Finite Group Databasebeta

← #33001985

Subgroups — #33001985

Order 18688.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4⋊Q8×Dic73C211yesmaximal, characteristic
2C2×#17988386C211yesmaximal, characteristic
3#22609925C212yesmaximal
4#22608057C212yesmaximal
5C2×#356×C73C211yesmaximal, characteristic
6C2×Q8×Dic73C411yescharacteristic
7C2×Q8×Dic73C2211yescharacteristic
8#17988610—22no—
9C2×#14556510C2212yes—
10#17988494C411yescharacteristic
11#17988494C2211yescharacteristic
12#17988486—22no—
13#17988457C411yescharacteristic
14C4⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic
16#17988386—22no—
17#17988355—22no—
18C2×C4⋊Q8×C73C2211yescharacteristic
19C2×#101×C73C2212yes—
20#356×C73—22no—
21C2×#11924012—22no—
22Q8×Dic73—41no—
23Q8×Dic73—41no—
24Q8×Dic73—41no—
25Q8×Dic73—41no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×C4⋊Dic73C2×C411yescharacteristic
30C2×C4⋊Dic73D412yes—
31C2×C4×Dic73—21no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73D412yes—
37#14556520—42no—
38#14556520—42no—
39#14556510—22no—
40#14556510—22no—
41#14556498—22no—
42#14556498—22no—
43C22×Q8×C73C2×C411yescharacteristic
44C22×Q8×C73C2311yescharacteristic
45C2×M16×C73—22no—
46C4⋊Q8×C73—22no—
47C4⋊Q8×C73D412yes—
48C4⋊Q8×C73Q812yes—
49C2×C4⋊C4×C73—22no—
50C2×C42×C73C2×C411yescharacteristic
51#101×C73—22no—
52#101×C73—22no—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73C22⋊C411yescharacteristic
56C22×Dic73C22⋊C412yes—
57C22×Dic73C22⋊C411yescharacteristic
58#11924012—42no—
59#11924012—42no—
60C2×#10596831—42no—
61C4⋊Dic73—41no—
62C4⋊Dic73—42no—
63C4⋊Dic73—42no—
64C4⋊Dic73—44no—
65C4⋊Dic73—41no—
66C4×Dic73—24no—
67C4×Dic73—22no—
68C4×Dic73—22no—
69C4×Dic73—21no—
70C4×Dic73—21no—
71C4×Dic73—42no—
72C4×Dic73—41no—
73C4×Dic73—41no—
74C4×Dic73—41no—
75C2×Q8×C73—21no—
76C2×Q8×C73—41no—
77C2×Q8×C73—41no—
78C2×Q8×C73—41no—
79C2×Q8×C73—41no—
80C2×Q8×C73C4⋊C412yes—
81C2×Q8×C73C2×D411yescharacteristic
82C2×Q8×C73C2×Q811yescharacteristic
83C2×Q8×C73#4112yes—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73C22×C411yescharacteristic
87C22×C4×C73C2×D412yes—
88M16×C73—42no—
89M16×C73—42no—
90C2×C8×C73—42no—
91C4⋊C4×C73—24no—
92C4⋊C4×C73—24no—
93C42×C73—21no—
94C42×C73C4⋊C412yes—
95C23×C73C2×C22⋊C411yescharacteristic
96Q8×C73—41no—
97Q8×C73—41no—
98Q8×C73—42no—
99Q8×C73—42no—
100Q8×C73—42no—
101C2×C4×C73—22no—
102C2×C4×C73—22no—
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73—21no—
106C2×C4×C73—22no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73C2×C4⋊C411yescharacteristic
112C2×C4×C73#11711yescharacteristic
113C2×C4×C73#12112yes—
114C2×C4×C73C22⋊Q812yes—
115C2×Dic73—22no—
116C2×Dic73—22no—
117C2×Dic73—21no—
118C2×Dic73—22no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—41no—
125C2×Dic73#9911yescharacteristic
126C2×Dic73#9911yescharacteristic
127C8×C73—82no—
128#10596831—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73#38011yescharacteristic
132C22×C73C2×#9911yescharacteristic
133C22×C73#41011yescharacteristic
134C4×C73—24no—
135C4×C73—22no—
136C4×C73—21no—
137C4×C73—21no—
138C4×C73—22no—
139C4×C73—42no—
140Dic73—21no—
141Dic73—42no—
142Dic73—41no—
143Dic73—42no—
144Dic73—81no—
145Dic73#35612yescomplemented
