Finite Group Databasebeta

← #32986493

Subgroups — #32986493

Order 18688.

568 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#17988920C211yesmaximal, characteristic
2#22614410C211yesmaximal, characteristic
3#22610917C211yesmaximal, characteristic
4#22610022C211yesmaximal, characteristic
5#22608503C211yesmaximal, characteristic
6#22607778C211yesmaximal, characteristic
7C73×#3189C211yesmaximal, characteristic
8C22×C4⋊Dic73C2211yescharacteristic
9#17988920—22no—
10#17988920—22no—
11C2×#14556524—21no—
12C2×#14556515—21no—
13C2×C22⋊#10596831C2211yescharacteristic
14#17988517—22no—
15#17988494—21no—
16#17988459—22no—
17#17988456—21no—
18#17988447C2211yescharacteristic
19C4×C22⋊Dic73C2211yescharacteristic
20#17988354—22no—
21#17988350—22no—
22#17988324—22no—
23C2×C22⋊Q8×C73C2211yescharacteristic
24C2×C22⋊Q8×C73C2211yescharacteristic
25#484×C73—21no—
26#484×C73—21no—
27#467×C73—22no—
28#451×C73—21no—
29#451×C73—21no—
30C2×C22⋊C8×C73C2211yescharacteristic
31C23×Dic73—21no—
32C22×#10596831—21no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73D412yes—
40C2×#11924007—24no—
41C2×C22⋊Dic73—21no—
42C2×C22⋊Dic73D412yes—
43C2×C4×Dic73—21no—
44#14556524—24no—
45#14556515—24no—
46C22⋊#10596831—21no—
47C22⋊#10596831D412yes—
48#14556498—24no—
49#14556498—21no—
50#14556498—21no—
51C22×Q8×C73D411yescharacteristic
52C22×Q8×C73D411yescharacteristic
53C23×C4×C73C2311yescharacteristic
54C2×Q16×C73—22no—
55C22×C8×C73D412yes—
56C22⋊Q8×C73—21no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—21no—
59C22⋊Q8×C73—21no—
60C22⋊Q8×C73—21no—
61C22⋊Q8×C73—21no—
62C22⋊Q8×C73—21no—
63C22⋊Q8×C73—21no—
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73D411yescharacteristic
67C2×C4⋊C4×C73D411yescharacteristic
68C2×C22⋊C4×C73—21no—
69C2×C22⋊C4×C73—21no—
70#107×C73—22no—
71#103×C73—22no—
72#103×C73—22no—
73C22⋊C8×C73—21no—
74C22⋊C8×C73—21no—
75C22×Dic73—22no—
76C22×Dic73—22no—
77C22×Dic73—41no—
78C22×Dic73—41no—
79C22×Dic73—41no—
80C22×Dic73—82no—
81C2×#10596831—22no—
82C2×#10596831—42no—
83C4⋊Dic73—22no—
84C4⋊Dic73—22no—
85C4⋊Dic73—41no—
86C4⋊Dic73—41no—
87C4⋊Dic73—42no—
88#11924007—84no—
89C22⋊Dic73—22no—
90C22⋊Dic73—42no—
91C22⋊Dic73Q1614yescomplemented
92C4×Dic73—42no—
93C24×C73C2×D411yescharacteristic
94C2×Q8×C73—21no—
95C2×Q8×C73—21no—
96C2×Q8×C73—21no—
97C2×Q8×C73—21no—
98C2×Q8×C73—42no—
99C2×Q8×C73—42no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73—42no—
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73#4111yescharacteristic
111Q16×C73—82no—
112C2×C8×C73—22no—
113C2×C8×C73—42no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C22⋊C4×C73—41no—
125C22⋊C4×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73#12011yescharacteristic
129C23×C73#12111yescharacteristic
130C23×C73#12711yescharacteristic
131C23×C73C2×Q1612yes—
132C23×C73#13712yes—
133Q8×C73—42no—
134Q8×C73—42no—
135Q8×C73—42no—
136Q8×C73—81no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—81no—
150C2×C4×C73—81no—
151C2×C4×C73#12011yescharacteristic
152C2×C4×C73#12011yescharacteristic
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×Dic73—24no—
156C2×Dic73—42no—
157C2×Dic73—42no—
158C2×Dic73—44no—
159C2×Dic73—42no—
160C2×Dic73—82no—
161C2×Dic73—82no—
162C2×Dic73—81no—
163C2×Dic73—81no—
164C2×Dic73—81no—
165C8×C73—42no—
166#10596831—42no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—42no—
174C22×C73—42no—
175C22×C73#39211yescharacteristic
176C22×C73#45112yes—
177C22×C73#46212yes—
178C22×C73#49411yescharacteristic
179C22×C73#49711yescharacteristic
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—81no—
184C4×C73—81no—
185C4×C73—81no—
186C4×C73—81no—
187C4×C73#45812yes—
188Dic73—44no—
189Dic73—82no—
190Dic73—161no—
