Finite Group Databasebeta

← #32995295

Subgroups — #32995295

Order 18688.

364 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 83 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#22618646C211yesmaximal, characteristic
2#22611550C212yesmaximal, complemented
3#22607881C212yesmaximal, complemented
4#22607869C211yesmaximal, characteristic, complemented
5C73×#1321C211yesmaximal, characteristic, complemented
6C2×#14556599C2211yescharacteristic
7C2×#14556598C411yescharacteristic
8#17989133C411yescharacteristic
9M32×D73—22no—
10#17988661—22no—
11#17988655—22no—
12#17988610C2211yescharacteristic
13#17988553C411yescharacteristic
14C2×#14556501C411yescharacteristic
15#17988369C2212yes—
16#17988368—22no—
17#408×C73C2211yescharacteristic
18#349×C73C2212yes—
19C2×#11924028C2×C411yescharacteristic
20#14556599—42no—
21#14556598—42no—
22C2×#11924012—21no—
23C2×#11924012C2×C411yescharacteristic
24C2×#11924011D412yes—
25C22×#10596831—21no—
26C22×#10596831C2×C411yescharacteristic
27C2×C8×D73D412yes—
28#14556589C811yescharacteristic
29#14556587C811yescharacteristic
30#14556580C811yescharacteristic
31#14556578C811yescharacteristic
32#14556563—22no—
33C4×C22⋊D73—22no—
34C2×C4×Dic73C2×C411yescharacteristic
35#14556531—22no—
36#14556530—42no—
37C16×D73—42no—
38#14556520—22no—
39#14556503—22no—
40#14556501—21no—
41#14556501—21no—
42C2×#41×C73C2×C411yescharacteristic
43C2×M16×C73C2311yescharacteristic
44C22×C8×C73C2×C411yescharacteristic
45M32×C73—22no—
46C22⋊C8×C73—22no—
47#11924028—41no—
48#11924028—41no—
49C2×Dic146C2×C811yescharacteristic
50C2×D292C2×C811yescharacteristic
51C2×C22⋊D73—22no—
52C2×C22⋊D73M1612yes—
53C22×Dic73—21no—
54C2×C4×D73C22⋊C412yes—
55#11924012—42no—
56#11924012—41no—
57#11924011—42no—
58C2×#10596831—22no—
59C2×#10596831—21no—
60C2×#10596831—42no—
61C2×#10596831C22⋊C411yescharacteristic
62C2×#10596831C22⋊C411yescharacteristic
63C8×D73—22no—
64C8×D73—22no—
65#11924007—22no—
66C22⋊Dic73—22no—
67#11924005—22no—
68C4×Dic73—21no—
69C4×Dic73C2×C811yescharacteristic
70C4×Dic73C2×C811yescharacteristic
71#11924003—42no—
72#41×C73—41no—
73#41×C73—41no—
74C2×Q8×C73C2×C811yescharacteristic
75C2×D4×C73—21no—
76C2×D4×C73C2×C811yescharacteristic
77C22×C4×C73—21no—
78C22×C4×C73C22×C411yescharacteristic
79M16×C73—41no—
80C2×C8×C73—21no—
81C2×C8×C73—41no—
82C2×C8×C73C2×D412yes—
83C16×C73—42no—
84C23×C73—21no—
85C23×C73C2×M1611yescharacteristic
86C22×D73C22⋊C812yes—
87Q8×C73—41no—
88D4×C73—41no—
89D4×C73—42no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73C2×C22⋊C411yescharacteristic
96C2×C4×C73C22×C811yescharacteristic
97C22⋊D73—42no—
98C22⋊D73—82no—
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—81no—
102C2×Dic73C22⋊C812yes—
103D292—41no—
104C4×D73—22no—
105C4×D73—22no—
106Dic146—41no—
107C8×C73—22no—
108C8×C73—41no—
109#10596831—21no—
110#10596831—41no—
111#10596831#9912yes—
112C22×C73—21no—
113C22×C73—41no—
114C22×C73—42no—
115C22×C73—41no—
116C22×C73C2×C22⋊C811yescharacteristic
117C2×D73—22no—
118C2×D73—42no—
119C4×C73—21no—
120C4×C73—41no—
121C4×C73C2×#9911yescharacteristic
122C4×C73#41311yescharacteristic
