Finite Group Databasebeta

Subgroups — #34033144

Order 18688.

178 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22627162	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989694	C₂	1	1	yes	maximal, characteristic
3	#22626143	C₂	1	1	yes	maximal, characteristic
4	#17989696	—	2	1	no	—
5	#17989695	C₄	1	2	yes	complemented
6	#17989694	—	2	1	no	—
7	#17989694	—	2	1	no	—
8	#17989694	—	2	1	no	—
9	C₂×#14556681	C₄	1	2	yes	complemented
10	C₂×#14556664	C₂²	1	1	yes	characteristic
11	#17989550	C₄	1	1	yes	characteristic
12	#17989474	—	2	1	no	—
13	C₁₆×#64408	—	2	1	no	—
14	M₃₂×D₇₃	C₄	1	1	yes	characteristic
15	#14556681	—	2	2	no	—
16	#14556681	—	2	2	no	—
17	D₇₃⋊M₁₆	C₈	1	1	yes	characteristic
18	#14556670	C₈	1	1	yes	characteristic
19	#14556664	D₄	1	1	yes	characteristic
20	#14556664	D₄	1	1	yes	characteristic
21	#14556664	D₄	1	1	yes	characteristic
22	#14556664	Q₈	1	1	yes	characteristic
23	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
25	#14556645	—	2	1	no	—
26	C₈×#64408	—	2	1	no	—
27	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	#14556531	C₈	1	1	yes	characteristic
29	#14556530	—	2	1	no	—
30	C₁₆×D₇₃	—	2	1	no	—
31	M₃₂×C₇₃	C₈	1	1	yes	characteristic
32	#11924038	C₁₆	1	2	yes	—
33	#11924037	—	2	1	no	—
34	#11924036	—	2	1	no	—
35	C₄⋊#64408	C₁₆	1	2	yes	—
36	C₂²⋊#64408	—	2	1	no	—
37	C₄×#64408	—	2	1	no	—
38	#11924031	C₂²⋊C₄	1	1	yes	characteristic
39	#11924031	C₂²⋊C₄	1	1	yes	characteristic
40	#11924031	C₄⋊C₄	1	1	yes	characteristic
41	#11924031	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×#10596831	C₄²	1	1	yes	characteristic
44	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₇₃	M₁₆	1	1	yes	characteristic
47	C₈×D₇₃	M₁₆	1	1	yes	characteristic
48	#11924003	—	2	1	no	—
49	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₁₆×C₇₃	—	2	1	no	—
51	C₂²×D₇₃	M₃₂	1	1	yes	characteristic
52	C₂×#64408	—	2	2	no	—
53	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	M₃₂	1	1	yes	characteristic
55	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
57	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
58	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
59	#10596833	—	2	2	no	—
60	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	#10596831	#95	1	1	yes	characteristic
63	#10596831	#97	1	1	yes	characteristic
64	C₂²×C₇₃	#346	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
67	C₂×D₇₃	C₄⋊C₁₆	1	1	yes	characteristic
68	#64408	—	4	2	no	—
69	C₄×C₇₃	#336	1	1	yes	characteristic
70	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
71	Dic₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
72	Dic₇₃	C₄⋊C₁₆	1	1	yes	characteristic
73	#7470	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	#1374	1	1	yes	characteristic
76	D₇₃	—	2	1	no	—
77	D₇₃	#1401	1	2	yes	—
78	C₂×M₆₄	—	73	1	no	—
79	C₂²×C₃₂	—	73	1	no	—
80	#2108	—	73	1	no	—
81	C₇₃	#7470	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	73	1	no	—
83	C₂²×C₁₆	—	73	1	no	—
84	#405	—	73	1	no	—
85	M₆₄	—	73	2	no	—
86	M₆₄	—	146	1	no	—
87	C₂×C₃₂	—	73	2	no	—
88	C₂×C₃₂	—	146	1	no	—
89	C₂×C₃₂	—	146	1	no	—
90	C₂×C₃₂	—	146	1	no	—
91	#346	—	146	1	no	—
92	C₄×C₁₆	—	146	1	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	#117	—	73	1	no	—
96	M₃₂	—	73	1	no	—
97	M₃₂	—	146	1	no	—
98	M₃₂	#10596842	1	1	yes	characteristic
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	73	1	no	—
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	—	73	1	no	—
105	C₂×C₁₆	—	146	1	no	—
106	#97	—	146	1	no	—
107	C₄×C₈	—	146	1	no	—
108	C₃₂	—	146	2	no	—
109	C₃₂	—	146	2	no	—
110	C₂²×C₄	—	73	1	no	—
111	M₁₆	—	73	2	no	—
112	M₁₆	—	146	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
120	C₄⋊C₄	—	73	2	no	—
121	C₂²⋊C₄	—	146	1	no	—
122	C₄²	—	146	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	73	1	no	—
125	C₁₆	—	73	1	no	—
126	C₁₆	—	73	1	no	—
127	C₁₆	—	73	1	no	—
128	C₁₆	—	146	1	no	—
129	C₂³	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
137	C₈	—	73	1	no	—
138	C₈	—	73	1	no	—
139	C₈	—	146	2	no	—
140	C₈	C₂²⋊#10596842	1	1	yes	characteristic
141	C₈	C₄⋊#10596842	1	1	yes	characteristic
142	C₂²	—	73	1	no	—
143	C₂²	—	73	1	no	—
144	C₂²	—	146	2	no	—
145	C₂²	#17989682	1	1	yes	characteristic
146	C₄	—	292	2	no	—
147	C₄	—	73	1	no	—
148	C₄	—	73	1	no	—
149	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
150	C₄	#64408⋊C₁₆	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	73	2	no	—
153	C₂	—	146	1	no	—
154	C₂	#22627002	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₈	C₂×#10596842
Fitting	M₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1374
prime radical	C₂²×D₇₃	M₃₂
cosocle	C₂×#14556664	C₂²
hypercenter	M₃₂	#10596842
hypocenter	C₇₃	#7470
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
5. #17989695	C₄	semidirect
9. C₂×#14556681	C₄	semidirect
81. C₇₃	#7470	semidirect

Maximal subgroups

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

subgroup	index
#22627162	2
C₂×#17989694	2
#22626143	2
#7470	73