Finite Group Databasebeta

← #34023795

Subgroups — #34023795

Order 18688.

1194 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 216 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#22626611C212yesmaximal, complemented
2#22626075C214yesmaximal, complemented
3#22626074C214yesmaximal, complemented
4#22625768C212yesmaximal
5#22625765C212yesmaximal, complemented
6C2×#17989159C211yesmaximal, characteristic
7#17989611—28no—
8#17989590C2214yes—
9C2×#14556652C2214yes—
10C2×#14556652C2218yes—
11C2×#14556651C2214yes—
12#17989557C2214yes—
13C4×#11924037C2214yes—
14C22×#11924027C2211yescharacteristic
15C23×Dic146C2211yescharacteristic
16#17989159—28no—
17C2×#14556590C2212yes—
18C2×#14556577C2212yes—
19C2×#14556575C411yescharacteristic
20C2×#14556574C411yescharacteristic
21C2×#14556572C411yescharacteristic
22C2×#14556569C412yes—
23C2×#14556568C412yes—
24C2×#14556547C2211yescharacteristic
25C2×#123×C73C411yescharacteristic
26C2×#11924037C2318yes—
27#14556658—24no—
28C4⋊#10596833—24no—
29#14556652—48no—
30#14556652—44no—
31#14556651—44no—
32C22⋊#10596833—28no—
33#14556646—24no—
34C4×#10596833—24no—
35C2×#11924027—24no—
36C2×#11924027—44no—
37C22×Dic146—24no—
38C22×Dic146—44no—
39C22×Dic146C2×C412yes—
40C22×Dic146C2311yescharacteristic
41C22×Dic146C2312yes—
42C22×C22⋊D73C2×C411yescharacteristic
43C22×C22⋊D73C2311yescharacteristic
44C23×Dic73C2×C411yescharacteristic
45C23×Dic73C2×C411yescharacteristic
46C23×Dic73C2311yescharacteristic
47C22×C4×D73C2×C411yescharacteristic
48#14556590—24no—
49#14556590—24no—
50#14556577—24no—
51#14556577—24no—
52#14556575—24no—
53#14556574—24no—
54#14556572—24no—
55#14556569—28no—
56#14556568—28no—
57#14556547D418yes—
58C2×C4⋊Dic73C2×C412yes—
59C2×#11924007C2×C412yes—
60C2×#11924007C2×C412yes—
61C2×#11924007C2×C412yes—
62C2×C22⋊Dic73C2×C412yes—
63C2×C22⋊Dic73C2×C411yescharacteristic
64C2×#11924005C2×C412yes—
65C2×#11924005C2×C412yes—
66C2×C4×Dic73C2312yes—
67C22×D4×C73C2×C411yescharacteristic
68C23×C4×C73C2×C411yescharacteristic
69#123×C73—24no—
70C2×C4⋊C4×C73C2×C412yes—
71C2×C22⋊C4×C73C2×C412yes—
72C2×C22⋊C4×C73C2×C411yescharacteristic
73#11924037—28no—
74#11924037—28no—
75C2×#10596833—28no—
76C23×D73C22×C411yescharacteristic
77#11924027—416no—
78C2×Dic146—24no—
79C2×Dic146—22no—
80C2×Dic146—22no—
81C2×Dic146—22no—
82C2×Dic146—28no—
83C2×Dic146—24no—
84C2×Dic146—44no—
85C2×Dic146—416no—
86C2×C22⋊D73—22no—
87C2×C22⋊D73—22no—
88C2×C22⋊D73—24no—
89C2×C22⋊D73—42no—
90C22×Dic73—24no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—42no—
94C22×Dic73—42no—
95C22×Dic73C22×C412yes—
96C22×Dic73C22×C411yescharacteristic
97C22×Dic73C22×C411yescharacteristic
98C22×Dic73C22×C412yes—
99C22×Dic73C2×D414yes—
100C22×Dic73C2411yescharacteristic
101C2×C4×D73—22no—
102C2×C4×D73—42no—
103C4⋊Dic73—24no—
104#11924007—24no—
105#11924007—24no—
106#11924007C22⋊C418yes—
