Finite Group Databasebeta

Subgroups — #34009685

Order 18688.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₄×#14556665	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989475	C₂	1	1	yes	maximal, characteristic
3	#22624849	C₂	1	4	yes	maximal, complemented
4	#22608215	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₄	1	2	yes	—
6	C₂²×#11924031	C₄	1	2	yes	—
7	#17989475	C₄	1	2	yes	—
8	#17989475	C₂²	1	2	yes	—
9	#17989475	C₂²	1	2	yes	—
10	C₄×#11924031	C₄	1	2	yes	—
11	#17988640	C₄	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	#17988309	C₂²	1	2	yes	—
14	C₈×#10596831	C₄	1	2	yes	—
15	#431×C₇₃	C₄	1	1	yes	characteristic
16	#14556665	—	2	4	no	—
17	C₂×#11924031	C₈	1	8	yes	complemented
18	C₂×#11924031	C₂×C₄	1	4	yes	—
19	C₂×#11924031	C₂×C₄	1	4	yes	—
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#10596831	—	2	1	no	—
23	C₂²⋊#10596831	—	2	1	no	—
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₄×#10596831	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	—	2	1	no	—
31	C₂²⋊C₈×C₇₃	—	2	1	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	#11924031	—	2	4	no	—
34	#11924031	C₂×C₈	1	8	yes	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₄²	1	1	yes	characteristic
37	C₂×#10596831	C₄²	1	1	yes	characteristic
38	C₂×#10596831	C₂×C₈	1	4	yes	—
39	C₂×#10596831	M₁₆	1	4	yes	—
40	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	2	no	—
46	C₄²×C₇₃	C₄²	1	1	yes	characteristic
47	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
49	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₇₃	#97	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₇₃	#97	1	2	yes	—
55	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₇₃	—	4	2	no	—
62	#10596831	—	4	2	no	—
63	#10596831	C₄×C₈	1	2	yes	—
64	#10596831	#97	1	2	yes	—
65	#10596831	C₂²×C₈	1	2	yes	—
66	#10596831	C₂×M₁₆	1	2	yes	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	#322	1	2	yes	—
70	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#322	1	2	yes	—
76	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
77	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
78	#11405	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	—	2	2	no	—
80	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
81	C₂×C₇₃	#2110	1	1	yes	characteristic
82	C₂×C₇₃	#2110	1	1	yes	characteristic
83	C₄×M₃₂	—	73	1	no	—
84	C₂×#346	—	73	1	no	—
85	#1450	—	73	1	no	—
86	#1313	—	73	4	no	—
87	C₇₃	#11405	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	—	73	2	no	—
89	C₂²×C₁₆	—	73	2	no	—
90	#431	—	73	1	no	—
91	#431	#64408	1	1	yes	characteristic
92	C₂×C₄×C₈	—	73	1	no	—
93	#346	—	73	2	no	—
94	#346	—	73	2	no	—
95	#346	—	73	2	no	—
96	C₄×C₁₆	—	73	2	no	—
97	#322	—	73	2	no	—
98	C₈²	—	73	2	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
102	M₃₂	—	146	4	no	—
103	C₂×C₁₆	—	73	4	no	—
104	C₂×C₁₆	—	73	8	no	—
105	C₂×C₁₆	—	73	4	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	C₂²⋊C₈	—	2	1	no	—
109	C₂²⋊C₈	—	146	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	2	no	—
115	C₄×C₈	C₂×#64408	1	2	yes	—
116	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
117	C₂²×C₄	#11924037	1	1	yes	characteristic
118	C₂²×C₄	#11924038	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	73	4	no	—
121	C₂×C₈	—	73	4	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	146	2	no	—
127	C₄²	C₄×#64408	1	1	yes	characteristic
128	C₄²	#11924037	1	1	yes	characteristic
129	C₄²	#11924038	1	1	yes	characteristic
130	C₄²	C₂²×#64408	1	1	yes	characteristic
131	C₁₆	—	73	8	no	—
132	C₁₆	—	146	4	no	—
133	C₂³	#14556646	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₈×#64408	1	2	yes	—
137	C₂×C₄	#14556645	1	2	yes	—
138	C₂×C₄	#14556646	1	1	yes	characteristic
139	C₂×C₄	#14556646	1	1	yes	characteristic
140	C₂×C₄	#14556646	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
142	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
143	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	292	2	no	—
146	C₈	—	73	2	no	—
147	C₈	—	73	2	no	—
148	C₈	—	73	2	no	—
149	C₈	—	73	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#17989429	1	2	yes	—
153	C₂²	C₂×#14556646	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#17989429	1	2	yes	—
159	C₄	C₂×C₈×#64408	1	1	yes	characteristic
160	C₄	C₂×#14556645	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#17989429	1	1	yes	characteristic
163	C₂	#22626152	1	1	yes	characteristic
164	C₂	#22626154	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#64408
Fitting	#431×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	#97
cosocle	C₄×#10596831	C₂³
hypercenter	#431	#64408
hypocenter	C₇₃	#11405
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #22624849	C₂	semidirect
17. C₂×#11924031	C₈	semidirect
87. C₇₃	#11405	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556665	2
C₂×#17989475	2
#22624849	2
#22608215	2
#11405	73