123Dic73—22no—
124Dic73—41no—
125Dic73—41no—
126#11486—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—41no—
129C2×C73—81no—
130C2×C73#145911yescharacteristic
131D73—42no—
132#2932—731no—
133#2115—732no—
134#1327—732no—
135#1321—731no—
136#1321D7311yescharacteristic, complemented
137C73#1148611yescharacteristic, complemented, Sylow
138C2×#131—731no—
139C2×#131—731no—
140#532—731no—
141C2×M32—1462no—
142#435—1462no—
143#435—1462no—
144#423—731no—
145#408—731no—
146#408C2×D7311yescharacteristic
147C2×#97—731no—
148#349—732no—
149#349—1462no—
150#349C2×D7312yes—
151C2×#41—731no—
152C2×#41C4×D7311yescharacteristic
153#131—2922no—
154#131—2922no—
155C2×M16—732no—
156C2×M16—731no—
157C2×M16—1461no—
158C2×M16C22×D7311yescharacteristic
159C22×C8—731no—
160C22×C8—732no—
161C22×C8—1461no—
162C22×C8C4×D7311yescharacteristic
163C4⋊Q8—731no—
164#127—731no—
165#124—731no—
166#124—731no—
167#121—1462no—
168C4×D4—1462no—
169C2×C42—731no—
170M32—22no—
171M32—2922no—
172M32—1462no—
173C2×C16—2922no—
174C4⋊C8—1462no—
175C22⋊C8—22no—
176C22⋊C8—1462no—
177#97—1461no—
178#97—1461no—
179#41—41no—
180#41—41no—
181#41—2921no—
182#41—2921no—
183C2×Q8—731no—
184C2×Q8C8×D7311yescharacteristic
185C2×D4—21no—
186C2×D4—732no—
187C2×D4—731no—
188C2×D4—1462no—
189C2×D4C8×D7311yescharacteristic
190C22×C4—21no—
191C22×C4—732no—
192C22×C4—1461no—
193C22×C4C2×C4×D7311yescharacteristic
194M16—41no—
195M16—2922no—
196M16—2922no—
197M16—2921no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—2922no—
201C2×C8—731no—
202C2×C8—731no—
203C2×C8—1462no—
204C2×C8—1462no—
205C2×C8—1461no—
206C2×C8—1462no—
207C2×C8D4×D7312yes—
208C4⋊C4—1462no—
209C22⋊C4—1462no—
210C22⋊C4—1462no—
211C42—731no—
212C42—731no—
213C42—1461no—
214C16—42no—
215C16—2922no—
216C23—21no—
217C23—732no—
218C23M16×D7311yescharacteristic
219Q8—41no—
220Q8—2921no—
221D4—41no—
222D4—42no—
223D4—2921no—
224D4—2922no—
225D4—5842no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—5841no—
232C2×C4—732no—
233C2×C4—1461no—
234C2×C4—1461no—
235C2×C4—1462no—
236C2×C4—1462no—
237C2×C4C22⋊C4×D7311yescharacteristic
238C2×C4C2×C8×D7311yescharacteristic
239C8—22no—
240C8—41no—
241C8—2921no—
242C8—732no—
243C8—1461no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22—2922no—
249C22—1462no—
250C22C22⋊C8×D7311yescharacteristic
251C4—21no—
252C4—41no—
253C4—2921no—
254C4—2921no—
255C4—1462no—
256C4#99×D7311yescharacteristic
257C4#1798856811yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—2922no—
262C2#2260824611yescharacteristic
subgroupG/H
centerC4#99×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×M16C22×D73
FittingC73×#1321C2
socleC2×C73#1459
prime radical#17989133C4
cosocleC2×M16×C73C23
hypercenter#1321D73
hypocenterC73#11486
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #22611550C2semidirect
3. #22607881C2semidirect
4. #22607869C2semidirect
5. C73×#1321C2semidirect
136. #1321D73semidirect
137. C73#11486semidirect

Maximal subgroups

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

subgroupindex
#226186462
#226115502
#226078812
#226078692
C73×#13212
#1148673