Subgroups — #34015658

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10596833	C₂	1	2	yes	maximal
2	#22625134	C₂	1	3	yes	maximal, complemented
3	M₁₆×#10596833	C₂	1	6	yes	maximal, complemented
4	C₂×#17989428	C₂	1	3	yes	maximal, complemented
5	#131×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	C₄	1	4	yes	—
7	D₄×#10596833	C₄	1	12	yes	—
8	#17989603	C₂²	1	6	yes	—
9	C₂×C₄×#10596833	C₂²	1	6	yes	—
10	#17989428	C₂²	1	1	yes	characteristic
11	#17989428	C₂²	1	6	yes	—
12	#17989428	C₂²	1	3	yes	—
13	C₈×#10596833	C₂²	1	6	yes	—
14	C₂×#14556598	C₄	1	1	yes	characteristic
15	#41×Dic₇₃	C₂²	1	1	yes	characteristic
16	#17988545	C₄	1	3	yes	—
17	#17988542	C₄	1	3	yes	—
18	M₁₆×Dic₇₃	C₂²	1	3	yes	—
19	C₂×C₈×Dic₇₃	C₂²	1	3	yes	—
20	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
21	C₂²×#10596833	C₂×C₄	1	12	yes	—
22	C₄⋊#10596833	C₂×C₄	1	12	yes	—
23	C₂²⋊#10596833	C₂×C₄	1	12	yes	—
24	C₄×#10596833	C₂×C₄	1	12	yes	—
25	C₄×#10596833	C₂³	1	6	yes	—
26	C₄×#10596833	C₂³	1	2	yes	—
27	#14556598	C₈	1	8	yes	complemented
28	C₂×#11924012	C₂×C₄	1	3	yes	—
29	C₂²×#10596831	C₂×C₄	1	3	yes	—
30	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
31	D₄×Dic₇₃	C₂×C₄	1	6	yes	—
32	#14556550	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₇₃	C₂³	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	3	yes	—
38	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#131×C₇₃	C₈	1	8	yes	complemented
41	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
43	C₂×#10596833	—	4	12	no	—
44	C₂×#10596833	C₂²×C₄	1	4	yes	—
45	C₂×#10596833	C₂²×C₄	1	12	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
47	#11924012	C₂×C₈	1	12	yes	complemented
48	C₂×#10596831	C₂×C₈	1	12	yes	complemented
49	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
50	C₂×#10596831	C₂²×C₄	1	3	yes	—
51	C₄⋊Dic₇₃	C₄²	1	6	yes	—
52	C₂²⋊Dic₇₃	C₄²	1	6	yes	—
53	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
54	C₄×Dic₇₃	C₂²×C₄	1	6	yes	—
55	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
56	#41×C₇₃	C₂×C₈	1	4	yes	—
57	#41×C₇₃	C₂×C₈	1	4	yes	—
58	C₂×Q₈×C₇₃	C₄²	1	1	yes	characteristic
59	C₂×D₄×C₇₃	C₄²	1	3	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
61	M₁₆×C₇₃	C₂×C₈	1	12	yes	complemented
62	C₂×C₈×C₇₃	C₂×C₈	1	12	yes	complemented
63	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
65	C₂³×C₇₃	C₂×C₄²	1	3	yes	—
66	Q₈×C₇₃	C₄×C₈	1	4	yes	—
67	D₄×C₇₃	C₄×C₈	1	12	yes	—
68	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
69	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
71	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₇₃	—	2	12	no	—
73	C₂×Dic₇₃	C₂×C₄²	1	6	yes	—
74	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
75	#10596833	—	2	4	no	—
76	#10596833	—	2	12	no	—
77	C₈×C₇₃	C₂²×C₈	1	2	yes	—
78	C₈×C₇₃	C₂²×C₈	1	6	yes	—
79	#10596831	C₂²×C₈	1	2	yes	—
80	#10596831	C₂²×C₈	1	6	yes	—
81	C₂²×C₇₃	—	2	6	no	—
82	C₂²×C₇₃	C₂×C₄×C₈	1	6	yes	—
83	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
84	C₄×C₇₃	C₂×C₄×C₈	1	6	yes	—
85	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
86	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
87	Dic₇₃	—	2	6	no	—
88	Dic₇₃	#568	1	4	yes	—
89	#17316	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	6	no	—
91	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
92	C₂×C₇₃	#2875	1	2	yes	—
93	C₈×#41	—	73	2	no	—
94	C₄×#131	—	73	1	no	—
