Finite Group Databasebeta

← #34011306

Subgroups — #34011306

Order 18688.

314 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626294C211yesmaximal, characteristic
2#22625853C211yesmaximal, characteristic
3#22624919C214yesmaximal
4#22620515C211yesmaximal, characteristic
5#17989531—21no—
6#17989508—21no—
7#17989482C2212yes—
8#17989475—21no—
9#17989461—21no—
10#17989461C2212yes—
11#17989142—21no—
12#17988992C411yescharacteristic, complemented
13#17988937—21no—
14#17988825—21no—
15#17988825C2212yes—
16Q16×Dic73—21no—
17Q16×Dic73C412yescomplemented
18C4×C8×D73C2211yescharacteristic
19#494×C73C411yescharacteristic, complemented
20C2×C4⋊#64408—22no—
21C2×#11924031—22no—
22(C2×#64408)⋊C4—41no—
23(C2×#64408)⋊C4—41no—
24C2×C8×D73D411yescharacteristic
25C2×C8×D73C2311yescharacteristic
26#14556591—21no—
27#14556582—41no—
28#14556571—41no—
29#14556567—22no—
30Q8×Dic73—22no—
31#14556556—41no—
32#14556549—41no—
33C42×D73D411yescharacteristic
34#14556525—21no—
35#14556525C2×C412yes—
36#14556515—22no—
37#14556515—22no—
38C4×#10596831D411yescharacteristic
39C8×Dic73C2×C411yescharacteristic
40C8×Dic73D411yescharacteristic
41C2×Q16×C73—21no—
42C2×Q16×C73C2×C412yes—
43C4⋊Q8×C73—21no—
44C4×C8×C73C2×C411yescharacteristic
45C22×#64408—42no—
46C4⋊#64408—42no—
47C4⋊#64408—42no—
48#11924031—42no—
49C2×Dic146—41no—
50C2×C4×D73D811yescharacteristic
51C2×C4×D73D811yescharacteristic
52C2×C4×D73C2×D411yescharacteristic
53C2×#10596831C22⋊C411yescharacteristic
54C2×#10596831C2×D411yescharacteristic
55C8×D73—41no—
56C8×D73—41no—
57C4⋊Dic73—22no—
58C4⋊Dic73—41no—
59C4⋊Dic73—42no—
60#11924007—42no—
61#11924005—42no—
62C4×Dic73—42no—
63C4×Dic73C22⋊C411yescharacteristic
64C4×Dic73SD1611yescharacteristic
65C4×Dic73SD1611yescharacteristic
66C2×Q8×C73—22no—
67Q16×C73—42no—
68Q16×C73—42no—
69C2×C8×C73C22⋊C411yescharacteristic
70C2×C8×C73C22×C411yescharacteristic
71C4⋊C4×C73—41no—
72C4⋊C4×C73—41no—
73C42×C73C22⋊C411yescharacteristic
74C22×D73C2×D811yescharacteristic
75C2×#64408—82no—
76C2×#64408—82no—
77C2×#64408—82no—
78Q8×C73—42no—
79Q8×C73—42no—
80C2×C4×C73—42no—
81C2×C4×C73#10211yescharacteristic
82C2×C4×C73#10211yescharacteristic
83C2×C4×C73C2×C22⋊C411yescharacteristic
84C2×Dic73—42no—
85C2×Dic73#10211yescharacteristic
86C2×Dic73#10211yescharacteristic
87C2×Dic73C2×SD1611yescharacteristic
88C4×D73—22no—
89C4×D73—21no—
90C4×D73—21no—
91C4×D73—41no—
92Dic146—82no—
93C8×C73—21no—
94C8×C73—21no—
95#10596831—21no—
96#10596831—21no—
97C22×C73C2×#10211yescharacteristic
98C2×D73—21no—
99C2×D73—21no—
100C2×D73#51012yes—
101#64408—162no—
102C4×C73—21no—
103C4×C73—21no—
104C4×C73—82no—
105C4×C73#41011yescharacteristic
106C4×C73#42111yescharacteristic
107Dic73—21no—
108Dic73—21no—
109Dic73—21no—
110Dic73—82no—
111#11859—731nomaximal, Sylow
112C2×C73#151711yescharacteristic