146#12802—731nomaximal, Sylow
147C2×C73—21no—
148C2×C73—21no—
149C2×C73#179911yescharacteristic
150C2×C73C2×#35611yescharacteristic
151C2×C73#213211yescharacteristic
152C4×C4⋊Q8—731no—
153C2×#356—731no—
154C2×#356D7311yescharacteristic
155#1801—732no—
156#1394—732no—
157C73#1280211yescharacteristic, complemented, Sylow
158C2×C4⋊Q8C2×D7311yescharacteristic
159C2×C4×Q8—731no—
160C2×C4×Q8—731no—
161#423—1462no—
162C2×#101—732no—
163C2×#101C2×D7312yes—
164#391—731no—
165#391—731no—
166#389—1462no—
167#384—731no—
168C4×C4⋊C4—1462no—
169C43—731no—
170#356—22no—
171#356—1462no—
172#343—1462no—
173C22×Q8C4×D7311yescharacteristic
174C22×Q8C22×D7311yescharacteristic
175C2×M16—22no—
176C2×M16—1462no—
177C4⋊Q8—22no—
178C4⋊Q8Dic14612yes—
179C4⋊Q8C22⋊D7312yes—
180C4×Q8—2921no—
181C4×Q8—2921no—
182C4×Q8—2921no—
183C4×Q8—2921no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—731no—
186C2×C4⋊C4—732no—
187C2×C4⋊C4—731no—
188C2×C4⋊C4—1461no—
189C2×C4⋊C4—1462no—
190C2×C42—731no—
191C2×C42—731no—
192C2×C42—732no—
193C2×C42—1461no—
194C2×C42—1461no—
195C2×C42—1461no—
196C2×C42C4×D7311yescharacteristic
197C4⋊C8—2922no—
198C4⋊C8—2922no—
199#101—22no—
200#101—22no—
201#101—1462no—
202#101—1462no—
203#95—1462no—
204#95—1462no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8#1192400712yes—
211C2×Q8C2×C22⋊D7311yescharacteristic
212C2×Q8C2×Dic14611yescharacteristic
213C2×Q8#1192402712yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—732no—
217C22×C4—731no—
218C22×C4—731no—
219C22×C4—1461no—
220C22×C4—1461no—
221C22×C4C2×C4×D7311yescharacteristic
222C22×C4D4×D7312yes—
223M16—42no—
224M16—42no—
225M16—2922no—
226M16—2922no—
227C2×C8—42no—
228C2×C8—2922no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—2921no—
232C4⋊C4—2924no—
233C4⋊C4—2921no—
234C4⋊C4—2922no—
235C4⋊C4—2922no—
236C42—21no—
237C42—2921no—
238C42—2922no—
239C42—2921no—
240C42—2921no—
241C42—1462no—
242C42—1462no—
243C42—1461no—
244C42—1461no—
245C42—1464no—
246C42#1192400712yes—
247C23C22⋊C4×D7311yescharacteristic
248Q8—41no—
249Q8—41no—
250Q8—42no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—2921no—
264C2×C4—2921no—
265C2×C4—2921no—
266C2×C4—2922no—
267C2×C4—2922no—
268C2×C4—2921no—
269C2×C4—731no—
270C2×C4—731no—
271C2×C4—1462no—
272C2×C4—1461no—
273C2×C4—1462no—
274C2×C4—1462no—
275C2×C4C2×#1192400711yescharacteristic
276C2×C4#1455654711yescharacteristic
277C2×C4#1455656312yes—
278C2×C4#1455656812yes—
279C8—82no—
280C8—5842no—
281C22—21no—
282C22—21no—
283C22#1798844511yescharacteristic
284C22#99×D7311yescharacteristic
285C22#1798855911yescharacteristic
286C4—24no—
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—2922no—
293C4—2921no—
294C4—2922no—
295C4—5841no—
296C4—732no—
297C4—1461no—
298C2—21no—
299C2—21no—
300C2#2260991711yescharacteristic
301C2#356×D7311yescharacteristic
302C2#2261161511yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC2×#356×C73C2
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Q8×C73C23
hypercenterC2×#356D73
hypocenterC73#12802
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
145. Dic73#356semidirect
157. C73#12802semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic732
C2×#179883862
#226099252
#226080572
C2×#356×C732
#1280273