Finite Group Databasebeta

Subgroups — #34012335

Order 18688.

326 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₂×#17989479	C₂	1	2	yes	maximal
2	#22624978	C₂	1	2	yes	maximal
3	#22624975	C₂	1	2	yes	maximal
4	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₂²	1	4	yes	—
6	#17989479	C₄	1	8	yes	complemented
7	C₂×#14556502	C₄	1	1	yes	characteristic
8	C₂×#14556502	C₄	1	1	yes	characteristic
9	C₂×#14556502	C₂²	1	1	yes	characteristic
10	C₂×#14556502	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₄³×C₇₃	C₄	1	1	yes	characteristic
14	#14556665	—	2	4	no	—
15	#14556665	C₂×C₄	1	8	yes	—
16	C₂×#11924031	—	2	4	no	—
17	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂³	1	1	yes	characteristic
21	#14556502	—	2	2	no	—
22	#14556502	—	2	2	no	—
23	#14556502	—	2	2	no	—
24	#14556502	C₂×C₄	1	4	yes	—
25	C₄×#10596831	—	2	2	no	—
26	C₄×#10596831	—	2	2	no	—
27	C₂×C₄²×C₇₃	—	2	2	no	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#11924031	—	2	8	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	2	1	no	—
37	C₂×#10596831	—	2	1	no	—
38	C₂×#10596831	C₄²	1	2	yes	—
39	C₂×#10596831	M₁₆	1	2	yes	—
40	C₂×#10596831	M₁₆	1	2	yes	—
41	C₂×#10596831	C₂²×C₄	1	2	yes	—
42	C₂×#10596831	#41	1	2	yes	—
43	C₂×#10596831	#41	1	2	yes	—
44	C₂²×C₄×C₇₃	—	2	1	no	—
45	C₂²×C₄×C₇₃	—	2	1	no	—
46	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	—	2	1	no	—
49	C₄²×C₇₃	—	2	1	no	—
50	C₄²×C₇₃	—	2	1	no	—
51	C₄²×C₇₃	—	2	1	no	—
52	C₄²×C₇₃	—	4	4	no	—
53	C₄²×C₇₃	C₄²	1	2	yes	—
54	C₄²×C₇₃	M₁₆	1	2	yes	—
55	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	4	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	#10596831	—	2	2	no	—
69	#10596831	—	2	2	no	—
70	#10596831	—	2	2	no	—
71	#10596831	#117	1	4	yes	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
74	C₂²×C₇₃	#432	1	1	yes	characteristic
75	C₂²×C₇₃	#432	1	1	yes	characteristic
76	C₄×C₇₃	—	2	4	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	—	4	2	no	—
79	C₄×C₇₃	#376	1	1	yes	characteristic
80	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₇₃	#432	1	1	yes	characteristic
82	C₄×C₇₃	#432	1	1	yes	characteristic
83	#12595	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	#1745	1	1	yes	characteristic
87	C₂×C₇₃	#2165	1	2	yes	—
88	C₂×#347	—	73	2	no	—
89	C₄×#97	—	73	1	no	—
90	#1377	—	73	2	no	—
91	#1376	—	73	2	no	—
92	C₇₃	#12595	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	73	4	no	—
94	C₂×#97	—	73	1	no	—
95	C₂×#97	—	73	1	no	—
96	C₂×#97	—	73	1	no	—
97	C₂×#97	—	73	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	C₂×C₄×C₈	—	73	1	no	—
100	C₄³	#64408	1	1	yes	characteristic
101	#347	—	73	8	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
108	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
109	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
110	M₃₂	—	73	8	no	—
111	M₃₂	—	146	4	no	—
112	C₂×C₁₆	—	146	4	no	—
113	#97	—	73	4	no	—
114	#97	—	146	2	no	—
115	#97	—	146	2	no	—
116	#97	—	146	2	no	—
117	C₄×C₈	—	146	2	no	—
118	C₄×C₈	—	146	2	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	#11924037	1	2	yes	—
122	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
123	C₂×C₈	—	73	2	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂×C₈	—	146	1	no	—
134	C₂×C₈	—	146	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₄×#64408	1	2	yes	—
141	C₄²	#11924038	1	2	yes	—
142	C₁₆	—	146	8	no	—
143	C₂³	C₂×#11924037	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
151	C₂×C₄	#14556670	1	1	yes	characteristic
152	C₂×C₄	#14556670	1	1	yes	characteristic
153	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
154	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
156	C₈	—	73	4	no	—
157	C₈	—	146	2	no	—
158	C₈	—	146	2	no	—
159	C₈	—	146	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	C₄×#11924037	1	1	yes	characteristic
162	C₂²	#17989607	1	1	yes	characteristic
163	C₂²	#17989608	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	2	no	—
167	C₄	#17989523	1	1	yes	characteristic
168	C₄	C₄×#11924038	1	1	yes	characteristic
169	C₄	#17989609	1	1	yes	characteristic
170	C₄	#17989610	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#22625648	1	1	yes	characteristic
174	C₂	#22626300	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556670
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₄³×C₇₃	C₄
socle	C₂²×C₇₃	#432
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12595
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. #17989479	C₄	semidirect
92. C₇₃	#12595	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989479	2
#22624978	2
#22624975	2
C₄×#14556502	2
#12595	73