107C22⋊Dic73—24no—
108C22⋊Dic73C22⋊C414yes—
109#11924005—24no—
110#11924005—24no—
111C4×Dic73C2×D414yes—
112C4×Dic73C2×D414yes—
113C24×C73C22×C411yescharacteristic
114C24×C73C22×C411yescharacteristic
115C2×D4×C73—22no—
116C2×D4×C73—42no—
117C22×C4×C73—22no—
118C22×C4×C73—42no—
119C22×C4×C73C22×C412yes—
120C22×C4×C73C22×C411yescharacteristic
121C22×C4×C73C22×C412yes—
122C4⋊C4×C73—24no—
123C22⋊C4×C73—24no—
124C22⋊C4×C73C22⋊C414yes—
125C23×C73—22no—
126C23×C73—21no—
127C23×C73—22no—
128C23×C73—42no—
129C23×C73—42no—
130C23×C73C2×C22⋊C412yes—
131C23×C73C23×C411yescharacteristic
132C22×D73—21no—
133C22×D73—22no—
134C22×D73—42no—
135D4×C73—44no—
136C2×C4×C73—22no—
137C2×C4×C73—21no—
138C2×C4×C73—24no—
139C2×C4×C73—22no—
140C2×C4×C73—42no—
141C2×C4×C73—44no—
142C2×C4×C73C2×C22⋊C414yes—
143C22⋊D73—44no—
144C22⋊D73—44no—
145C2×Dic73—22no—
146C2×Dic73—28no—
147C2×Dic73—22no—
148C2×Dic73—21no—
149C2×Dic73—22no—
150C2×Dic73—24no—
151C2×Dic73—42no—
152C2×Dic73—44no—
153C2×Dic73—44no—
154C2×Dic73C2×C22⋊C414yes—
155C2×Dic73C2×C22⋊C412yes—
156C2×Dic73C22×D412yes—
157C2×Dic73H(5,𝔽2)14yes—
158C4×D73—44no—
159Dic146—48no—
160Dic146—44no—
161Dic146—44no—
162Dic146—44no—
163#10596833—48no—
164C22×C73—22no—
165C22×C73—22no—
166C22×C73—24no—
167C22×C73—44no—
168C22×C73—41no—
169C22×C73—42no—
170C22×C73—82no—
171C22×C73C22×C22⋊C411yescharacteristic
172C22×C73#51812yes—
173C2×D73—41no—
174C2×D73—42no—
175C2×D73—82no—
176C4×C73—24no—
177C4×C73—42no—
178C4×C73—41no—
179Dic73—22no—
180Dic73—24no—
181Dic73—42no—
182Dic73—41no—
183Dic73#53514yes—
184#21802—731nomaximal, Sylow
185C2×C73—22no—
186C2×C73—42no—
187C2×C73—81no—
188C2×C73#230911yescharacteristic
189C2×C73#288912yes—
190D73—81no—
191C2×#536—731no—
192#2921—732no—
193#1967—734no—
194#1966—734no—
195#1785—732no—
196#1784—732no—
197C73#2180211yescharacteristic, complemented, Sylow
198C22×#41—731no—
199C23×Q8—731no—
200#536—1468no—
201C2×C4⋊Q8—732no—
202C2×#124—732no—
203C2×#123—731no—
204C2×#123—731no—
205C2×#123—731no—
206C2×#123#6440811yescharacteristic
207C2×C22⋊Q8—732no—
208C2×C22⋊Q8—732no—
209C2×#117—731no—
210#433—1468no—
211#423—734no—
212C2×#101—734no—
213C2×#101—738no—
214C2×#100—734no—
215#407—734no—
216C4×M16—734no—
217C2×#41—2924no—
218C2×#41—1464no—
219C22×Q8—2924no—
220C22×Q8—732no—
221C22×Q8—732no—
222C22×Q8—731no—
223C22×Q8—1464no—
224C22×D4—731no—
225C22×D4—731no—
226C22×D4C2×#6440811yescharacteristic
227C23×C4—731no—
228C23×C4—731no—
229C23×C4—731no—
230C23×C4—731no—
231C23×C4C2×#6440811yescharacteristic
232C2×M16—738no—
233C4⋊Q8—1464no—
234C4⋊Q8—1464no—
235#124—1464no—
236#124—1464no—
237#123—24no—
238#123—1464no—
239#123—1464no—
240#123—1464no—
241C22⋊Q8—1468no—
242C22⋊Q8—1468no—
243#117—738no—
244C2×C4⋊C4—732no—
245C2×C4⋊C4—732no—
246C2×C4⋊C4—732no—
247C2×C4⋊C4—732no—
248C2×C4⋊C4C2×#6440812yes—
