Finite Group Databasebeta

← #32983361

Subgroups — #32983361

Order 18688.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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#22619011C211yesmaximal, characteristic
2#22615147C211yesmaximal, characteristic
3#22609924C211yesmaximal, characteristic
4#22608616C211yesmaximal, characteristic
5#22607833C211yesmaximal, characteristic
6#22607832C211yesmaximal, characteristic
7C73×#1549C211yesmaximal, characteristic
8C2×C4×Dic146C2211yescharacteristic
9C8×Dic146—22no—
10#17988711—22no—
11#17988640C2211yescharacteristic
12#17988637—22no—
13#17988611C2211yescharacteristic
14C2×#14556509—21no—
15#17988525C414yes—
16#17988494—22no—
17#17988492—21no—
18#17988452—21no—
19C4×C4⋊Dic73C2211yescharacteristic
20#17988355—21no—
21#17988333—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23#431×C73C2211yescharacteristic
24#423×C73C2211yescharacteristic
25#332×C73—22no—
26C22×Dic146D411yescharacteristic
27C2×#11924012—21no—
28C4×Dic146—21no—
29C4×Dic146—21no—
30C4×Dic146—21no—
31C4×Dic146—21no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73D411yescharacteristic
34C2×#11924007C2×C412yes—
35C2×C4×Dic73—21no—
36C2×C4×Dic73D412yes—
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831D412yes—
40#14556520—24no—
41#14556509—42no—
42C22⋊#10596831—21no—
43C4×#10596831—21no—
44C8×Dic73—24no—
45#14556498—24no—
46#14556498—24no—
47#14556498C2×C412yes—
48C22×Q8×C73—21no—
49C2×M16×C73—21no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73C2×C412yes—
56C2×C42×C73C2311yescharacteristic
57C4⋊C8×C73—21no—
58C4⋊C8×C73—21no—
59C4⋊C8×C73D412yes—
60C22⋊C8×C73—21no—
61C4×C8×C73—21no—
62C2×Dic146—21no—
63C2×Dic146—21no—
64C2×Dic146—21no—
65C2×Dic146—21no—
66C2×Dic146SD1612yes—
67C2×Dic146Q1612yes—
68C22×Dic73—22no—
69C22×Dic73C22⋊C412yes—
70C22×Dic73C22⋊C412yes—
71#11924012—41no—
72#11924012—41no—
73C2×#10596831—22no—
74C2×#10596831—41no—
75C4⋊Dic73—21no—
76C4⋊Dic73—21no—
77C4⋊Dic73—42no—
78#11924007—22no—
79#11924007—22no—
80C4×Dic73—22no—
81C4×Dic73—22no—
82C4×Dic73—41no—
83C4×Dic73—41no—
84C2×Q8×C73—44no—
85C2×Q8×C73—41no—
86C2×Q8×C73—41no—
87C22×C4×C73—22no—
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91M16×C73—41no—
92M16×C73—41no—
93C2×C8×C73—22no—
94C2×C8×C73—41no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73—22no—
97C4⋊C4×C73—41no—
98C4⋊C4×C73—41no—
99C42×C73C2×D411yescharacteristic
100C42×C73C2×D411yescharacteristic
101C42×C73#4111yescharacteristic
102C42×C73#4111yescharacteristic
103C23×C73C2×C22⋊C411yescharacteristic
104Q8×C73—82no—
105Q8×C73—82no—
106C2×C4×C73—22no—
107C2×C4×C73—41no—
108C2×C4×C73—44no—
109C2×C4×C73—42no—
110C2×C4×C73C4×D411yescharacteristic
111C2×C4×C73C4×D411yescharacteristic
112C2×C4×C73#12011yescharacteristic
113C2×C4×C73#12111yescharacteristic
114C2×C4×C73#12111yescharacteristic
115C2×C4×C73#12311yescharacteristic
116C2×C4×C73C2×SD1611yescharacteristic
117C2×C4×C73C2×Q1611yescharacteristic
118C2×C4×C73#13511yescharacteristic
119C2×C4×C73#13511yescharacteristic
120C2×Dic73—22no—
121C2×Dic73—22no—
122C2×Dic73—22no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73#10314yes—
128Dic146—22no—
129Dic146—22no—
130Dic146—22no—
131Dic146—22no—
132C8×C73—42no—
133C8×C73—81no—
134#10596831—42no—
135#10596831—81no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73#38611yescharacteristic
139C22×C73C2×#10311yescharacteristic
140C22×C73#41611yescharacteristic
141C4×C73—21no—
142C4×C73—21no—
143C4×C73—21no—
144C4×C73—82no—
145C4×C73#45412yes—
146C4×C73#46511yescharacteristic
147C4×C73#46511yescharacteristic
148C4×C73#46711yescharacteristic
149C4×C73#46711yescharacteristic
150Dic73—24no—
151Dic73—42no—
152Dic73—82no—
