Finite Group Databasebeta

Subgroups — #34000577

Order 18688.

196 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 52 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	#22626148	C₂	1	1	yes	maximal, characteristic
2	#22625857	C₂	1	1	yes	maximal, characteristic
3	#347×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989643	—	2	1	no	—
5	#17989592	C₄	1	1	yes	characteristic
6	#17989591	C₄	1	1	yes	characteristic
7	#17989480	—	2	1	no	—
8	#17989479	C₄	1	2	yes	—
9	#17989463	—	2	1	no	—
10	#17989463	—	2	1	no	—
11	M₃₂×D₇₃	—	2	1	no	—
12	#97×D₇₃	C₂²	1	1	yes	characteristic
13	#17988365	C₄	1	1	yes	characteristic
14	#17988364	—	2	1	no	—
15	#347×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	2	1	no	—
17	C₂×#11924038	—	2	1	no	—
18	#14556666	—	2	2	no	—
19	#14556665	—	2	2	no	—
20	#14556664	—	2	2	no	—
21	#14556658	—	4	1	no	—
22	C₄⋊#10596833	—	4	1	no	—
23	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
26	#14556531	—	2	1	no	—
27	#14556530	—	2	2	no	—
28	C₁₆×D₇₃	—	2	2	no	—
29	#14556502	C₂×C₄	1	1	yes	characteristic
30	#14556501	D₄	1	1	yes	characteristic
31	#14556501	Q₈	1	1	yes	characteristic
32	M₃₂×C₇₃	—	2	1	no	—
33	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924038	—	4	1	no	—
35	#11924038	—	4	1	no	—
36	#11924038	—	4	1	no	—
37	#11924038	—	4	1	no	—
38	C₂×#10596833	—	4	1	no	—
39	C₂×#10596833	—	4	1	no	—
40	#11924031	—	2	4	no	—
41	C₂×C₄×D₇₃	—	2	1	no	—
42	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₇₃	—	2	1	no	—
46	C₈×D₇₃	—	2	1	no	—
47	C₈×D₇₃	—	2	1	no	—
48	C₈×D₇₃	—	2	1	no	—
49	C₄×Dic₇₃	—	2	1	no	—
50	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
51	#11924003	—	2	2	no	—
52	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₄²	1	1	yes	characteristic
55	C₁₆×C₇₃	—	2	2	no	—
56	C₂²×D₇₃	#101	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	#100	1	1	yes	characteristic
61	C₄×D₇₃	—	8	1	no	—
62	C₄×D₇₃	#104	1	1	yes	characteristic
63	C₄×D₇₃	#104	1	1	yes	characteristic
64	C₄×D₇₃	#104	1	1	yes	characteristic
65	C₄×D₇₃	#104	1	1	yes	characteristic
66	#10596833	—	8	1	no	—
67	#10596833	—	8	1	no	—
68	C₈×C₇₃	—	2	1	no	—
69	C₈×C₇₃	—	2	1	no	—
70	#10596831	—	2	1	no	—
71	#10596831	—	2	1	no	—
72	C₂²×C₇₃	#343	1	1	yes	characteristic
73	C₂×D₇₃	—	4	1	no	—
74	C₂×D₇₃	#330	1	1	yes	characteristic
75	C₂×D₇₃	#330	1	1	yes	characteristic
76	C₄×C₇₃	—	4	1	no	—
77	C₄×C₇₃	#339	1	1	yes	characteristic
78	C₄×C₇₃	#339	1	1	yes	characteristic
79	Dic₇₃	—	4	1	no	—
80	Dic₇₃	#329	1	1	yes	characteristic
81	Dic₇₃	#329	1	1	yes	characteristic
82	#7267	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	#1295	1	1	yes	characteristic
85	D₇₃	—	2	1	no	—
86	D₇₃	—	2	1	no	—
87	C₂×#347	—	73	1	no	—
88	C₂×#347	—	73	1	no	—
89	#1838	—	73	1	no	—
90	C₇₃	#7267	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	146	1	no	—
92	C₂×M₃₂	—	146	1	no	—
93	#423	—	73	1	no	—
94	#423	—	73	1	no	—
95	C₂×#97	—	73	1	no	—
96	#347	—	73	2	no	—
97	#347	—	73	1	no	—
98	#347	—	146	1	no	—
99	#347	—	146	1	no	—
100	#347	#64408	1	1	yes	characteristic
101	#341	—	146	1	no	—
102	#341	—	146	1	no	—
103	C₂×M₁₆	—	146	1	no	—
104	C₂×M₁₆	—	146	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄²	—	73	1	no	—
108	M₃₂	—	2	1	no	—
109	M₃₂	—	146	2	no	—
110	M₃₂	—	146	2	no	—
111	M₃₂	—	146	2	no	—
112	M₃₂	—	146	1	no	—
113	C₂×C₁₆	—	146	2	no	—
114	C₂×C₁₆	—	146	2	no	—
115	C₄⋊C₈	—	292	1	no	—
116	C₄⋊C₈	—	292	1	no	—
117	#97	—	73	1	no	—
118	#97	—	73	1	no	—
119	#97	—	73	1	no	—
120	#97	C₂×#64408	1	1	yes	characteristic
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	146	1	no	—
123	M₁₆	—	292	1	no	—
124	M₁₆	—	292	1	no	—
125	M₁₆	—	292	1	no	—
126	M₁₆	—	292	1	no	—
127	C₂×C₈	—	292	1	no	—
128	C₂×C₈	—	292	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	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	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
136	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
137	C₄²	—	73	1	no	—
138	C₄²	—	146	1	no	—
139	C₄²	C₄×#64408	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	146	2	no	—
142	C₁₆	—	146	4	no	—
143	C₂³	—	73	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	584	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	584	1	no	—
156	C₈	—	584	1	no	—
157	C₈	—	146	1	no	—
158	C₈	—	146	1	no	—
159	C₂²	—	292	1	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	73	1	no	—
162	C₂²	#17989466	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	292	1	no	—
165	C₄	—	73	1	no	—
166	C₄	—	73	1	no	—
167	C₄	#17989458	1	1	yes	characteristic
168	C₄	#17989459	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	146	1	no	—
171	C₂	—	146	1	no	—
172	C₂	#22624808	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22624808
commutator	C₄²×C₇₃	C₄²
Frattini	#97	C₂×#64408
Fitting	#347×C₇₃	C₄
socle	C₂×C₇₃	#1295
prime radical	C₂²×D₇₃	#101
cosocle	#97×D₇₃	C₂²
hypercenter	#347	#64408
hypocenter	C₇₃	#7267
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
90. C₇₃	#7267	semidirect

Maximal subgroups

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

subgroup	index
#22626148	2
#22625857	2
#347×D₇₃	2
#7267	73