Subgroups — #32972919

Order 18688.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988363	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	—	2	1	no	—
5	C₂×#14556525	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₇₃	—	2	1	no	—
7	#17988363	—	2	1	no	—
8	#17988363	C₄	1	2	yes	complemented
9	#17988350	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#334×C₇₃	—	2	1	no	—
13	#334×C₇₃	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₇₃	—	2	2	no	—
15	C₂×C₄×Dic₇₃	—	4	1	no	—
16	C₂×#11924003	—	2	4	no	—
17	C₂×#11924003	—	2	2	no	—
18	#14556525	—	2	2	no	—
19	#14556525	C₈	1	4	yes	complemented
20	#14556498	—	4	1	no	—
21	C₂²×C₈×C₇₃	—	4	1	no	—
22	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₇₃	—	2	2	no	—
26	C₄⋊C₈×C₇₃	—	4	1	no	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	C₂²×Dic₇₃	—	4	2	no	—
32	C₄⋊Dic₇₃	—	2	4	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	C₄×Dic₇₃	—	8	2	no	—
35	#11924003	—	2	4	no	—
36	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	—	4	2	no	—
41	C₂×C₈×C₇₃	—	8	2	no	—
42	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
48	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
51	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
52	C₂³×C₇₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#104	1	2	yes	—
62	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
64	C₂×Dic₇₃	—	4	4	no	—
65	C₂×Dic₇₃	—	8	2	no	—
66	C₂×Dic₇₃	—	8	2	no	—
67	C₈×C₇₃	—	2	1	no	—
68	C₈×C₇₃	—	2	1	no	—
69	C₈×C₇₃	—	8	2	no	—
70	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
71	C₈×C₇₃	#97	1	1	yes	characteristic
72	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
73	C₂²×C₇₃	D₄⋊C₈	1	2	yes	—
74	C₂²×C₇₃	#340	1	1	yes	characteristic
75	C₂²×C₇₃	#360	1	2	yes	—
76	C₂²×C₇₃	#361	1	1	yes	characteristic
77	C₂²×C₇₃	#362	1	1	yes	characteristic
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
83	C₄×C₇₃	#336	1	1	yes	characteristic
84	C₄×C₇₃	#339	1	1	yes	characteristic
85	Dic₇₃	—	8	4	no	—
86	#7178	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#1279	1	1	yes	characteristic
88	C₂×C₇₃	#1333	1	2	yes	—
89	C₂×C₇₃	#1334	1	2	yes	—
90	C₂×C₇₃	#1386	1	1	yes	characteristic
91	C₂×C₇₃	#1387	1	1	yes	characteristic
92	C₂×#346	—	73	1	no	—
93	C₄×#107	—	73	1	no	—
94	C₂×#334	D₇₃	1	1	yes	characteristic
95	C₇₃	#7178	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	146	1	no	—
97	C₂×#107	—	73	2	no	—
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	146	1	no	—
101	#346	—	73	2	no	—
102	#346	—	146	1	no	—
103	#340	—	146	1	no	—
104	#334	—	2	1	no	—
105	#334	Dic₇₃	1	2	yes	complemented
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
108	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	146	2	no	—
110	C₂×C₄²	—	292	1	no	—
111	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂×C₁₆	—	146	4	no	—
113	C₂×C₁₆	—	146	2	no	—
114	#107	—	73	4	no	—
115	#107	—	146	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
122	#95	—	292	1	no	—
123	C₂²×C₄	—	292	2	no	—
124	C₂²×C₄	#11924005	1	1	yes	characteristic
125	C₂²×C₄	#11924013	1	1	yes	characteristic
126	C₂²×C₄	#11924015	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	#11924005	1	1	yes	characteristic
132	C₂×C₈	#11924007	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
134	C₂×C₈	C₈×D₇₃	1	2	yes	—
135	C₂×C₈	#11924011	1	2	yes	—
136	C₄⋊C₄	—	292	2	no	—
137	C₄⋊C₄	—	146	4	no	—
138	C₄²	—	584	2	no	—
139	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
140	C₄²	#11924007	1	1	yes	characteristic
141	C₄²	#11924016	1	1	yes	characteristic
142	C₄²	#11924018	1	1	yes	characteristic
143	C₁₆	—	146	4	no	—
144	C₂³	#14556511	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	292	4	no	—
149	C₂×C₄	—	584	2	no	—
150	C₂×C₄	—	584	2	no	—
151	C₂×C₄	#14556498	1	1	yes	characteristic
152	C₂×C₄	#14556503	1	2	yes	—
153	C₂×C₄	#14556512	1	1	yes	characteristic
154	C₂×C₄	#14556514	1	1	yes	characteristic
155	C₂×C₄	#14556515	1	1	yes	characteristic
156	C₂×C₄	#14556517	1	2	yes	—
157	C₂×C₄	#14556522	1	1	yes	characteristic
158	C₂×C₄	#14556524	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	2	no	—
162	C₈	C₈×Dic₇₃	1	1	yes	characteristic
163	C₈	#14556501	1	1	yes	characteristic
164	C₈	#14556520	1	2	yes	—
165	C₂²	#17988314	1	2	yes	—
166	C₂²	#17988349	1	1	yes	characteristic
167	C₂²	#17988396	1	2	yes	—
168	C₂²	#17988399	1	1	yes	characteristic
169	C₂²	#17988402	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	584	4	no	—
175	C₄	#17988317	1	2	yes	—
176	C₄	#17988341	1	1	yes	characteristic
177	C₄	#17988348	1	1	yes	characteristic
178	C₂	#22607770	1	1	yes	characteristic
179	C₂	#22607897	1	2	yes	—
180	C₂	#22607900	1	2	yes	—
181	C₂	#22608038	1	1	yes	characteristic
182	C₂	#22608041	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556511
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#7178
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #17988363	C₄	semidirect
13. #334×C₇₃	C₄	semidirect
19. #14556525	C₈	semidirect
95. C₇₃	#7178	semidirect
105. #334	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988363	2
C₄×#14556525	2
C₂×#334×C₇₃	2
#7178	73