Subgroups — #34009729

Order 18688.

260 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	#22626288	C₂	1	1	yes	maximal, characteristic
2	#22626288	C₂	1	1	yes	maximal, characteristic
3	#22624846	C₂	1	1	yes	maximal, characteristic, complemented
4	#22624844	C₂	1	1	yes	maximal, characteristic, complemented
5	#22624843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608215	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#11924031	C₄	1	2	yes	—
9	C₄⋊#11924031	C₄	1	2	yes	—
10	C₂²⋊#11924031	C₄	1	2	yes	—
11	C₂²⋊#11924031	C₄	1	2	yes	—
12	C₄×#11924031	C₂²	1	1	yes	characteristic
13	C₄×#11924031	C₂²	1	1	yes	characteristic
14	C₄×#11924031	C₂²	1	1	yes	characteristic
15	C₄×#11924031	C₂²	1	1	yes	characteristic
16	#17988640	C₄	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	#17988309	C₄	1	1	yes	characteristic
19	#17988309	C₄	1	1	yes	characteristic
20	C₈×#10596831	C₂²	1	1	yes	characteristic
21	C₈×#10596831	C₂²	1	1	yes	characteristic
22	#431×C₇₃	C₄	1	1	yes	characteristic
23	C₂×#11924031	C₂×C₄	1	2	yes	—
24	C₂×#11924031	C₂×C₄	1	2	yes	—
25	C₂×#11924031	C₂×C₄	1	2	yes	—
26	C₂×#11924031	C₂×C₄	1	2	yes	—
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₄⋊#10596831	C₈	1	2	yes	—
30	C₂²⋊#10596831	C₈	1	2	yes	—
31	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₄×#10596831	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₈×C₇₃	C₈	1	2	yes	—
39	C₂²⋊C₈×C₇₃	C₈	1	2	yes	—
40	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#11924031	—	2	2	no	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	2	no	—
45	#11924031	—	2	2	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	C₄²	1	1	yes	characteristic
49	C₂×#10596831	C₄²	1	1	yes	characteristic
50	C₂×#10596831	C₂×C₈	1	2	yes	—
51	C₂×#10596831	C₂×C₈	1	2	yes	—
52	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
59	C₄²×C₇₃	C₄²	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	1	no	—
65	C₂×C₄×C₇₃	—	2	1	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₇₃	—	2	2	no	—
75	C₈×C₇₃	—	2	2	no	—
76	#10596831	—	2	1	no	—
77	#10596831	—	2	1	no	—
78	#10596831	—	2	2	no	—
79	#10596831	—	2	2	no	—
80	#10596831	#131	1	2	yes	—
81	#10596831	#131	1	2	yes	—
82	C₂²×C₇₃	—	2	1	no	—
83	C₂²×C₇₃	—	2	1	no	—
84	C₂²×C₇₃	—	2	1	no	—
85	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	#405	1	1	yes	characteristic
90	C₄×C₇₃	#405	1	1	yes	characteristic
91	C₄×C₇₃	#504	1	2	yes	—
92	C₄×C₇₃	#504	1	2	yes	—
93	#11412	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	4	1	no	—
95	C₂×C₇₃	#1450	1	1	yes	characteristic
96	C₂×C₇₃	#2108	1	1	yes	characteristic
97	C₂×C₇₃	#2108	1	1	yes	characteristic
98	#2162	—	73	1	no	—
99	#2162	—	73	1	no	—
100	#1450	—	73	1	no	—
101	#1312	—	73	1	no	—
102	#1312	—	73	1	no	—
103	#1311	—	73	1	no	—
104	C₈×C₁₆	—	73	1	no	—
105	C₇₃	#11412	1	1	yes	characteristic, complemented, Sylow
106	#431	—	73	1	no	—
107	#431	#64408	1	1	yes	characteristic
108	C₂×C₄×C₈	—	73	1	no	—
109	C₄⋊C₁₆	—	73	2	no	—
110	C₄⋊C₁₆	—	73	2	no	—
111	C₂²⋊C₁₆	—	73	2	no	—
112	C₂²⋊C₁₆	—	73	2	no	—
113	C₄×C₁₆	—	73	1	no	—
114	C₄×C₁₆	—	73	1	no	—
115	C₄×C₁₆	—	73	1	no	—
116	C₄×C₁₆	—	73	1	no	—
117	#322	—	73	1	no	—
118	#322	—	73	1	no	—
119	C₈²	—	73	1	no	—
120	C₈²	—	73	1	no	—
121	C₂²×C₈	—	73	1	no	—
122	C₂²×C₈	—	73	1	no	—
123	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
124	C₂×C₁₆	—	73	2	no	—
125	C₂×C₁₆	—	73	2	no	—
126	C₂×C₁₆	—	73	2	no	—
127	C₂×C₁₆	—	73	2	no	—
128	C₄⋊C₈	—	73	2	no	—
129	C₄⋊C₈	#10596833	1	2	yes	—
130	C₂²⋊C₈	—	73	2	no	—
131	C₂²⋊C₈	#10596833	1	2	yes	—
132	C₄×C₈	—	73	1	no	—
133	C₄×C₈	—	73	1	no	—
134	C₄×C₈	—	73	1	no	—
135	C₄×C₈	—	73	1	no	—
136	C₄×C₈	—	73	1	no	—
137	C₄×C₈	—	73	1	no	—
138	C₄×C₈	C₂×#64408	1	1	yes	characteristic
139	C₄×C₈	C₂×#64408	1	1	yes	characteristic
140	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
141	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
142	C₂²×C₄	#11924036	1	1	yes	characteristic
143	C₂×C₈	—	73	1	no	—
144	C₂×C₈	—	73	1	no	—
145	C₂×C₈	—	73	2	no	—
146	C₂×C₈	—	73	2	no	—
147	C₂×C₈	—	73	1	no	—
148	C₂×C₈	—	73	1	no	—
149	C₂×C₈	—	146	2	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	C₂×#10596833	1	2	yes	—
152	C₂×C₈	C₂×#10596833	1	2	yes	—
153	C₄²	C₄×#64408	1	1	yes	characteristic
154	C₄²	C₂×#10596833	1	1	yes	characteristic
155	C₄²	#11924036	1	1	yes	characteristic
156	C₄²	C₂²×#64408	1	1	yes	characteristic
157	C₁₆	—	146	2	no	—
158	C₁₆	—	146	2	no	—
159	C₁₆	—	146	2	no	—
160	C₁₆	—	146	2	no	—
161	C₂³	C₄×#10596833	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
167	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
168	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
170	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₈	—	73	2	no	—
175	C₈	—	73	2	no	—
176	C₈	—	146	2	no	—
177	C₈	—	146	2	no	—
178	C₈	—	146	1	no	—
179	C₈	—	146	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	#17989550	1	1	yes	characteristic
188	C₄	#17989550	1	1	yes	characteristic
189	C₄	#17989645	1	2	yes	—
190	C₄	#17989646	1	2	yes	—
191	C₂	—	4	1	no	—
192	C₂	#22625134	1	1	yes	characteristic
193	C₂	#22626140	1	1	yes	characteristic
194	C₂	#22626142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#431×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	#431	#64408
hypocenter	C₇₃	#11412
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #22624846	C₂	semidirect
4. #22624844	C₂	semidirect
5. #22624843	C₂	semidirect
6. C₈×#11924031	C₂	semidirect
105. C₇₃	#11412	semidirect

Maximal subgroups

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

subgroup	index
#22626288	2
#22624846	2
#22624844	2
#22624843	2
C₈×#11924031	2
#22608215	2
#11412	73