Subgroups — #34030070

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989645	C₂	1	2	yes	maximal
2	#22626268	C₂	1	6	yes	maximal, complemented
3	#22626265	C₂	1	3	yes	maximal, complemented
4	C₂×#17989508	C₂	1	3	yes	maximal, complemented
5	#131×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989645	—	2	8	no	—
7	C₂×#14556665	C₂²	1	6	yes	—
8	C₂²×#11924031	C₂²	1	6	yes	—
9	#17989508	C₂²	1	1	yes	characteristic
10	#17989508	C₂²	1	3	yes	—
11	#17989508	C₂²	1	6	yes	—
12	#17989508	C₂²	1	6	yes	—
13	C₂×#14556598	C₂²	1	1	yes	characteristic
14	#41×Dic₇₃	C₄	1	1	yes	characteristic
15	#17988545	C₂²	1	3	yes	—
16	#17988542	C₄	1	3	yes	—
17	M₁₆×Dic₇₃	C₄	1	3	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	3	yes	—
19	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
20	#14556665	—	2	12	no	—
21	C₂×#11924031	—	2	12	no	—
22	C₂×#11924031	C₂³	1	2	yes	—
23	C₂×#11924031	C₂³	1	6	yes	—
24	#14556598	D₄	1	4	yes	—
25	#14556598	Q₈	1	4	yes	—
26	C₂×#11924012	C₂×C₄	1	3	yes	—
27	C₂²×#10596831	C₂³	1	3	yes	—
28	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
29	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
30	D₄×Dic₇₃	C₈	1	3	yes	—
31	D₄×Dic₇₃	C₈	1	3	yes	—
32	#14556550	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
34	#14556501	C₂×C₄	1	6	yes	—
35	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
36	C₈×Dic₇₃	C₂×C₄	1	3	yes	—
37	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂³	1	3	yes	—
39	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#131×C₇₃	—	2	4	no	—
41	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	6	no	—
45	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
46	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
47	#11924012	C₄⋊C₄	1	12	yes	—
48	C₂×#10596831	C₂²×C₄	1	3	yes	—
49	C₂×#10596831	C₂×D₄	1	6	yes	—
50	C₂×#10596831	C₂×Q₈	1	6	yes	—
51	C₂×#10596831	C₂⁴	1	1	yes	characteristic
52	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
53	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
54	C₂²⋊Dic₇₃	C₂×C₈	1	6	yes	—
55	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
56	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
57	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
59	#41×C₇₃	C₄⋊C₄	1	4	yes	—
60	#41×C₇₃	M₁₆	1	4	yes	—
61	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
64	M₁₆×C₇₃	—	2	6	no	—
65	C₂×C₈×C₇₃	—	2	6	no	—
66	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
68	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
69	Q₈×C₇₃	C₄⋊C₈	1	4	yes	—
70	D₄×C₇₃	C₄⋊C₈	1	12	yes	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
74	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₇₃	—	2	6	no	—
76	C₂×Dic₇₃	—	4	3	no	—
77	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
79	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
80	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
81	C₈×C₇₃	—	2	1	no	—
82	C₈×C₇₃	—	2	3	no	—
83	#10596831	C₂×C₄⋊C₄	1	6	yes	—
84	#10596831	C₂²×D₄	1	1	yes	characteristic
85	#10596831	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₇₃	—	2	3	no	—
87	C₂²×C₇₃	—	2	3	no	—
88	C₂²×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
89	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
91	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
93	Dic₇₃	—	2	3	no	—
94	Dic₇₃	—	2	1	no	—
95	Dic₇₃	—	2	3	no	—
