Finite Group Databasebeta

← #32995312

Subgroups — #32995312

Order 18688.

356 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 87 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#22618639C211yesmaximal, characteristic
2#22611731C212yesmaximal, complemented
3#22607857C211yesmaximal, characteristic, complemented
4#22607856C212yesmaximal, complemented
5C73×#1315C211yesmaximal, characteristic, complemented
6C2×#14556599C411yescharacteristic
7C2×#14556598C411yescharacteristic
8#17989133C411yescharacteristic
9#17988662—22no—
10#17988650—22no—
11#17988610C411yescharacteristic
12#17988553C2211yescharacteristic
13C2×C8×Dic73C2211yescharacteristic
14#17988405—22no—
15C22⋊#11924003C2212yes—
16#17988362—22no—
17#408×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C2×#11924028C2×C411yescharacteristic
20#14556599—42no—
21#14556598—42no—
22C2×#11924012—21no—
23C2×#11924012C2×C411yescharacteristic
24C2×#11924011—21no—
25C22×#10596831—21no—
26C22×#10596831C2×C411yescharacteristic
27C2×C8×D73—21no—
28#14556589C811yescharacteristic
29#14556587C811yescharacteristic
30#14556580C811yescharacteristic
31#14556578C811yescharacteristic
32#14556563—22no—
33C4×C22⋊D73—22no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×#11924003—22no—
36#14556520—22no—
37#14556503—22no—
38C8×Dic73D412yes—
39C8×Dic73D412yes—
40C2×#41×C73C2×C411yescharacteristic
41C2×M16×C73C2×C411yescharacteristic
42C22×C8×C73C2311yescharacteristic
43C2×C16×C73—22no—
44C22⋊C8×C73—22no—
45#11924028—41no—
46#11924028—41no—
47C2×Dic146C2×C811yescharacteristic
48C2×D292C2×C811yescharacteristic
49C2×C22⋊D73—21no—
50C2×C22⋊D73—22no—
51C22×Dic73M1612yes—
52C2×C4×D73—21no—
53#11924012—42no—
54#11924012—41no—
55#11924011—42no—
56C2×#10596831—22no—
57C2×#10596831—21no—
58C2×#10596831—42no—
59C2×#10596831C22⋊C411yescharacteristic
60C2×#10596831C22⋊C411yescharacteristic
61C8×D73—42no—
62#11924007—22no—
63C22⋊Dic73—22no—
64#11924005—22no—
65C4×Dic73C22⋊C412yes—
66C4×Dic73C2×C811yescharacteristic
67C4×Dic73C2×C811yescharacteristic
68#11924003—42no—
69#41×C73—41no—
70#41×C73—41no—
71C2×Q8×C73C2×C811yescharacteristic
72C2×D4×C73—21no—
73C2×D4×C73C2×C811yescharacteristic
74C22×C4×C73—21no—
75C22×C4×C73C22×C411yescharacteristic
76M16×C73—41no—
77C2×C8×C73—21no—
78C2×C8×C73—41no—
79C2×C8×C73C2×D412yes—
80C16×C73—42no—
81C23×C73—21no—
82C23×C73C2×M1611yescharacteristic
83C22×D73—21no—
84Q8×C73—41no—
85D4×C73—41no—
86D4×C73—42no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73C2×C22⋊C411yescharacteristic
93C2×C4×C73C22×C811yescharacteristic
94C22⋊D73—44no—
95C22⋊D73—42no—
96C2×Dic73—21no—
97C2×Dic73—42no—
98C2×Dic73C22⋊C812yes—
99C2×Dic73C22⋊C812yes—
100D292—41no—
101C4×D73—41no—
102C4×D73—41no—
103Dic146—41no—
104C8×C73—22no—
105C8×C73—41no—
106#10596831—21no—
107#10596831—41no—
108#10596831#10012yes—
109C22×C73—22no—
110C22×C73—21no—
111C22×C73—42no—
112C22×C73—41no—
113C22×C73C2×C22⋊C811yescharacteristic
114C2×D73—42no—
115C2×D73—41no—
116C4×C73—21no—
117C4×C73—41no—
118C4×C73#41011yescharacteristic
119C4×C73C2×#10011yescharacteristic
