Subgroups — #34030433

Order 18688.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	6	yes	maximal, complemented
2	Q₈×#11924031	C₂	1	2	yes	maximal, complemented
3	#22626292	C₂	1	3	yes	maximal
4	#22626288	C₂	1	3	yes	maximal
5	C₂×Q₈×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11924031	C₂²	1	6	yes	—
7	C₄⋊#11924031	C₂²	1	6	yes	—
8	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
9	C₂²⋊#11924031	C₂²	1	3	yes	—
10	#17989475	C₂²	1	6	yes	—
11	C₄×#11924031	C₂²	1	6	yes	—
12	Q₈×#10596831	C₄	1	3	yes	—
13	Q₈×#10596831	C₄	1	1	yes	characteristic
14	Q₈×#10596831	C₂²	1	1	yes	characteristic
15	Q₈×#10596831	C₂²	1	3	yes	—
16	C₂×C₄⋊#10596831	C₄	1	3	yes	—
17	C₂×C₄×#10596831	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
19	C₂×#11924031	C₂³	1	2	yes	—
20	C₂×#11924031	C₂³	1	6	yes	—
21	C₂²×#10596831	C₂×C₄	1	3	yes	—
22	C₂²×#10596831	C₂³	1	1	yes	characteristic
23	C₄⋊#10596831	C₂×C₄	1	3	yes	—
24	C₄⋊#10596831	C₂×C₄	1	6	yes	—
25	C₄⋊#10596831	C₂×C₄	1	3	yes	—
26	C₄×#10596831	C₂×C₄	1	3	yes	—
27	C₄×#10596831	C₂×C₄	1	3	yes	—
28	C₄×#10596831	C₂³	1	3	yes	—
29	C₄×#10596831	C₂³	1	3	yes	—
30	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
31	C₄×Q₈×C₇₃	C₈	1	3	yes	—
32	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₇₃	C₂×C₄	1	3	yes	—
34	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₃	C₈	1	3	yes	—
36	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
37	#11924031	—	2	2	no	—
38	#11924031	—	2	6	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	4	3	no	—
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₂×#10596831	C₂²×C₄	1	3	yes	—
43	C₂×#10596831	C₂²×C₄	1	3	yes	—
44	C₂×#10596831	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₇₃	—	2	4	no	—
46	C₂×Q₈×C₇₃	C₂×C₈	1	3	yes	—
47	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₇₃	C₂×C₈	1	6	yes	—
52	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
53	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
54	C₄²×C₇₃	C₂×C₈	1	3	yes	—
55	C₄²×C₇₃	C₂×C₈	1	3	yes	—
56	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
57	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
58	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₇₃	—	2	6	no	—
60	Q₈×C₇₃	M₃₂	1	4	yes	—
61	C₂×C₄×C₇₃	—	2	6	no	—
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	—	2	1	no	—
64	C₂×C₄×C₇₃	—	2	6	no	—
65	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
71	#10596831	—	2	1	no	—
72	#10596831	—	2	3	no	—
73	#10596831	—	2	3	no	—
74	#10596831	2₋¹⁺⁴	1	2	yes	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	—	2	1	no	—
77	C₂²×C₇₃	#504	1	2	yes	—
78	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
79	C₄×C₇₃	—	2	3	no	—
80	C₄×C₇₃	—	2	3	no	—
81	C₄×C₇₃	—	2	3	no	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	C₂×M₃₂	1	6	yes	—
84	C₄×C₇₃	#519	1	1	yes	characteristic
85	C₄×C₇₃	#568	1	1	yes	characteristic
86	#33498	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	—	2	2	no	—
88	C₂×C₇₃	#2982	1	1	yes	characteristic
89	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
90	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
91	C₂×C₈×Q₈	—	73	1	no	—
92	#2183	—	73	6	no	—
93	Q₈×C₁₆	—	73	2	no	—
94	#2163	—	73	3	no	—
95	#2162	—	73	3	no	—
96	C₇₃	#33498	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
98	C₈×Q₈	—	73	3	no	—
99	C₈×Q₈	—	73	1	no	—
100	C₈×Q₈	—	73	1	no	—
101	C₈×Q₈	—	73	3	no	—
102	C₂×C₄⋊C₈	—	73	3	no	—
103	C₂×C₄×C₈	—	73	3	no	—
104	C₄⋊C₁₆	—	73	6	no	—
105	C₄⋊C₁₆	—	73	6	no	—
106	C₂²⋊C₁₆	—	73	1	no	—
107	C₂²⋊C₁₆	—	73	3	no	—
108	#346	—	73	6	no	—
109	C₄×C₁₆	—	73	6	no	—
110	C₂²×Q₈	#10596833	1	1	yes	characteristic
111	C₂²×C₈	—	73	3	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₄×Q₈	#10596833	1	3	yes	—
114	C₄×Q₈	#10596833	1	1	yes	characteristic
115	C₄×Q₈	C₂×#64408	1	3	yes	—
116	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
117	C₂×C₄⋊C₄	#10596833	1	3	yes	—
118	C₂×C₄²	C₂×#64408	1	3	yes	—
119	C₂×C₁₆	—	73	2	no	—
120	C₂×C₁₆	—	73	6	no	—
121	C₄⋊C₈	—	73	3	no	—
122	C₄⋊C₈	—	73	6	no	—
123	C₄⋊C₈	—	73	3	no	—
124	C₄×C₈	—	73	3	no	—
125	C₄×C₈	—	73	3	no	—
126	C₄×C₈	—	73	3	no	—
127	C₄×C₈	—	73	3	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	C₂×#10596833	1	3	yes	—
130	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
131	C₂²×C₄	C₂×#10596833	1	3	yes	—
132	C₂²×C₄	C₂×#10596833	1	3	yes	—
133	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
134	C₂×C₈	—	292	3	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	3	no	—
138	C₂×C₈	—	73	3	no	—
139	C₂×C₈	—	146	2	no	—
140	C₄⋊C₄	C₂×#10596833	1	6	yes	—
141	C₄⋊C₄	C₂×#10596833	1	3	yes	—
142	C₄⋊C₄	C₂×#10596833	1	3	yes	—
143	C₄²	C₂×#10596833	1	3	yes	—
144	C₄²	C₂×#10596833	1	3	yes	—
145	C₄²	C₂²×#64408	1	3	yes	—
146	C₄²	C₂²×#64408	1	3	yes	—
147	C₁₆	—	146	2	no	—
148	C₁₆	—	146	6	no	—
149	C₂³	C₂²×#10596833	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#14556665	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
157	C₂×C₄	C₂²×#10596833	1	3	yes	—
158	C₂×C₄	C₂²×#10596833	1	3	yes	—
159	C₂×C₄	C₂²×#10596833	1	3	yes	—
160	C₂×C₄	C₂²×#10596833	1	3	yes	—
161	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
162	C₈	—	73	2	no	—
163	C₈	—	146	1	no	—
164	C₈	—	146	3	no	—
165	C₈	—	146	3	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#17989645	1	2	yes	—
169	C₂²	C₂³×#10596833	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂×#14556665	1	6	yes	—
175	C₄	#17989655	1	1	yes	characteristic
176	C₄	#17989667	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#22626824	1	1	yes	characteristic
179	C₂	C₂²×#14556665	1	1	yes	characteristic
180	C₂	C₂×#17989645	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₇₃	C₄
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	#33498
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. Q₈×#11924031	C₂	semidirect
96. C₇₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
Q₈×#11924031	2
#22626292	2
#22626288	2
C₂×Q₈×#10596831	2
#33498	73