Subgroups — C₂²×#17989479 #34029811

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556665	C₂	1	2	yes	maximal
2	C₂×#17989479	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₄	1	12	yes	complemented
5	C₂×#14556665	C₂²	1	12	yes	—
6	C₂²×#11924031	C₄	1	4	yes	complemented
7	#17989479	C₂²	1	16	yes	direct product factor
8	C₂³×#10596831	C₄	1	1	yes	characteristic
9	C₂³×#10596831	C₂²	1	1	yes	characteristic
10	C₂×#14556502	C₄	1	6	yes	—
11	C₂×#14556502	C₂²	1	6	yes	—
12	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
13	#14556665	C₂×C₄	1	24	yes	complemented
14	#14556665	C₂³	1	8	yes	—
15	C₂×#11924031	C₂×C₄	1	24	yes	complemented
16	C₂²×#10596831	—	2	3	no	—
17	C₂²×#10596831	—	2	1	no	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	C₂²×#10596831	C₂×C₄	1	6	yes	—
20	C₂²×#10596831	C₂×C₄	1	6	yes	—
21	C₂²×#10596831	C₂×C₄	1	3	yes	—
22	C₂²×#10596831	C₂³	1	3	yes	—
23	#14556502	C₂×C₄	1	12	yes	—
24	#14556502	C₂³	1	4	yes	—
25	C₂³×C₄×C₇₃	—	2	1	no	—
26	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	—	2	3	no	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	6	yes	—
29	#11924031	C₂²×C₄	1	16	yes	complemented
30	C₂×#10596831	—	2	6	no	—
31	C₂×#10596831	—	2	6	no	—
32	C₂×#10596831	C₄²	1	6	yes	—
33	C₂×#10596831	C₄²	1	6	yes	—
34	C₂×#10596831	C₂²×C₄	1	4	yes	—
35	C₂×#10596831	C₂²×C₄	1	3	yes	—
36	C₂×#10596831	C₂²×C₄	1	6	yes	—
37	C₂×#10596831	C₂²×C₄	1	6	yes	—
38	C₂×#10596831	C₂⁴	1	1	yes	characteristic
39	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	—	2	6	no	—
41	C₂²×C₄×C₇₃	—	4	4	no	—
42	C₂²×C₄×C₇₃	C₄²	1	3	yes	—
43	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	M₁₆	1	3	yes	—
45	C₂²×C₄×C₇₃	M₁₆	1	3	yes	—
46	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
48	C₄²×C₇₃	—	2	6	no	—
49	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
50	C₂³×C₇₃	—	2	4	no	—
51	C₂³×C₇₃	#97	1	3	yes	—
52	C₂³×C₇₃	#97	1	1	yes	characteristic
53	C₂³×C₇₃	C₂×M₁₆	1	3	yes	—
54	C₂×C₄×C₇₃	—	2	4	no	—
55	C₂×C₄×C₇₃	—	4	12	no	—
56	C₂×C₄×C₇₃	#97	1	6	yes	—
57	C₂×C₄×C₇₃	#97	1	3	yes	—
58	C₂×C₄×C₇₃	#97	1	3	yes	—
59	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
62	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
65	#10596831	—	2	4	no	—
66	#10596831	C₂×C₄²	1	6	yes	—
67	#10596831	C₂³×C₄	1	2	yes	—
68	C₂²×C₇₃	—	2	6	no	—
69	C₂²×C₇₃	—	2	6	no	—
70	C₂²×C₇₃	#347	1	4	yes	complemented
71	C₂²×C₇₃	C₂×#97	1	3	yes	—
72	C₂²×C₇₃	C₂×#97	1	3	yes	—
73	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
74	C₄×C₇₃	—	4	4	no	—
75	C₄×C₇₃	C₂×#97	1	3	yes	—
76	C₄×C₇₃	C₂×#97	1	3	yes	—
77	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
78	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
79	C₂²×#347	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	3	no	—
82	C₂×C₇₃	C₂×#347	1	6	yes	complemented
83	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
84	C₂²×M₃₂	—	73	2	no	—
85	C₂²×#97	—	73	1	no	—
86	C₂×#347	—	73	12	no	—
87	C₇₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
88	C₂³×C₈	—	73	1	no	—
89	C₂³×C₈	—	73	1	no	—
90	C₂²×C₄²	#64408	1	1	yes	characteristic
