Subgroups — #34009702

Order 18688.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626288	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22624846	C₂	1	2	yes	maximal, complemented
4	#22624843	C₂	1	2	yes	maximal, complemented
5	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924031	C₄	1	2	yes	—
7	C₄⋊#11924031	C₄	1	2	yes	—
8	C₂²⋊#11924031	C₄	1	2	yes	—
9	C₄×#11924031	C₄	1	2	yes	—
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	C₄×#11924031	C₂²	1	2	yes	—
12	C₄×#11924031	C₂²	1	1	yes	characteristic
13	#17988640	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	#17988308	C₄	1	2	yes	—
16	C₈×#10596831	C₂²	1	2	yes	—
17	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924031	—	2	4	no	—
19	C₂×#11924031	C₂×C₄	1	2	yes	—
20	C₂×#11924031	C₂×C₄	1	4	yes	—
21	C₂×#11924031	C₂×C₄	1	2	yes	—
22	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#10596831	C₈	1	2	yes	—
25	C₂²⋊#10596831	C₈	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₄×#10596831	C₂×C₄	1	2	yes	—
30	C₄×#10596831	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	C₈	1	2	yes	—
32	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#97×C₇₃	C₈	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	#11924031	—	2	2	no	—
36	#11924031	—	2	4	no	—
37	#11924031	M₁₆	1	4	yes	complemented
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	C₄²	1	1	yes	characteristic
41	C₂×#10596831	C₄²	1	1	yes	characteristic
42	C₂×#10596831	C₂×C₈	1	4	yes	—
43	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
44	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	4	no	—
49	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
50	C₄²×C₇₃	C₄²	1	1	yes	characteristic
51	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₇₃	—	2	4	no	—
66	#10596831	—	2	1	no	—
67	#10596831	—	2	1	no	—
68	#10596831	—	2	4	no	—
69	#10596831	C₂×M₁₆	1	2	yes	—
70	#10596831	#131	1	2	yes	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	#346	1	2	yes	—
74	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	#346	1	2	yes	—
78	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
79	C₄×C₇₃	#405	1	1	yes	characteristic
80	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
81	C₄×C₇₃	#504	1	2	yes	—
82	#11408	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	2	no	—
84	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
86	C₂×C₇₃	#2108	1	1	yes	characteristic
87	#2162	—	73	1	no	—
88	C₂×C₄×C₁₆	—	73	1	no	—
89	C₈×M₁₆	—	73	1	no	—
90	#1312	—	73	2	no	—
91	#1311	—	73	2	no	—
92	C₇₃	#11408	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	73	2	no	—
94	#431	—	73	1	no	—
95	C₄×M₁₆	#64408	1	1	yes	characteristic
96	C₂×C₄×C₈	—	73	1	no	—
97	C₄⋊C₁₆	—	73	2	no	—
98	C₂²⋊C₁₆	—	73	2	no	—
99	C₄×C₁₆	—	73	1	no	—
100	C₄×C₁₆	—	73	2	no	—
101	C₄×C₁₆	—	73	1	no	—
102	C₄×C₁₆	—	73	2	no	—
103	#322	—	73	2	no	—
104	C₈²	—	73	2	no	—
105	C₂×M₁₆	#10596833	1	2	yes	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
109	C₂×C₁₆	—	73	2	no	—
110	C₂×C₁₆	—	73	2	no	—
111	C₂×C₁₆	—	73	4	no	—
112	C₂×C₁₆	—	146	4	no	—
113	C₄⋊C₈	—	73	2	no	—
114	C₂²⋊C₈	—	73	2	no	—
115	#97	#10596833	1	2	yes	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	2	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	C₂×#64408	1	2	yes	—
122	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
123	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
124	C₂²×C₄	#11924036	1	1	yes	characteristic
125	M₁₆	—	2	4	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	4	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	146	2	no	—
132	C₂×C₈	—	146	2	no	—
133	C₂×C₈	C₂×#10596833	1	4	yes	—
134	C₄²	C₄×#64408	1	1	yes	characteristic
135	C₄²	C₂×#10596833	1	1	yes	characteristic
136	C₄²	#11924036	1	1	yes	characteristic
137	C₄²	C₂²×#64408	1	1	yes	characteristic
138	C₁₆	—	73	4	no	—
139	C₁₆	—	146	4	no	—
140	C₁₆	—	146	2	no	—
141	C₂³	C₄×#10596833	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
146	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
147	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
148	C₂×C₄	#14556665	1	4	yes	—
149	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
150	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
151	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	73	2	no	—
154	C₈	—	73	2	no	—
155	C₈	—	146	4	no	—
156	C₈	—	146	1	no	—
157	C₈	—	146	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#17989475	1	2	yes	—
161	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#17989475	1	2	yes	—
165	C₄	M₁₆×#64408	1	1	yes	characteristic
166	C₄	#17989553	1	1	yes	characteristic
167	C₄	C₂×#14556665	1	2	yes	—
168	C₄	#17989646	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	M₁₆×#10596833	1	1	yes	characteristic
171	C₂	C₂×#17989475	1	1	yes	characteristic
172	C₂	#22626142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₄×M₁₆×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#11408
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22624846	C₂	semidirect
4. #22624843	C₂	semidirect
37. #11924031	M₁₆	semidirect
92. C₇₃	#11408	semidirect

Maximal subgroups

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

subgroup	index
#22626288	2
C₂×C₄×#11924031	2
#22624846	2
#22624843	2
M₁₆×#10596831	2
#11408	73