95	#1450	—	73	3	no	—
96	C₈×M₁₆	—	73	6	no	—
97	C₂×#322	—	73	3	no	—
98	C₇₃	#17316	1	1	yes	characteristic, complemented, Sylow
99	C₂×#131	—	73	1	no	—
100	C₂×#131	#64408	1	1	yes	characteristic
101	C₄×#41	—	73	1	no	—
102	C₈×Q₈	—	73	4	no	—
103	C₈×D₄	—	73	12	no	—
104	#431	—	73	6	no	—
105	#405	—	73	3	no	—
106	#405	—	73	3	no	—
107	C₄×M₁₆	—	73	3	no	—
108	C₂×C₄×C₈	—	73	6	no	—
109	C₂×C₄×C₈	—	73	3	no	—
110	#322	—	73	1	no	—
111	#322	—	73	6	no	—
112	#322	—	73	3	no	—
113	C₈²	—	73	6	no	—
114	C₂×#41	C₂×#64408	1	1	yes	characteristic
115	#131	—	73	8	no	—
116	#131	#10596833	1	8	yes	complemented
117	C₂×M₁₆	—	73	3	no	—
118	C₂×M₁₆	C₂×#64408	1	3	yes	—
119	C₂²×C₈	—	73	12	no	—
120	C₂²×C₈	—	73	3	no	—
121	C₂²×C₈	C₂×#64408	1	3	yes	—
122	C₄×Q₈	—	73	2	no	—
123	C₄×D₄	—	73	6	no	—
124	#117	—	73	3	no	—
125	C₂×C₄²	—	73	3	no	—
126	C₄⋊C₈	—	73	12	no	—
127	C₂²⋊C₈	—	73	12	no	—
128	#97	—	73	6	no	—
129	C₄×C₈	—	73	6	no	—
130	C₄×C₈	—	73	2	no	—
131	C₄×C₈	—	73	12	no	—
132	C₄×C₈	—	73	3	no	—
133	C₄×C₈	—	73	1	no	—
134	C₄×C₈	—	73	3	no	—
135	C₄×C₈	—	73	3	no	—
136	#41	C₂×#10596833	1	4	yes	—
137	#41	#11924036	1	4	yes	—
138	C₂×Q₈	C₄×#64408	1	1	yes	characteristic
139	C₂×D₄	C₄×#64408	1	3	yes	—
140	C₂²×C₄	—	73	6	no	—
141	C₂²×C₄	C₂²×#64408	1	3	yes	—
142	M₁₆	—	73	12	no	—
143	M₁₆	C₂×#10596833	1	12	yes	complemented
144	C₂×C₈	—	292	12	no	—
145	C₂×C₈	—	73	12	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	3	no	—
148	C₂×C₈	—	73	4	no	—
149	C₂×C₈	—	73	12	no	—
150	C₂×C₈	C₂×#10596833	1	12	yes	complemented
151	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
152	C₂×C₈	C₂²×#64408	1	3	yes	—
153	C₄⋊C₄	—	73	6	no	—
154	C₂²⋊C₄	—	73	6	no	—
155	C₄²	—	73	3	no	—
156	C₄²	—	73	1	no	—
157	C₄²	—	73	6	no	—
158	C₂³	C₂×C₄×#64408	1	3	yes	—
159	Q₈	C₄×#10596833	1	4	yes	—
160	D₄	C₄×#10596833	1	12	yes	—
161	C₂×C₄	—	73	2	no	—
162	C₂×C₄	—	73	6	no	—
163	C₂×C₄	—	146	12	no	—
164	C₂×C₄	C₂×C₄×#64408	1	3	yes	—
165	C₂×C₄	C₂²×#10596833	1	6	yes	—
166	C₂×C₄	C₂×#11924036	1	6	yes	—
167	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
168	C₈	—	73	2	no	—
169	C₈	—	73	6	no	—
170	C₈	—	146	4	no	—
171	C₈	—	146	12	no	—
172	C₈	C₂²×#10596833	1	2	yes	—
173	C₈	C₂²×#10596833	1	6	yes	—
174	C₂²	—	2	6	no	—
175	C₂²	C₂×C₄×#10596833	1	6	yes	—
176	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
177	C₄	—	73	4	no	—
178	C₄	—	146	6	no	—
179	C₄	C₂×C₄×#10596833	1	6	yes	—
180	C₄	C₂³×#10596833	1	1	yes	characteristic
181	C₄	C₂²×#11924036	1	1	yes	characteristic
182	C₂	—	2	6	no	—
183	C₂	C₂²×C₄×#10596833	1	1	yes	characteristic
184	C₂	#22626501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Fitting	C₂×#131×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄²
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	#17316
LCS	G ▷ C₂×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. #22625134	C₂	semidirect
3. M₁₆×#10596833	C₂	semidirect
4. C₂×#17989428	C₂	semidirect
27. #14556598	C₈	semidirect
40. #131×C₇₃	C₈	semidirect
47. #11924012	C₂×C₈	semidirect
48. C₂×#10596831	C₂×C₈	semidirect
61. M₁₆×C₇₃	C₂×C₈	semidirect
62. C₂×C₈×C₇₃	C₂×C₈	semidirect
98. C₇₃	#17316	semidirect
116. #131	#10596833	semidirect
143. M₁₆	C₂×#10596833	semidirect
150. C₂×C₈	C₂×#10596833	semidirect

Maximal subgroups

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

subgroup	index
#41×#10596833	2
#22625134	2
M₁₆×#10596833	2
C₂×#17989428	2
#131×Dic₇₃	2
#17316	73