191#9959—731nomaximal, Sylow
192C2×C73—22no—
193C2×C73—21no—
194C2×C73—22no—
195C2×C73—21no—
196C2×C73#200811yescharacteristic
197C2×C73#201611yescharacteristic
198C2×C73#202211yescharacteristic
199#3189D7311yescharacteristic
200C2×#484—731no—
201#2502—731no—
202#1996—731no—
203#1824—731no—
204#1520—731no—
205#1282—731no—
206C73#995911yescharacteristic, complemented, Sylow
207C2×C22⋊Q8C2×D7311yescharacteristic
208C2×C22⋊Q8C2×D7311yescharacteristic
209C22×C4⋊C4—731no—
210#484—21no—
211#484—21no—
212#484—1462no—
213#484—1462no—
214#467—22no—
215#451—21no—
216#451—21no—
217C2×#107—1461no—
218C2×#103—1461no—
219C2×C22⋊C8—731no—
220C2×C22⋊C8C2×D7311yescharacteristic
221#398—1462no—
222#391—1461no—
223#385—1462no—
224#384—1461no—
225#380—731no—
226C4×C22⋊C4—731no—
227#342—1462no—
228#340—1462no—
229#328—1462no—
230C22×Q8C22⋊D7311yescharacteristic
231C22×Q8C22⋊D7311yescharacteristic
232C23×C4—1461no—
233C23×C4C22×D7311yescharacteristic
234C2×Q16—22no—
235C22×C8—1461no—
236C22×C8C22⋊D7312yes—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—732no—
248C2×C4⋊C4—1461no—
249C2×C4⋊C4—1461no—
250C2×C4⋊C4—1461no—
251C2×C4⋊C4—1461no—
252C2×C4⋊C4—1461no—
253C2×C4⋊C4—1461no—
254C2×C4⋊C4—1464no—
255C2×C4⋊C4C22⋊D7311yescharacteristic
256C2×C4⋊C4C22⋊D7311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—732no—
260C2×C22⋊C4—1461no—
261C2×C42—1461no—
262#107—22no—
263#107—1464no—
264#103—22no—
265#103—22no—
266#103—1464no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—732no—
270C22⋊C8—1461no—
271#95—1464no—
272#95—1461no—
273#95—1461no—
274C24D4×D7311yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—2921no—
287C22×C4—2921no—
288C22×C4—2921no—
289C22×C4—5842no—
290C22×C4—1462no—
291C22×C4—1462no—
292C22×C4D4×D7311yescharacteristic
293C22×C4D4×D7311yescharacteristic
294C22×C4C2×C22⋊D7311yescharacteristic
295C22×C4C2×C22⋊D7311yescharacteristic
296C22×C4C2×C22⋊D7311yescharacteristic
297C22×C4#1192402711yescharacteristic
298Q16—82no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—2922no—
302C2×C8—1462no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—2922no—
312C4⋊C4—2921no—
313C4⋊C4—2921no—
314C4⋊C4—5844no—
315C4⋊C4—1462no—
316C4⋊C4—1462no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—2922no—
322C22⋊C4—734no—
323C22⋊C4—1462no—
324C42—2922no—
325C23—41no—
326C23—41no—
327C23#1455655911yescharacteristic
328C23#1455656311yescharacteristic
329C23#1455658711yescharacteristic
330C23Q16×D7312yes—
331C23#1455662312yes—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—2922no—
351C2×C4—2922no—
352C2×C4—2924no—
353C2×C4—2922no—
354C2×C4—5842no—
355C2×C4—5842no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4—1464no—
360C2×C4#1455655911yescharacteristic
361C2×C4#1455656011yescharacteristic
362C2×C4#1455656211yescharacteristic
363C2×C4#1455656311yescharacteristic
364C8—42no—
365C8—2922no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1798849511yescharacteristic
375C22#1798874112yes—
376C22#1798879512yes—
377C22#1798896511yescharacteristic
378C22#1798898011yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—2924no—
387C4—5842no—
388C4—11681no—
389C4#1798877712yes—
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#2261098111yescharacteristic
395C2#2261102911yescharacteristic
396C2#2261106011yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#3189C2
socleC22×C73#392
prime radicalC24×C73C2×D4
hypercenter#3189D73
hypocenterC73#9959
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
91. C22⋊Dic73Q16semidirect
206. C73#9959semidirect

Maximal subgroups

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

subgroupindex
C2×#179889202
#226144102
#226109172
#226100222
#226085032
#226077782
C73×#31892
#995973