Finite Group Databasebeta

← #32983391

Subgroups — #32983391

Order 18688.

486 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 109 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#22619014C211yesmaximal, characteristic
2#22614920C211yesmaximal, characteristic
3#22609924C211yesmaximal, characteristic
4#22608603C211yesmaximal, characteristic
5#22607833C211yesmaximal, characteristic
6#22607832C211yesmaximal, characteristic
7C73×#1546C211yesmaximal, characteristic
8C2×#14556589—21no—
9C2×C4×Dic146C2211yescharacteristic
10#17988717—22no—
11#17988714—22no—
12#17988641—22no—
13#17988640C2211yescharacteristic
14#17988611C2211yescharacteristic
15C2×#14556509—21no—
16#17988518C414yes—
17#17988494—21no—
18#17988483—22no—
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—
28#14556589—42no—
29#14556589—42no—
30C4×Dic146—21no—
31C4×Dic146—21no—
32C4×Dic146—21no—
33C4×Dic146—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73D411yescharacteristic
36C2×#11924007—24no—
37C2×#11924007C2×C412yes—
38C2×C4×Dic73—21no—
39C2×C4×Dic73D412yes—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831D412yes—
43#14556520—24no—
44#14556509—42no—
45C22⋊#10596831—21no—
46#14556501—24no—
47C4×#10596831—21no—
48#14556498—24no—
49#14556498C2×C412yes—
50C22×Q8×C73—21no—
51C2×M16×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73C2×C412yes—
58C2×C42×C73C2311yescharacteristic
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62C22⋊C8×C73—21no—
63C4×C8×C73—21no—
64C2×Dic146—21no—
65C2×Dic146—21no—
66C2×Dic146—21no—
67C2×Dic146—21no—
68C2×Dic146—21no—
69C2×Dic146—21no—
70C22×Dic73—22no—
71C22×Dic73C22⋊C412yes—
72C22×Dic73C22⋊C412yes—
73#11924012—41no—
74#11924012—41no—
75C2×#10596831—22no—
76C2×#10596831—41no—
77C4⋊Dic73—24no—
78C4⋊Dic73—21no—
79C4⋊Dic73—21no—
80#11924007—22no—
81#11924007—22no—
82#11924007—48no—
83C4×Dic73—22no—
84C4×Dic73—22no—
85C4×Dic73—41no—
86C4×Dic73—41no—
87C2×Q8×C73—41no—
88C2×Q8×C73—41no—
89C2×Q8×C73—41no—
90C2×Q8×C73—41no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C22×C4×C73—22no—
94C22×C4×C73C22×C411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97M16×C73—41no—
98M16×C73—41no—
99C2×C8×C73—22no—
100C2×C8×C73—41no—
101C4⋊C4×C73—22no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—41no—
104C4⋊C4×C73—41no—
105C42×C73C2×D411yescharacteristic
106C42×C73C2×D411yescharacteristic
107C42×C73#4111yescharacteristic
108C42×C73#4111yescharacteristic
109C23×C73C2×C22⋊C411yescharacteristic
110Q8×C73—81no—
111Q8×C73—81no—
112Q8×C73—81no—
113Q8×C73—81no—
114C2×C4×C73—22no—
115C2×C4×C73—41no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73C4×D411yescharacteristic
120C2×C4×C73C4×D411yescharacteristic
121C2×C4×C73#12011yescharacteristic
122C2×C4×C73#12111yescharacteristic
123C2×C4×C73#12111yescharacteristic
124C2×C4×C73#12311yescharacteristic
125C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C73#13711yescharacteristic
129C2×Dic73—24no—
130C2×Dic73—22no—
131C2×Dic73—22no—
132C2×Dic73—22no—
133C2×Dic73—22no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137Dic146—41no—
138Dic146—41no—
139Dic146—41no—
140Dic146—41no—
141C8×C73—42no—
142C8×C73—81no—
143#10596831—42no—
144#10596831—81no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73#38611yescharacteristic
148C22×C73#41711yescharacteristic
149C22×C73#41811yescharacteristic
150C4×C73—21no—
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—82no—
154C4×C73#45612yes—
155C4×C73#46811yescharacteristic
156C4×C73#46811yescharacteristic
157C4×C73#46911yescharacteristic
158C4×C73#46911yescharacteristic
159Dic73—44no—
160Dic73—42no—
161Dic73—42no—
162#9402—731nomaximal, Sylow
163C2×C73—21no—
164C2×C73—21no—
165C2×C73#188611yescharacteristic
