Subgroups — C₂²×#17988407 #33238979

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556531	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988407	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10596831	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
7	C₂×#14556531	—	2	3	no	—
8	C₂×#14556531	C₂²	1	6	yes	—
9	C₂²×#11924003	—	2	1	no	—
10	C₂×C₈×Dic₇₃	C₄	1	6	yes	—
11	C₂×C₈×Dic₇₃	C₂²	1	6	yes	—
12	#17988407	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₇₃	—	2	3	no	—
15	C₂×M₃₂×C₇₃	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₇₃	—	2	1	no	—
17	C₂³×Dic₇₃	—	2	1	no	—
18	C₂²×#10596831	C₈	1	6	yes	—
19	C₂²×#10596831	C₈	1	2	yes	—
20	C₂²×#10596831	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₇₃	—	2	3	no	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
23	#14556531	—	2	6	no	—
24	#14556531	C₂³	1	4	yes	—
25	C₂×#11924003	—	2	6	no	—
26	C₈×Dic₇₃	C₂×C₄	1	12	yes	—
27	C₈×Dic₇₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₇₃	D₄	1	3	yes	—
31	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	Q₈	1	3	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
35	M₃₂×C₇₃	—	2	6	no	—
36	M₃₂×C₇₃	C₂³	1	4	yes	—
37	C₂×C₁₆×C₇₃	—	2	6	no	—
38	C₂²×Dic₇₃	—	2	8	no	—
39	C₂²×Dic₇₃	—	2	6	no	—
40	C₂×#10596831	C₂×C₈	1	12	yes	—
41	C₂×#10596831	C₂×C₈	1	12	yes	—
42	C₂×#10596831	C₂²×C₄	1	4	yes	—
43	C₄×Dic₇₃	—	2	6	no	—
44	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
45	#11924003	—	2	4	no	—
46	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	M₁₆	1	3	yes	—
52	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₇₃	C₄⋊C₄	1	6	yes	—
54	C₂×C₈×C₇₃	C₄⋊C₄	1	3	yes	—
55	C₂×C₈×C₇₃	C₄⋊C₄	1	3	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₇₃	C₂×D₄	1	3	yes	—
59	C₂×C₈×C₇₃	C₂×Q₈	1	3	yes	—
60	C₂×C₈×C₇₃	C₂×Q₈	1	3	yes	—
61	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
62	C₁₆×C₇₃	—	2	4	no	—
63	C₂³×C₇₃	—	2	4	no	—
64	C₂³×C₇₃	C₄⋊C₈	1	3	yes	—
65	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂³×C₇₃	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₇₃	C₄⋊C₈	1	6	yes	—
68	C₂×C₄×C₇₃	C₄⋊C₈	1	3	yes	—
69	C₂×C₄×C₇₃	C₄⋊C₈	1	3	yes	—
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₇₃	—	2	24	no	—
77	C₂×Dic₇₃	—	2	4	no	—
78	C₈×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₈×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
81	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
82	#10596831	C₂²×C₈	1	8	yes	—
83	C₂²×C₇₃	—	2	6	no	—
84	C₂²×C₇₃	—	2	6	no	—
85	C₂²×C₇₃	#364	1	4	yes	complemented
86	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
87	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
88	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
89	C₄×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
90	C₄×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
91	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
93	Dic₇₃	—	2	8	no	—
94	C₂²×#364	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	3	no	—
97	C₂×C₇₃	C₂×#364	1	6	yes	complemented
98	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×M₃₂	—	73	1	no	—
100	C₂²×M₃₂	D₇₃	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	73	1	no	—
102	C₂×#364	—	73	12	no	—
103	C₇₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₈	—	73	1	no	—
105	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
106	C₂²×C₄²	—	73	1	no	—
107	C₂×M₃₂	—	2	3	no	—
108	C₂×M₃₂	—	73	6	no	—