91	C₂×M₃₂	—	73	12	no	—
92	C₂×M₃₂	—	73	12	no	—
93	C₂²×C₁₆	—	73	4	no	—
94	C₂×#97	—	73	6	no	—
95	C₂×#97	—	73	6	no	—
96	#347	—	73	16	no	—
97	C₂³×C₄	—	2	1	no	—
98	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
99	C₂²×C₈	—	73	6	no	—
100	C₂²×C₈	—	73	2	no	—
101	C₂²×C₈	—	73	3	no	—
102	C₂²×C₈	—	73	3	no	—
103	C₂²×C₈	—	73	6	no	—
104	C₂²×C₈	—	146	1	no	—
105	C₂²×C₈	—	146	3	no	—
106	C₂×C₄²	—	2	3	no	—
107	C₂×C₄²	C₂×#64408	1	6	yes	—
108	M₃₂	—	73	8	no	—
109	M₃₂	—	73	24	no	—
110	C₂×C₁₆	—	73	24	no	—
111	#97	—	73	12	no	—
112	#97	—	73	4	no	—
113	C₂⁴	#11924037	1	1	yes	characteristic
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	C₄×#64408	1	3	yes	—
117	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
118	C₂²×C₄	#11924037	1	3	yes	—
119	C₂²×C₄	#11924038	1	1	yes	characteristic
120	C₂²×C₄	#11924038	1	3	yes	—
121	C₂²×C₄	C₂²×#64408	1	3	yes	—
122	C₂×C₈	—	73	3	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	6	no	—
125	C₂×C₈	—	73	6	no	—
126	C₂×C₈	—	73	6	no	—
127	C₂×C₈	—	73	6	no	—
128	C₂×C₈	—	73	4	no	—
129	C₂×C₈	—	146	6	no	—
130	C₂×C₈	—	146	6	no	—
131	C₄²	—	2	6	no	—
132	C₄²	C₂²×#64408	1	4	yes	—
133	C₁₆	—	73	16	no	—
134	C₂³	—	2	4	no	—
135	C₂³	#14556646	1	3	yes	—
136	C₂³	#14556646	1	1	yes	characteristic
137	C₂³	C₂×#11924037	1	3	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	#14556646	1	6	yes	—
141	C₂×C₄	#14556646	1	3	yes	—
142	C₂×C₄	#14556646	1	3	yes	—
143	C₂×C₄	C₂×C₄×#64408	1	3	yes	—
144	C₂×C₄	C₂×C₄×#64408	1	3	yes	—
145	C₂×C₄	C₂×#11924037	1	3	yes	—
146	C₂×C₄	C₂×#11924038	1	3	yes	—
147	C₂×C₄	C₂×#11924038	1	3	yes	—
148	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
149	C₈	—	73	2	no	—
150	C₈	—	73	6	no	—
151	C₈	—	146	4	no	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#17989479	1	4	yes	direct product factor
155	C₂²	C₂×#14556646	1	3	yes	—
156	C₂²	C₂×#14556646	1	3	yes	—
157	C₂²	C₂²×#11924037	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×#14556646	1	3	yes	—
160	C₄	C₂×#14556646	1	3	yes	—
161	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
162	C₄	C₂²×#11924038	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#17989479	1	6	yes	direct product factor
166	C₂	C₂²×#14556646	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#64408
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle	C₂³×C₇₃	#97
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂²×C₄²	#64408
hypocenter	C₇₃	C₂²×#347
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#17989479	C₂	direct (N × H)
4. C₂×#14556665	C₄	semidirect
6. C₂²×#11924031	C₄	semidirect
7. #17989479	C₂²	direct (N × H)
13. #14556665	C₂×C₄	semidirect
15. C₂×#11924031	C₂×C₄	semidirect
29. #11924031	C₂²×C₄	semidirect
70. C₂²×C₇₃	#347	semidirect
82. C₂×C₇₃	C₂×#347	semidirect
87. C₇₃	C₂²×#347	semidirect
154. C₂²	#17989479	direct (N × H)
165. C₂	C₂×#17989479	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556665	2
C₂×#17989479	2
C₂²×#14556502	2
C₂²×#347	73

Subgroups — C22×#17989479 #34029811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989479 #34029811