Subgroups — #34011739

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626261	C₂	1	2	yes	maximal
2	#22624961	C₂	1	2	yes	maximal, complemented
3	#22624960	C₂	1	2	yes	maximal, complemented
4	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	—	2	2	no	—
6	#17989508	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	4	yes	—
8	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₈×#10596831	C₄	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈²×C₇₃	C₄	1	1	yes	characteristic
15	#14556664	—	4	4	no	—
16	C₂×#11924031	—	2	4	no	—
17	C₂×C₈×D₇₃	—	2	1	no	—
18	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
19	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₂³	1	1	yes	characteristic
26	C₈×Dic₇₃	—	2	1	no	—
27	C₈×Dic₇₃	C₈	1	1	yes	characteristic
28	C₈×Dic₇₃	C₈	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924031	—	4	4	no	—
35	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
40	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
41	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
42	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
43	C₈×D₇₃	—	2	4	no	—
44	C₈×D₇₃	—	2	2	no	—
45	C₈×D₇₃	—	2	2	no	—
46	C₈×D₇₃	C₁₆	1	4	yes	complemented
47	C₈×D₇₃	D₈	1	2	yes	—
48	C₈×D₇₃	Q₁₆	1	2	yes	—
49	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
51	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	#107	1	2	yes	—
67	C₄×D₇₃	#108	1	2	yes	—
68	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
69	C₄×D₇₃	M₃₂	1	2	yes	—
70	C₈×C₇₃	—	2	2	no	—
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	—	2	1	no	—
73	C₈×C₇₃	#107	1	2	yes	—
74	C₈×C₇₃	C₂×C₁₆	1	2	yes	complemented
75	#10596831	—	2	2	no	—
76	#10596831	—	2	1	no	—
77	#10596831	—	2	1	no	—
78	#10596831	C₂×C₁₆	1	2	yes	complemented
79	#10596831	C₂×D₈	1	1	yes	characteristic
80	#10596831	C₂×Q₁₆	1	1	yes	characteristic
81	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×D₇₃	#335	1	2	yes	—
83	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
84	C₂×D₇₃	#364	1	2	yes	—
85	C₄×C₇₃	—	2	1	no	—
86	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
87	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
88	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
90	Dic₇₃	—	2	1	no	—
91	Dic₇₃	#335	1	2	yes	—
92	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
93	C₂×#1370	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
95	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
96	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
97	D₇₃	#1370	1	4	yes	complemented
98	C₂×C₄⋊C₁₆	—	73	2	no	—
99	C₂×C₈²	—	73	1	no	—
100	#1370	—	73	2	no	—
101	#1370	—	73	2	no	—
102	C₇₃	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	146	2	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	C₂×C₄×C₈	—	73	1	no	—
107	C₄⋊C₁₆	—	73	4	no	—
108	C₄⋊C₁₆	—	146	2	no	—
109	C₈²	—	73	1	no	—
110	C₈²	—	73	1	no	—
111	C₈²	—	73	1	no	—
112	C₈²	#64408	1	1	yes	characteristic
113	C₂²×C₈	—	73	1	no	—
114	C₂²×C₈	—	73	1	no	—
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	—	73	1	no	—
117	C₂²×C₈	—	146	1	no	—
118	C₂×C₄²	—	73	1	no	—
119	C₂×C₁₆	—	292	4	no	—
120	C₂×C₁₆	—	146	4	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	1	no	—
123	C₄×C₈	—	73	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	—	73	1	no	—
127	C₄×C₈	—	73	1	no	—
128	C₄×C₈	—	146	1	no	—
129	C₄×C₈	C₂×#64408	1	1	yes	characteristic
130	C₄×C₈	C₂×#64408	1	1	yes	characteristic
131	C₄×C₈	C₂×#64408	1	1	yes	characteristic
132	C₂²×C₄	—	73	1	no	—
133	C₂²×C₄	—	73	1	no	—
134	C₂²×C₄	—	73	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	73	4	no	—
141	C₂×C₈	—	73	1	no	—
142	C₂×C₈	—	73	1	no	—
143	C₂×C₈	—	146	2	no	—
144	C₂×C₈	—	146	2	no	—
145	C₂×C₈	—	146	4	no	—
146	C₂×C₈	—	146	1	no	—
147	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
148	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
149	C₂×C₈	#11924036	1	1	yes	characteristic
150	C₂×C₈	#11924036	1	1	yes	characteristic
151	C₄²	—	73	1	no	—
152	C₄²	—	73	1	no	—
153	C₄²	—	73	1	no	—
154	C₄²	C₂²×#64408	1	1	yes	characteristic
155	C₁₆	—	292	4	no	—
156	C₂³	—	73	1	no	—
157	C₂×C₄	—	73	1	no	—
158	C₂×C₄	—	73	1	no	—
159	C₂×C₄	—	73	2	no	—
160	C₂×C₄	—	73	2	no	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	73	2	no	—
163	C₂×C₄	—	73	1	no	—
164	C₂×C₄	—	146	2	no	—
165	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
166	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
167	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	73	2	no	—
172	C₈	—	73	1	no	—
173	C₈	—	73	1	no	—
174	C₈	—	146	2	no	—
175	C₈	—	146	1	no	—
176	C₈	—	146	1	no	—
177	C₈	#14556660	1	2	yes	—
178	C₈	#14556664	1	2	yes	complemented
179	C₂²	—	73	2	no	—
180	C₂²	—	73	2	no	—
181	C₂²	—	73	2	no	—
182	C₂²	#17989588	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	73	2	no	—
185	C₄	—	73	2	no	—
186	C₄	—	146	1	no	—
187	C₄	C₂×#14556660	1	1	yes	characteristic
188	C₄	D₇₃⋊#108	1	1	yes	characteristic
189	C₄	C₂×#14556664	1	1	yes	characteristic
190	C₄	#17989643	1	1	yes	characteristic
191	C₂	—	73	4	no	—
192	C₂	#22625476	1	1	yes	characteristic
193	C₂	#22626261	1	1	yes	characteristic
194	C₂	#22626270	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	C₂×#1370
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #22624961	C₂	semidirect
3. #22624960	C₂	semidirect
46. C₈×D₇₃	C₁₆	semidirect
74. C₈×C₇₃	C₂×C₁₆	semidirect
78. #10596831	C₂×C₁₆	semidirect
97. D₇₃	#1370	semidirect
102. C₇₃	C₂×#1370	semidirect
178. C₈	#14556664	semidirect

Maximal subgroups

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

subgroup	index
#22626261	2
#22624961	2
#22624960	2
C₈²×D₇₃	2
C₂×#1370	73