Finite Group Databasebeta

← #33001981

Subgroups — #33001981

Order 18688.

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

normal subgroups only

This group has 500 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
1#124×Dic73C211yesmaximal, characteristic
2C2×#17988384C211yesmaximal, characteristic
3#22609925C212yesmaximal, complemented
4#22608057C212yesmaximal, complemented
5C2×#355×C73C211yesmaximal, characteristic
6C2×Q8×Dic73C2211yescharacteristic
7C2×D4×Dic73C411yescharacteristic
8#17988610—22no—
9C2×#14556510C2212yes—
10#17988494C2211yescharacteristic
11#17988490C411yescharacteristic
12#17988480—22no—
13#17988453C411yescharacteristic
14C22⋊C4×Dic73—22no—
15C42×Dic73C411yescharacteristic
16#17988384—22no—
17#17988355—22no—
18C2×#124×C73C2211yescharacteristic
19C2×#101×C73C2212yes—
20#355×C73—22no—
21C23×Dic73—21no—
22C2×#11924012—22no—
23Q8×Dic73—41no—
24Q8×Dic73—41no—
25D4×Dic73—41no—
26D4×Dic73—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×C4⋊Dic73D412yes—
30C2×C22⋊Dic73—22no—
31C2×C22⋊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×C73C2311yescharacteristic
44C22×D4×C73C2×C411yescharacteristic
45C2×M16×C73—22no—
46#124×C73—22no—
47#124×C73D412yes—
48#124×C73Q812yes—
49C2×C22⋊C4×C73—22no—
50C2×C42×C73C2×C411yescharacteristic
51#101×C73—22no—
52#101×C73—22no—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73—44no—
56C22×Dic73—41no—
57C22×Dic73—41no—
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C22⋊C412yes—
60C22×Dic73C22⋊C411yescharacteristic
61#11924012—42no—
62#11924012—42no—
63C2×#10596831—42no—
64C4⋊Dic73—41no—
65C4⋊Dic73—42no—
66C4⋊Dic73—41no—
67C22⋊Dic73—44no—
68C22⋊Dic73—42no—
69C4×Dic73—22no—
70C4×Dic73—22no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C4×Dic73—42no—
74C4×Dic73—42no—
75C4×Dic73—41no—
76C24×C73—21no—
77C2×Q8×C73—41no—
78C2×Q8×C73—41no—
79C2×Q8×C73C2×D411yescharacteristic
80C2×Q8×C73C2×Q811yescharacteristic
81C2×Q8×C73#4112yes—
82C2×D4×C73—21no—
83C2×D4×C73—41no—
84C2×D4×C73—41no—
85C2×D4×C73C4⋊C412yes—
86C22×C4×C73—21no—
87C22×C4×C73C22×C411yescharacteristic
88C22×C4×C73C2×D412yes—
89M16×C73—42no—
90M16×C73—42no—
91C2×C8×C73—42no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—24no—
94C42×C73—21no—
95C42×C73C4⋊C412yes—
96C23×C73—22no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73C2×C22⋊C411yescharacteristic
102Q8×C73—41no—
103Q8×C73—41no—
104Q8×C73—42no—
105D4×C73—42no—
106D4×C73—42no—
107C2×C4×C73—22no—
108C2×C4×C73—21no—
109C2×C4×C73—22no—
110C2×C4×C73—21no—
111C2×C4×C73—22no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73C2×C4⋊C411yescharacteristic
115C2×C4×C73#11711yescharacteristic
116C2×C4×C73#12112yes—
117C2×C4×C73C22⋊Q812yes—
118C2×Dic73—22no—
119C2×Dic73—22no—
120C2×Dic73—21no—
121C2×Dic73—22no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—41no—
128C2×Dic73—82no—
129C2×Dic73—82no—
130C2×Dic73#9911yescharacteristic
131C2×Dic73#9911yescharacteristic
132C8×C73—82no—
133#10596831—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—44no—
137C22×C73—42no—
138C22×C73—82no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73#38011yescharacteristic
142C22×C73C2×#9911yescharacteristic
143C22×C73#41011yescharacteristic
144C4×C73—22no—
145C4×C73—21no—
146C4×C73—21no—
147C4×C73—22no—
148C4×C73—42no—
149Dic73—21no—
150Dic73—42no—
151Dic73—41no—
152Dic73—42no—
153Dic73—81no—
154Dic73#35512yescomplemented