96	Dic₇₃	#504	1	2	yes	complemented
97	#33398	—	73	1	no	maximal, Sylow
98	C₂×C₇₃	—	2	6	no	—
99	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
101	C₂×C₇₃	#3407	1	1	yes	characteristic
102	C₂×#504	—	73	2	no	—
103	C₄×#131	—	73	1	no	—
104	#2151	—	73	6	no	—
105	#2150	—	73	3	no	—
106	C₂×C₄⋊C₁₆	—	73	3	no	—
107	C₇₃	#33398	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	—	73	1	no	—
109	C₂×#131	#64408	1	1	yes	characteristic
110	C₄×#41	—	73	1	no	—
111	#504	—	146	8	no	—
112	C₂×M₃₂	—	73	6	no	—
113	C₂²×C₁₆	—	73	6	no	—
114	#405	—	73	3	no	—
115	#405	—	73	3	no	—
116	C₄×M₁₆	—	73	3	no	—
117	C₂×C₄×C₈	—	73	3	no	—
118	C₄⋊C₁₆	—	73	1	no	—
119	C₄⋊C₁₆	—	73	3	no	—
120	C₄⋊C₁₆	—	73	6	no	—
121	C₄⋊C₁₆	—	73	6	no	—
122	C₂×#41	C₂×#64408	1	1	yes	characteristic
123	#131	—	2	4	no	—
124	#131	—	73	4	no	—
125	#131	—	73	4	no	—
126	C₂×M₁₆	—	73	3	no	—
127	C₂×M₁₆	C₂×#64408	1	3	yes	—
128	C₂²×C₈	—	73	3	no	—
129	C₂²×C₈	C₂×#64408	1	3	yes	—
130	C₄×Q₈	—	73	1	no	—
131	C₄×Q₈	—	73	1	no	—
132	C₄×D₄	—	73	3	no	—
133	C₄×D₄	—	73	3	no	—
134	#117	—	73	3	no	—
135	C₂×C₄²	—	73	3	no	—
136	M₃₂	—	146	12	no	—
137	C₂×C₁₆	—	73	2	no	—
138	C₂×C₁₆	—	73	6	no	—
139	C₂×C₁₆	—	146	12	no	—
140	#97	—	73	6	no	—
141	C₄×C₈	—	73	3	no	—
142	C₄×C₈	—	73	3	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	3	no	—
145	#41	C₄⋊#64408	1	4	yes	—
146	#41	#11924037	1	4	yes	—
147	C₂×Q₈	#11924036	1	1	yes	characteristic
148	C₂×D₄	#11924036	1	3	yes	—
149	C₂²×C₄	—	73	3	no	—
150	C₂²×C₄	—	73	3	no	—
151	C₂²×C₄	C₂²×#64408	1	3	yes	—
152	M₁₆	—	2	6	no	—
153	M₁₆	—	73	12	no	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	—	73	3	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	6	no	—
158	C₂×C₈	—	73	6	no	—
159	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
160	C₂×C₈	C₂²×#64408	1	3	yes	—
161	C₄⋊C₄	—	73	3	no	—
162	C₄⋊C₄	—	73	3	no	—
163	C₂²⋊C₄	—	73	6	no	—
164	C₄²	—	73	3	no	—
165	C₄²	—	73	3	no	—
166	C₄²	—	73	1	no	—
167	C₄²	—	73	3	no	—
168	C₁₆	—	146	2	no	—
169	C₁₆	—	146	6	no	—
170	C₂³	C₂×#11924036	1	3	yes	—
171	Q₈	#14556658	1	4	yes	—
172	D₄	#14556658	1	12	yes	—
173	C₂×C₄	—	292	3	no	—
174	C₂×C₄	—	73	1	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	3	no	—
177	C₂×C₄	—	73	3	no	—
178	C₂×C₄	—	146	6	no	—
179	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
180	C₂×C₄	C₂×#11924036	1	3	yes	—
181	C₂×C₄	C₂×#11924037	1	6	yes	—
182	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	—	73	1	no	—
186	C₈	—	73	1	no	—
187	C₈	—	73	6	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×#14556658	1	6	yes	—
191	C₂²	C₂²×#11924036	1	1	yes	characteristic
192	C₄	—	73	2	no	—
193	C₄	—	146	3	no	—
194	C₄	—	146	1	no	—
195	C₄	—	146	3	no	—
196	C₄	C₂×#14556658	1	6	yes	—
197	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
198	C₄	C₂²×#11924037	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×#14556658	1	1	yes	characteristic
201	C₂	#22626896	1	1	yes	characteristic
202	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂×#131×C₇₃	C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	#33398
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #22626268	C₂	semidirect
3. #22626265	C₂	semidirect
4. C₂×#17989508	C₂	semidirect
96. Dic₇₃	#504	semidirect
107. C₇₃	#33398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989645	2
#22626268	2
#22626265	2
C₂×#17989508	2
#131×Dic₇₃	2
#33398	73