109	C₂×M₃₂	—	146	3	no	—
110	C₂×M₃₂	C₂×D₇₃	1	6	yes	—
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	146	1	no	—
113	C₂×C₄×C₈	—	73	6	no	—
114	C₂×C₄×C₈	—	73	6	no	—
115	#364	—	73	16	no	—
116	C₂³×C₄	—	146	1	no	—
117	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
118	C₂²×C₈	—	73	6	no	—
119	C₂²×C₈	—	73	2	no	—
120	C₂²×C₈	—	73	6	no	—
121	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₄₆	1	3	yes	—
123	C₂²×C₈	C₄×D₇₃	1	3	yes	—
124	C₂²×C₈	C₂²⋊D₇₃	1	3	yes	—
125	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
127	C₂×C₄²	—	73	6	no	—
128	C₂×C₄²	—	146	3	no	—
129	M₃₂	—	2	6	no	—
130	M₃₂	—	73	4	no	—
131	M₃₂	—	146	6	no	—
132	M₃₂	C₂²×D₇₃	1	4	yes	—
133	C₂×C₁₆	—	2	6	no	—
134	C₂×C₁₆	—	146	6	no	—
135	C₄×C₈	—	73	12	no	—
136	C₄×C₈	—	73	4	no	—
137	C₂⁴	#11924011	1	1	yes	characteristic
138	C₂²×C₄	—	146	8	no	—
139	C₂²×C₄	—	146	6	no	—
140	C₂²×C₄	#11924007	1	3	yes	—
141	C₂²×C₄	#11924007	1	1	yes	characteristic
142	C₂²×C₄	C₈×D₇₃	1	3	yes	—
143	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
144	C₂²×C₄	#11924011	1	3	yes	—
145	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
146	C₂×C₈	—	73	12	no	—
147	C₂×C₈	—	73	12	no	—
148	C₂×C₈	—	73	4	no	—
149	C₂×C₈	#11924007	1	3	yes	—
150	C₂×C₈	#11924007	1	6	yes	—
151	C₂×C₈	#11924007	1	3	yes	—
152	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
153	C₂×C₈	C₂×C₂²⋊D₇₃	1	3	yes	—
154	C₂×C₈	C₂×C₂²⋊D₇₃	1	3	yes	—
155	C₂×C₈	C₂×Dic₁₄₆	1	3	yes	—
156	C₂×C₈	C₂×Dic₁₄₆	1	3	yes	—
157	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
158	C₄²	—	73	4	no	—
159	C₄²	—	146	6	no	—
160	C₁₆	—	2	4	no	—
161	C₁₆	—	146	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	#14556520	1	3	yes	—
164	C₂³	#14556520	1	1	yes	characteristic
165	C₂³	C₂×#11924011	1	3	yes	—
166	C₂×C₄	—	146	24	no	—
167	C₂×C₄	—	146	4	no	—
168	C₂×C₄	#14556520	1	6	yes	—
169	C₂×C₄	#14556520	1	3	yes	—
170	C₂×C₄	#14556520	1	3	yes	—
171	C₂×C₄	C₂×#11924007	1	3	yes	—
172	C₂×C₄	C₂×#11924007	1	3	yes	—
173	C₂×C₄	C₂×C₈×D₇₃	1	3	yes	—
174	C₂×C₄	C₂×C₈×D₇₃	1	3	yes	—
175	C₂×C₄	C₂×#11924011	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
177	C₈	—	73	8	no	—
178	C₈	C₂×#11924007	1	3	yes	—
179	C₈	C₂×#11924007	1	3	yes	—
180	C₈	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
181	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#17988407	1	4	yes	direct product factor
185	C₂²	C₂×#14556520	1	3	yes	—
186	C₂²	C₂×#14556520	1	3	yes	—
187	C₂²	C₂²×#11924011	1	1	yes	characteristic
188	C₄	—	146	8	no	—
189	C₄	C₂×#14556520	1	3	yes	—
190	C₄	C₂×#14556520	1	3	yes	—
191	C₄	C₂²×#11924007	1	1	yes	characteristic
192	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#17988407	1	6	yes	direct product factor
196	C₂	C₂²×#14556520	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂²×M₃₂×C₇₃	C₂
socle	C₂³×C₇₃	C₄⋊C₈
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂²×M₃₂	D₇₃
hypocenter	C₇₃	C₂²×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#17988407	C₂	direct (N × H)
12. #17988407	C₂²	direct (N × H)
85. C₂²×C₇₃	#364	semidirect
97. C₂×C₇₃	C₂×#364	semidirect
103. C₇₃	C₂²×#364	semidirect
184. C₂²	#17988407	direct (N × H)
195. C₂	C₂×#17988407	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×#14556531	2
C₂²×C₈×Dic₇₃	2
C₂×#17988407	2
C₂²×M₃₂×C₇₃	2
C₂²×#364	73

Subgroups — C22×#17988407 #33238979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988407 #33238979