Subgroups — #34030429

Order 18688.

486 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	#22626346	C₂	1	8	yes	maximal, complemented
2	#22626292	C₂	1	4	yes	maximal
3	#22626288	C₂	1	2	yes	maximal
4	C₂×Q₈×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11924031	C₂²	1	4	yes	—
6	C₄⋊#11924031	C₂²	1	8	yes	—
7	C₂²⋊#11924031	C₂²	1	4	yes	—
8	#17989475	C₂²	1	8	yes	—
9	C₄×#11924031	C₂²	1	4	yes	—
10	Q₈×#10596831	C₄	1	4	yes	—
11	Q₈×#10596831	C₂²	1	4	yes	—
12	C₂×C₄⋊#10596831	C₄	1	2	yes	—
13	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10596831	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924031	C₂³	1	8	yes	—
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	2	yes	—
20	C₂²×#10596831	C₂³	1	1	yes	characteristic
21	C₄⋊#10596831	C₂×C₄	1	8	yes	—
22	C₄⋊#10596831	C₂×C₄	1	4	yes	—
23	C₄×#10596831	C₂×C₄	1	2	yes	—
24	C₄×#10596831	C₂×C₄	1	4	yes	—
25	C₄×#10596831	C₂³	1	2	yes	—
26	C₄×#10596831	C₂³	1	4	yes	—
27	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
28	C₄×Q₈×C₇₃	C₈	1	4	yes	—
29	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₇₃	C₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924031	—	2	8	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	—	4	1	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
39	C₂×#10596831	C₂²×C₄	1	2	yes	—
40	C₂×#10596831	C₂²×C₄	1	4	yes	—
41	C₂×#10596831	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₇₃	—	2	4	no	—
43	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
50	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
51	C₄²×C₇₃	C₂×C₈	1	4	yes	—
52	C₄²×C₇₃	C₂×C₈	1	2	yes	—
53	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
54	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
55	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	Q₈×C₇₃	—	2	8	no	—
57	C₂×C₄×C₇₃	—	2	4	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
63	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
67	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
69	#10596831	—	2	1	no	—
70	#10596831	—	2	2	no	—
71	#10596831	—	2	4	no	—
72	#10596831	2₋¹⁺⁴	1	2	yes	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
76	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
77	C₄×C₇₃	—	2	4	no	—
78	C₄×C₇₃	—	2	4	no	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	C₂×M₃₂	1	4	yes	—
82	C₄×C₇₃	#519	1	1	yes	characteristic
83	C₄×C₇₃	#568	1	1	yes	characteristic
84	#33497	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	2	no	—
86	C₂×C₇₃	#2982	1	1	yes	characteristic
87	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
88	C₂×C₇₃	#3407	1	1	yes	characteristic
89	C₂×C₈×Q₈	—	73	1	no	—
90	#2183	—	73	8	no	—
91	#2163	—	73	4	no	—
92	#2162	—	73	2	no	—
93	C₇₃	#33497	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
95	C₈×Q₈	—	73	4	no	—
96	C₈×Q₈	—	73	4	no	—
97	C₂×C₄⋊C₈	—	73	2	no	—
98	C₂×C₄⋊C₈	—	73	1	no	—
99	C₂×C₄×C₈	—	73	1	no	—
100	C₂×C₄×C₈	—	73	2	no	—
101	C₄⋊C₁₆	—	73	4	no	—
102	C₄⋊C₁₆	—	73	8	no	—
103	C₂²⋊C₁₆	—	73	4	no	—
104	#346	—	73	8	no	—
105	C₄×C₁₆	—	73	4	no	—
106	C₂²×Q₈	#10596833	1	1	yes	characteristic
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	—	73	2	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₄×Q₈	#10596833	1	4	yes	—
111	C₄×Q₈	C₂×#64408	1	4	yes	—
112	C₂×C₄⋊C₄	#10596833	1	2	yes	—
113	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
114	C₂×C₄²	C₂×#64408	1	2	yes	—
115	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
116	C₂×C₁₆	—	73	8	no	—
117	C₄⋊C₈	—	73	8	no	—
118	C₄⋊C₈	—	73	4	no	—
119	C₄×C₈	—	73	2	no	—
120	C₄×C₈	—	73	4	no	—
121	C₄×C₈	—	73	2	no	—
122	C₄×C₈	—	73	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×#10596833	1	4	yes	—
125	C₂²×C₄	C₂×#10596833	1	2	yes	—
126	C₂²×C₄	C₂×#10596833	1	2	yes	—
127	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
128	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
130	C₂×C₈	—	292	1	no	—
131	C₂×C₈	—	292	2	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	2	no	—
135	C₂×C₈	—	73	4	no	—
136	C₂×C₈	—	146	2	no	—
137	C₄⋊C₄	C₂×#10596833	1	8	yes	—
138	C₄⋊C₄	C₂×#10596833	1	4	yes	—
139	C₄²	C₂×#10596833	1	4	yes	—
140	C₄²	C₂×#10596833	1	2	yes	—
141	C₄²	C₂²×#64408	1	4	yes	—
142	C₄²	C₂²×#64408	1	2	yes	—
143	C₁₆	—	146	8	no	—
144	C₂³	C₂²×#10596833	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	#14556665	1	4	yes	—
152	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
153	C₂×C₄	C₂²×#10596833	1	4	yes	—
154	C₂×C₄	C₂²×#10596833	1	2	yes	—
155	C₂×C₄	C₂²×#10596833	1	4	yes	—
156	C₂×C₄	C₂²×#10596833	1	2	yes	—
157	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
158	C₈	—	73	2	no	—
159	C₈	—	146	1	no	—
160	C₈	—	146	2	no	—
161	C₈	—	146	4	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#14556665	1	2	yes	—
165	C₂²	C₂³×#10596833	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×#14556665	1	4	yes	—
171	C₄	#17989655	1	1	yes	characteristic
172	C₄	#17989667	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#22626824	1	1	yes	characteristic
175	C₂	C₂²×#14556665	1	1	yes	characteristic
176	C₂	#22626898	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	#33497
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
93. C₇₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
#22626292	2
#22626288	2
C₂×Q₈×#10596831	2
#33497	73