155#12801—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73—82no—
159C2×C73#179911yescharacteristic
160C2×C73C2×#35511yescharacteristic
161C2×C73#213211yescharacteristic
162C4×#124—731no—
163C2×#355—731no—
164C2×#355D7311yescharacteristic
165#1801—732no—
166#1394—732no—
167C73#1280111yescharacteristic, complemented, Sylow
168C2×#124C2×D7311yescharacteristic
169C2×C4×Q8—731no—
170C2×C4×D4—731no—
171#423—1462no—
172C2×#101—732no—
173C2×#101C2×D7312yes—
174#391—731no—
175#390—731no—
176#388—1462no—
177#382—731no—
178C4×C22⋊C4—1462no—
179C43—731no—
180#355—22no—
181#355—1462no—
182#343—1462no—
183C22×Q8C22×D7311yescharacteristic
184C22×D4C4×D7311yescharacteristic
185C23×C4—1461no—
186C2×M16—22no—
187C2×M16—1462no—
188#124—22no—
189#124Dic14612yes—
190#124C22⋊D7312yes—
191C4×Q8—2921no—
192C4×Q8—2921no—
193C4×D4—2921no—
194C4×D4—2921no—
195C2×C4⋊C4—731no—
196C2×C4⋊C4—732no—
197C2×C4⋊C4—731no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1461no—
200C2×C22⋊C4—1462no—
201C2×C42—731no—
202C2×C42—731no—
203C2×C42—732no—
204C2×C42—1461no—
205C2×C42—1461no—
206C2×C42C4×D7311yescharacteristic
207C4⋊C8—2922no—
208C4⋊C8—2922no—
209#101—22no—
210#101—22no—
211#101—1462no—
212#101—1462no—
213#95—1462no—
214#95—1462no—
215C24—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8C2×C22⋊D7311yescharacteristic
219C2×Q8C2×Dic14611yescharacteristic
220C2×Q8#1192402712yes—
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1192400712yes—
225C22×C4—21no—
226C22×C4—2921no—
227C22×C4—2921no—
228C22×C4—2924no—
229C22×C4—732no—
230C22×C4—731no—
231C22×C4—731no—
232C22×C4—1461no—
233C22×C4—1461no—
234C22×C4C2×C4×D7311yescharacteristic
235C22×C4D4×D7312yes—
236M16—42no—
237M16—42no—
238M16—2922no—
239M16—2922no—
240C2×C8—42no—
241C2×C8—2922no—
242C4⋊C4—2921no—
243C4⋊C4—2921no—
244C4⋊C4—2922no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—2924no—
248C22⋊C4—2922no—
249C42—21no—
250C42—2921no—
251C42—2922no—
252C42—2922no—
253C42—1462no—
254C42—1462no—
255C42—1461no—
256C42—1461no—
257C42#1192400712yes—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23C22⋊C4×D7311yescharacteristic
264Q8—41no—
265Q8—41no—
266Q8—42no—
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2922no—
280C2×C4—2922no—
281C2×C4—2921no—
282C2×C4—5842no—
283C2×C4—5842no—
284C2×C4—731no—
285C2×C4—731no—
286C2×C4—1462no—
287C2×C4—1461no—
288C2×C4—1462no—
289C2×C4—1462no—
290C2×C4C2×#1192400711yescharacteristic
291C2×C4#1455654711yescharacteristic
292C2×C4#1455656312yes—
293C2×C4#1455656812yes—
294C8—82no—
295C8—5842no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22#1798844511yescharacteristic
304C22#99×D7311yescharacteristic
305C22#1798855911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—42no—
311C4—2922no—
312C4—2921no—
313C4—2922no—
314C4—5841no—
315C4—732no—
316C4—1461no—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#2260991711yescharacteristic
321C2#355×D7311yescharacteristic
322C2#2261161311yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC2×#355×C73C2
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC22×Q8×C73C23
hypercenterC2×#355D73
hypocenterC73#12801
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #22609925C2semidirect
4. #22608057C2semidirect
154. Dic73#355semidirect
167. C73#12801semidirect

Maximal subgroups

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

subgroupindex
#124×Dic732
C2×#179883842
#226099252
#226080572
C2×#355×C732
#1280173