166C2×C73#189511yescharacteristic
167C2×C73#191511yescharacteristic
168#2983—731no—
169#2546—731no—
170#1801—731no—
171#1546—731no—
172#1546D7311yescharacteristic
173#1303—731no—
174#1303—731no—
175C73#940211yescharacteristic, complemented, Sylow
176C2×C4⋊Q8—1461no—
177C2×C4⋊Q8C2×D7311yescharacteristic
178C2×C4×Q8—731no—
179#446—1462no—
180#446—1462no—
181#432—1462no—
182#431—731no—
183#431C2×D7311yescharacteristic
184#423—731no—
185#423C2×D7311yescharacteristic
186C2×#101—1461no—
187#398—734no—
188#391—1461no—
189#389—1462no—
190C4×C4⋊C4—731no—
191#343—1461no—
192#332—22no—
193#332—1462no—
194C22×Q8—21no—
195C22×Q8—731no—
196C2×M16—21no—
197C2×M16—1461no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—2922no—
203C4⋊Q8—2922no—
204C4×Q8—1461no—
205C4×Q8—1461no—
206C4×Q8—1461no—
207C4×Q8—1461no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—732no—
210C2×C4⋊C4—731no—
211C2×C4⋊C4—1464no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4C4×D7312yes—
214C2×C42—732no—
215C2×C42—1461no—
216C2×C42C22×D7311yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—732no—
220C4⋊C8—1464no—
221C4⋊C8—1461no—
222C4⋊C8—1461no—
223C4⋊C8C22⋊D7312yes—
224#101—2922no—
225C22⋊C8—21no—
226C22⋊C8—1461no—
227#97—1464no—
228C4×C8—21no—
229C4×C8—1461no—
230#95—732no—
231#95—1464no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—1461no—
239C2×Q8—1461no—
240C2×Q8—1461no—
241C2×Q8—1461no—
242C2×Q8—1461no—
243C2×Q8—1461no—
244C22×C4—22no—
245C22×C4—732no—
246C22×C4—732no—
247C22×C4—1462no—
248C22×C4C2×C4×D7311yescharacteristic
249C22×C4D4×D7311yescharacteristic
250C22×C4D4×D7311yescharacteristic
251M16—41no—
252M16—41no—
253M16—2921no—
254M16—2921no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—2921no—
258C2×C8—1462no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—2928no—
264C4⋊C4—1464no—
265C4⋊C4—1462no—
266C4⋊C4—1462no—
267C4⋊C4—1461no—
268C4⋊C4—1461no—
269C42—2921no—
270C42—2921no—
271C42—1462no—
272C42—1462no—
273C42D4×D7311yescharacteristic
274C42C2×C22⋊D7311yescharacteristic
275C42#1192402711yescharacteristic
276C42#1192402811yescharacteristic
277C23C22⋊C4×D7311yescharacteristic
278Q8—81no—
279Q8—81no—
280Q8—81no—
281Q8—81no—
282Q8—2921no—
283Q8—2921no—
284Q8—2921no—
285Q8—2921no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—2922no—
292C2×C4—2922no—
293C2×C4—2922no—
294C2×C4—1464no—
295C2×C4—1462no—
296C2×C4—1462no—
297C2×C4—1462no—
298C2×C4—1462no—
299C2×C4#1455655111yescharacteristic
300C2×C4C4×C22⋊D7311yescharacteristic
301C2×C4#1455655911yescharacteristic
302C2×C4#1455656211yescharacteristic
303C2×C4#1455656411yescharacteristic
304C2×C4#1455657311yescharacteristic
305C2×C4#1455661511yescharacteristic
306C2×C4#1455661511yescharacteristic
307C2×C4#1455661811yescharacteristic
308C2×C4#1455662011yescharacteristic
309C8—42no—
310C8—81no—
311C8—2922no—
312C8—5841no—
313C22—21no—
314C22—21no—
315C22#1798846311yescharacteristic
316C22#1798858511yescharacteristic
317C22#1798859011yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—82no—
322C4—2924no—
323C4—2922no—
324C4—2922no—
325C4#1798876812yes—
326C4#1798883011yescharacteristic
327C4#1798883211yescharacteristic
328C4#1798883711yescharacteristic
329C4#1798883911yescharacteristic
330C2—21no—
331C2—21no—
332C2#2261032111yescharacteristic
333C2#2261036611yescharacteristic
334C2#2261048011yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1546C2
socleC22×C73#386
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1546D73
hypocenterC73#9402
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
175. C73#9402semidirect

Maximal subgroups

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

subgroupindex
#226190142
#226149202
#226099242
#226086032
#226078332
#226078322
C73×#15462
#940273