924031	1	4	yes	complemented
129	C₄²	C₄⋊#64408	1	2	yes	—
130	C₄²	#11924038	1	2	yes	—
131	C₁₆	—	292	4	no	—
132	C₂³	C₂²×#10596833	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊#10596833	1	2	yes	—
139	C₂×C₄	C₄⋊#10596833	1	2	yes	—
140	C₂×C₄	C₂×#11924031	1	4	yes	—
141	C₂×C₄	C₂×#11924031	1	2	yes	—
142	C₂×C₄	#14556665	1	4	yes	—
143	C₂×C₄	#14556665	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₄	#14556670	1	2	yes	—
146	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
147	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
148	C₈	—	73	2	no	—
149	C₈	—	73	2	no	—
150	C₈	—	146	2	no	—
151	C₈	—	146	4	no	—
152	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
153	C₂²	#17989603	1	2	yes	—
154	C₂²	C₂²×#11924031	1	1	yes	characteristic
155	C₂²	C₂×#14556665	1	1	yes	characteristic
156	C₂²	C₂×#14556665	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊#11924031	1	4	yes	—
161	C₄	#17989531	1	1	yes	characteristic
162	C₄	#17989592	1	1	yes	characteristic
163	C₄	#17989609	1	2	yes	—
164	C₂	#22625846	1	1	yes	characteristic
165	C₂	C₂×C₄⋊#11924031	1	1	yes	characteristic
166	C₂	#22626263	1	1	yes	characteristic
167	C₂	#22626288	1	2	yes	—
168	C₂	#22626292	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄³×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12879
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
44. C₄²×C₇₃	C₁₆	semidirect
88. C₇₃	#12879	semidirect
128. C₄²	#11924031	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924031	2
#22624969	2
C₄²×#10596831	2
#12879	73