Finite Group Databasebeta

Subgroups — #34032886

Order 18688.

254 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627134	C₂	1	1	yes	maximal, characteristic
2	#22627130	C₂	1	1	yes	maximal, characteristic
3	#22625589	C₂	1	1	yes	maximal, characteristic
4	#17989690	C₄	1	2	yes	complemented
5	C₄⋊#11924040	C₄	1	2	yes	complemented
6	#17989684	—	2	1	no	—
7	#17989684	—	2	1	no	—
8	#17989681	—	2	1	no	—
9	C₄×#11924040	—	2	1	no	—
10	C₂×#14556645	—	2	1	no	—
11	C₄×#11924036	C₂²	1	1	yes	characteristic
12	C₂×C₈×#64408	—	2	1	no	—
13	#17989523	C₄	1	1	yes	characteristic
14	#97×D₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924040	—	2	2	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924036	D₄	1	1	yes	characteristic
18	C₂×#11924036	D₄	1	1	yes	characteristic
19	C₂×C₄×#64408	—	2	1	no	—
20	#14556645	—	2	4	no	—
21	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596833	D₄	1	1	yes	characteristic
24	C₄×#10596833	Q₈	1	1	yes	characteristic
25	C₈×#64408	—	2	4	no	—
26	(C₂×#64408)⋊C₄	—	2	1	no	—
27	(C₂×#64408)⋊C₄	C₈	1	2	yes	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
30	#14556502	C₈	1	1	yes	characteristic
31	#14556501	—	2	1	no	—
32	#97×C₇₃	C₈	1	1	yes	characteristic
33	#11924040	—	4	2	no	—
34	#11924040	—	4	2	no	—
35	C₂²×#64408	—	2	2	no	—
36	#11924036	—	2	2	no	—
37	#11924036	—	2	2	no	—
38	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
42	C₄×#64408	—	2	2	no	—
43	C₄×#64408	—	2	2	no	—
44	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
47	C₂×#10596831	—	2	1	no	—
48	C₈×D₇₃	—	2	2	no	—
49	C₈×D₇₃	—	2	2	no	—
50	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
51	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×#64408	—	2	4	no	—
57	C₂×#64408	—	4	2	no	—
58	C₂×#64408	—	4	2	no	—
59	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×Dic₇₃	#95	1	1	yes	characteristic
63	C₂×Dic₇₃	#97	1	1	yes	characteristic
64	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	—	4	1	no	—
69	#10596833	—	2	1	no	—
70	#10596833	—	2	1	no	—
71	#10596833	#104	1	2	yes	—
72	#10596833	#104	1	2	yes	—
73	C₈×C₇₃	—	2	2	no	—
74	#10596831	—	2	2	no	—
75	C₂²×C₇₃	#336	1	1	yes	characteristic
76	C₂×D₇₃	—	2	1	no	—
77	C₂×D₇₃	—	2	1	no	—
78	C₂×D₇₃	#350	1	2	yes	—
79	#64408	—	4	4	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	1	no	—
82	C₄×C₇₃	—	2	1	no	—
83	Dic₇₃	—	2	1	no	—
84	Dic₇₃	—	2	1	no	—
85	Dic₇₃	#339	1	1	yes	characteristic
86	Dic₇₃	#339	1	1	yes	characteristic
87	#7150	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	#1276	1	1	yes	characteristic
89	C₂×C₇₃	#1378	1	1	yes	characteristic
90	C₂×C₇₃	#1379	1	1	yes	characteristic
91	D₇₃	—	2	2	no	—
92	#2163	—	73	1	no	—
93	#2162	—	73	1	no	—
94	#1726	—	73	1	no	—
95	C₇₃	#7150	1	1	yes	characteristic, complemented, Sylow
96	C₂×#97	—	73	1	no	—
97	C₂×#97	—	146	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	C₂×C₄×C₈	—	146	1	no	—
100	#376	—	73	1	no	—
101	C₄⋊C₁₆	—	73	2	no	—
102	C₄⋊C₁₆	—	73	2	no	—
103	C₂²⋊C₁₆	—	146	1	no	—
104	C₂²⋊C₁₆	—	146	1	no	—
105	#346	—	146	1	no	—
106	C₄×C₁₆	—	146	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	—	146	1	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₄²	—	146	1	no	—
112	C₂×C₁₆	—	146	2	no	—
113	C₂×C₁₆	—	146	2	no	—
114	#97	—	73	1	no	—
115	#97	—	146	1	no	—
116	#97	—	146	4	no	—
117	#97	#10596842	1	1	yes	characteristic
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	146	4	no	—
123	#95	—	73	2	no	—
124	#95	—	146	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂²×C₄	—	73	1	no	—
127	C₂²×C₄	—	73	1	no	—
128	C₂²×C₄	—	146	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₂×C₈	—	146	2	no	—
137	C₂×C₈	—	146	2	no	—
138	C₂×C₈	—	146	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	73	1	no	—
142	C₄²	—	146	2	no	—
143	C₄²	—	146	2	no	—
144	C₄²	C₂×#10596842	1	1	yes	characteristic
145	C₁₆	—	292	2	no	—
146	C₁₆	—	292	2	no	—
147	C₂³	—	73	1	no	—
148	C₂×C₄	—	292	1	no	—
149	C₂×C₄	—	292	2	no	—
150	C₂×C₄	—	292	2	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	146	1	no	—
155	C₂×C₄	—	146	1	no	—
156	C₂×C₄	—	146	2	no	—
157	C₂×C₄	—	146	4	no	—
158	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
160	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	73	2	no	—
163	C₈	—	73	2	no	—
164	C₈	—	146	2	no	—
165	C₈	—	146	1	no	—
166	C₈	—	146	1	no	—
167	C₂²	—	73	2	no	—
168	C₂²	—	146	1	no	—
169	C₂²	—	146	1	no	—
170	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	292	4	no	—
175	C₄	—	73	1	no	—
176	C₄	—	73	1	no	—
177	C₄	—	146	1	no	—
178	C₄	—	146	1	no	—
179	C₂	—	146	2	no	—
180	C₂	#22626936	1	1	yes	characteristic
181	C₂	#22627010	1	1	yes	characteristic
182	C₂	#22627012	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄²	C₂×#10596842
Fitting	#97×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₂²⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	#97	#10596842
hypocenter	C₇₃	#7150
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
4. #17989690	C₄	semidirect
5. C₄⋊#11924040	C₄	semidirect
95. C₇₃	#7150	semidirect

Maximal subgroups

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

subgroup	index
#22627134	2
#22627130	2
#22625589	2
#7150	73