153#9396—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73#188811yescharacteristic
157C2×C73#189211yescharacteristic
158C2×C73#191511yescharacteristic
159#2982—731no—
160#2568—731no—
161#1801—731no—
162#1549—731no—
163#1549D7311yescharacteristic
164#1303—731no—
165#1303—731no—
166C73#939611yescharacteristic, complemented, Sylow
167C2×C4⋊Q8C2×D7311yescharacteristic
168C2×C4×Q8—731no—
169C8×Q8—1462no—
170C8×Q8—1462no—
171#431—731no—
172#431—1462no—
173#431C2×D7311yescharacteristic
174#423—731no—
175#423C2×D7311yescharacteristic
176C2×#101—1461no—
177#400—734no—
178#391—1462no—
179#391—1461no—
180#382—1461no—
181C4×C4⋊C4—731no—
182#343—1461no—
183#332—22no—
184#332—1462no—
185C22×Q8—21no—
186C22×Q8—731no—
187C2×M16—21no—
188C2×M16—1461no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4×Q8—1461no—
194C4×Q8—1461no—
195C4×Q8—1461no—
196C4×Q8—1461no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—732no—
199C2×C4⋊C4—731no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4C4×D7312yes—
202C2×C42—732no—
203C2×C42—1461no—
204C2×C42C22×D7311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—732no—
208C4⋊C8—1464no—
209C4⋊C8—1461no—
210C4⋊C8—1461no—
211C4⋊C8C22⋊D7312yes—
212#101—2922no—
213C22⋊C8—21no—
214C22⋊C8—1461no—
215C4×C8—21no—
216C4×C8—1464no—
217C4×C8—1461no—
218#95—732no—
219#95—1464no—
220#95—1464no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—732no—
225C2×Q8—732no—
226C2×Q8—1461no—
227C2×Q8—1461no—
228C2×Q8—1461no—
229C2×Q8—1461no—
230C22×C4—22no—
231C22×C4—732no—
232C22×C4—732no—
233C22×C4—1462no—
234C22×C4C2×C4×D7311yescharacteristic
235C22×C4D4×D7311yescharacteristic
236C22×C4D4×D7311yescharacteristic
237M16—41no—
238M16—41no—
239M16—2921no—
240M16—2921no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—2921no—
244C2×C8—1462no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—2922no—
250C4⋊C4—1462no—
251C4⋊C4—1462no—
252C4⋊C4—1461no—
253C4⋊C4—1461no—
254C42—2921no—
255C42—2921no—
256C42—1462no—
257C42—1462no—
258C42D4×D7311yescharacteristic
259C42C2×C22⋊D7311yescharacteristic
260C42#1192402711yescharacteristic
261C42#1192402811yescharacteristic
262C23C22⋊C4×D7311yescharacteristic
263Q8—82no—
264Q8—82no—
265Q8—1462no—
266Q8—1462no—
267Q8—1462no—
268Q8—1462no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—2922no—
274C2×C4—2922no—
275C2×C4—2922no—
276C2×C4—2922no—
277C2×C4—734no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4#1455655111yescharacteristic
282C2×C4C4×C22⋊D7311yescharacteristic
283C2×C4#1455655911yescharacteristic
284C2×C4#1455656211yescharacteristic
285C2×C4#1455656411yescharacteristic
286C2×C4#1455657311yescharacteristic
287C2×C4SD16×D7311yescharacteristic
288C2×C4Q16×D7311yescharacteristic
289C2×C4#1455661211yescharacteristic
290C2×C4#1455661411yescharacteristic
291C8—42no—
292C8—81no—
293C8—2922no—
294C8—5841no—
295C22—21no—
296C22—21no—
297C22#1798846311yescharacteristic
298C22#103×D7311yescharacteristic
299C22#1798858211yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—2922no—
305C4—5842no—
306C4—1464no—
307C4#1798875812yes—
308C4#1798881311yescharacteristic
309C4#1798881911yescharacteristic
310C4#1798882511yescharacteristic
311C4#1798882711yescharacteristic
312C2—21no—
313C2—21no—
314C2#2261033111yescharacteristic
315C2#2261035111yescharacteristic
316C2#2261048011yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1549C2
socleC22×C73#386
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1549D73
hypocenterC73#9396
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
166. C73#9396semidirect

Maximal subgroups

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

subgroupindex
#226190112
#226151472
#226099242
#226086162
#226078332
#226078322
C73×#15492
#939673