113C2×C73#214111yescharacteristic
114C2×C73#214511yescharacteristic
115D73—22no—
116#3170—731no—
117#2163—731no—
118#1835—731no—
119#1348—734no—
120C73#1185911yescharacteristic, complemented, Sylow
121#533—1461no—
122#500—731no—
123#494#6440811yescharacteristic, complemented
124#487—1461no—
125#467—732no—
126#467—1461no—
127C4×Q16—732no—
128C4×Q16—1461no—
129C2×C4×C8—731no—
130#382—1461no—
131C4⋊C16—1461no—
132C22⋊C16—732no—
133#346—1461no—
134#340—732no—
135#340—1461no—
136C2×Q16—21no—
137C2×Q16C2×#6440812yes—
138C22×C8—731no—
139C22×C8—731no—
140C4⋊Q8—21no—
141C4⋊Q8—1461no—
142#125—2921no—
143C22⋊Q8—2921no—
144C22⋊Q8—1462no—
145C4×Q8—2921no—
146C4×Q8—1462no—
147#117—2921no—
148C2×C4⋊C4—1462no—
149C2×C42—731no—
150C2×C16—1462no—
151#107—732no—
152#107—1461no—
153#103—1462no—
154#103—1462no—
155C4×C8—731no—
156C4×C8—731no—
157C4×C8—731no—
158C4×C8C2×#6440811yescharacteristic
159#95—2921no—
160#95—2921no—
161C2×Q8—22no—
162C2×Q8—2921no—
163C22×C4—2922no—
164C22×C4—731no—
165C22×C4—731no—
166C22×C4—731no—
167Q16—42no—
168Q16—42no—
169C2×C8—2921no—
170C2×C8—2921no—
171C2×C8—731no—
172C2×C8—731no—
173C2×C8C22⋊#6440811yescharacteristic
174C2×C8C22×#6440811yescharacteristic
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—2922no—
178C4⋊C4—2922no—
179C4⋊C4—2921no—
180C4⋊C4—2922no—
181C4⋊C4—2922no—
182C4⋊C4—1462no—
183C22⋊C4—2922no—
184C42—2922no—
185C42—731no—
186C42—731no—
187C42—731no—
188C42C22⋊#6440811yescharacteristic
189C16—2922no—
190C23—731no—
191Q8—42no—
192Q8—42no—
193Q8—5842no—
194C2×C4—42no—
195C2×C4—2921no—
196C2×C4—2922no—
197C2×C4—5842no—
198C2×C4—5842no—
199C2×C4—5842no—
200C2×C4—731no—
201C2×C4—731no—
202C2×C4—731no—
203C2×C4—1462no—
204C2×C4—1461no—
205C2×C4—1461no—
206C2×C4#1455665311yescharacteristic
207C2×C4#1455665311yescharacteristic
208C2×C4C2×C22⋊#6440811yescharacteristic
209C8—21no—
210C8—21no—
211C8—1461no—
212C8—1461no—
213C22—732no—
214C22—1461no—
215C22—1461no—
216C22C2×#1455665311yescharacteristic
217C4—21no—
218C4—21no—
219C4—82no—
220C4—5842no—
221C4—11682no—
222C4—1461no—
223C4—1461no—
224C4—1461no—
225C4#1798956711yescharacteristic
226C4#1798958511yescharacteristic
227C2—1462no—
228C2#2262536011yescharacteristic
229C2#2262624111yescharacteristic
230C2#2262625311yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#494×C73C4
socleC22×C73C2×#102
prime radicalC22×D73C2×D8
cosocleC2×C8×D73C23
hypercenter#494#64408
hypocenterC73#11859
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #494

Complements

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

NH ≅ G/Ntype
12. #17988992C4semidirect
17. Q16×Dic73C4semidirect
19. #494×C73C4semidirect
120. C73#11859semidirect
123. #494#64408semidirect

Maximal subgroups

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

subgroupindex
#226262942
#226258532
#226249192
#226205152
#1185973