249C2×C22⋊C4—731no—
250C2×C22⋊C4—732no—
251C2×C22⋊C4—732no—
252C2×C22⋊C4—732no—
253C2×C22⋊C4C2×#6440812yes—
254C2×C22⋊C4C2×#6440811yescharacteristic
255C2×C42—732no—
256C4⋊C8—1464no—
257C4⋊C8—1464no—
258#101—2928no—
259#101—2924no—
260#100—2924no—
261C22⋊C8—1468no—
262#97—1464no—
263C4×C8—1464no—
264C24—731no—
265C24C22×#6440811yescharacteristic
266C24C22×#6440811yescharacteristic
267#41—29216no—
268C2×Q8—2924no—
269C2×Q8—29216no—
270C2×Q8—1468no—
271C2×Q8—1464no—
272C2×Q8—1464no—
273C2×Q8—1462no—
274C2×Q8—1462no—
275C2×Q8—1462no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—2922no—
279C2×D4—1464no—
280C2×D4—1462no—
281C2×D4—1462no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—2922no—
285C22×C4—2922no—
286C22×C4—2922no—
287C22×C4—732no—
288C22×C4—731no—
289C22×C4—731no—
290C22×C4—734no—
291C22×C4—732no—
292C22×C4—731no—
293C22×C4—1462no—
294C22×C4—1462no—
295C22×C4—1462no—
296C22×C4—1464no—
297C22×C4C22×#6440812yes—
298C22×C4C22×#6440811yescharacteristic
299C22×C4C22×#6440812yes—
300M16—1468no—
301M16—1468no—
302C2×C8—1468no—
303C4⋊C4—24no—
304C4⋊C4—738no—
305C4⋊C4—1464no—
306C4⋊C4—1464no—
307C4⋊C4—1464no—
308C22⋊C4—24no—
309C22⋊C4—734no—
310C22⋊C4—1464no—
311C22⋊C4—1464no—
312C22⋊C4—1464no—
313C22⋊C4C22⋊#6440814yes—
314C42—734no—
315C42—734no—
316C23—22no—
317C23—21no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—2922no—
322C23—1461no—
323C23—1462no—
324C23C2×C22⋊#6440812yes—
325C23C23×#6440811yescharacteristic
326Q8—2924no—
327Q8—2924no—
328Q8—2924no—
329Q8—2928no—
330D4—44no—
331D4—2924no—
332D4—2924no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—2924no—
340C2×C4—2924no—
341C2×C4—2924no—
342C2×C4—2922no—
343C2×C4—732no—
344C2×C4—734no—
345C2×C4—734no—
346C2×C4—732no—
347C2×C4—1462no—
348C2×C4—1461no—
349C2×C4—1462no—
350C2×C4—1464no—
351C2×C4—1468no—
352C2×C4—1462no—
353C2×C4C2×C22⋊#6440814yes—
354C8—2928no—
355C22—22no—
356C22—22no—
357C22—24no—
358C22—44no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—2921no—
363C22—2922no—
364C22—5842no—
365C22C22×C22⋊#6440811yescharacteristic
366C22#1798965412yes—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—2921no—
371C4—2922no—
372C4—734no—
373C4—1462no—
374C4—1464no—
375C2—22no—
376C2—42no—
377C2—81no—
378C2—5841no—
379C2#2262640411yescharacteristic
380C2#2262654312yes—
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#123×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924027C22
cosocleC22×Dic73C24
hypercenterC2×#123#64408
hypocenterC73#21802
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #22626611C2semidirect
2. #22626075C2semidirect
3. #22626074C2semidirect
5. #22625765C2semidirect
197. C73#21802semidirect

Maximal subgroups

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

subgroupindex
#226266112
#226260752
#226260742
#226257682
#226257652
C2×#179891592
#2180273