120Dic73—22no—
121Dic73—22no—
122Dic73—41no—
123#11491—731nomaximal, Sylow
124C2×C73—41no—
125C2×C73—81no—
126C2×C73#146011yescharacteristic
127C2×C73#211712yes—
128D73—81no—
129#2931—731no—
130#2163—732no—
131#1315—731no—
132#1315—732no—
133#1315D7311yescharacteristic, complemented
134C73#1149111yescharacteristic, complemented, Sylow
135C2×#131—731no—
136C2×#131—731no—
137#532—731no—
138#436—1462no—
139C8×D4—1462no—
140#423—731no—
141#408—731no—
142#408C2×D7311yescharacteristic
143C2×C4×C8—731no—
144C4⋊C16—1462no—
145C22⋊C16—732no—
146C22⋊C16C2×D7312yes—
147#346—1462no—
148C2×#41—731no—
149C2×#41C4×D7311yescharacteristic
150#131—2922no—
151#131—2922no—
152C2×M16—731no—
153C2×M16—1461no—
154C2×M16—1461no—
155C2×M16C4×D7311yescharacteristic
156C22×C8—731no—
157C22×C8—1461no—
158C22×C8—1461no—
159C22×C8C22×D7311yescharacteristic
160C4⋊Q8—731no—
161#127—731no—
162#124—731no—
163#124—731no—
164#121—1462no—
165C4×D4—1462no—
166C2×C42—731no—
167C2×C16—22no—
168C2×C16—1462no—
169C4⋊C8—1462no—
170C22⋊C8—22no—
171C22⋊C8—1462no—
172C4×C8—732no—
173C4×C8—732no—
174#41—41no—
175#41—41no—
176#41—2921no—
177#41—2921no—
178C2×Q8—731no—
179C2×Q8C8×D7311yescharacteristic
180C2×D4—21no—
181C2×D4—731no—
182C2×D4—1462no—
183C2×D4—1461no—
184C2×D4C8×D7311yescharacteristic
185C22×C4—21no—
186C22×C4—732no—
187C22×C4—1461no—
188C22×C4C2×C4×D7311yescharacteristic
189M16—41no—
190M16—2922no—
191M16—2922no—
192M16—2921no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8—2922no—
196C2×C8—2922no—
197C2×C8—731no—
198C2×C8—731no—
199C2×C8—1461no—
200C2×C8—1462no—
201C2×C8D4×D7312yes—
202C4⋊C4—1462no—
203C22⋊C4—1462no—
204C22⋊C4—1462no—
205C42—732no—
206C42—731no—
207C42—731no—
208C16—42no—
209C16—2922no—
210C23—21no—
211C23—1461no—
212C23M16×D7311yescharacteristic
213Q8—41no—
214Q8—2921no—
215D4—41no—
216D4—42no—
217D4—2921no—
218D4—2922no—
219D4—2924no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2921no—
226C2×C4—2921no—
227C2×C4—2922no—
228C2×C4—732no—
229C2×C4—732no—
230C2×C4—1461no—
231C2×C4C22⋊C4×D7311yescharacteristic
232C2×C4C2×C8×D7311yescharacteristic
233C8—22no—
234C8—41no—
235C8—2921no—
236C8—732no—
237C8—1461no—
238C22—22no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—2922no—
243C22—2921no—
244C22C22⋊C8×D7311yescharacteristic
245C4—21no—
246C4—41no—
247C4—2921no—
248C4—1462no—
249C4—1462no—
250C4#1798855911yescharacteristic
251C4#100×D7311yescharacteristic
252C2—41no—
253C2—81no—
254C2—5841no—
255C2#2260825011yescharacteristic
256C2#2261155712yes—
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1315C2
socleC22×C73C2×C22⋊C8
prime radical#17989133C4
cosocleC22×C8×C73C23
hypercenter#1315D73
hypocenterC73#11491
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #22611731C2semidirect
3. #22607857C2semidirect
4. #22607856C2semidirect
5. C73×#1315C2semidirect
133. #1315D73semidirect
134. C73#11491semidirect

Maximal subgroups

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

subgroupindex
#226186392
#226117312
#226078572
#226078562
C73×#13152
#1149173