Finite Group Databasebeta

← #33001973

Subgroups — #33001973

Order 18688.

588 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 101 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#22612657C211yesmaximal, characteristic
2C2×#17988380C211yesmaximal, characteristic
3#22609917C212yesmaximal, complemented
4#22608055C212yesmaximal, complemented
5C2×#353×C73C211yesmaximal, characteristic
6C2×D4×Dic73C411yescharacteristic, complemented
7C2×D4×Dic73C2211yescharacteristic
8C2×#14556506C2212yes—
9#17988490C411yescharacteristic, complemented
10#17988490C2211yescharacteristic
11#17988467—22no—
12#17988467—22no—
13#17988453C411yescharacteristic, complemented
14#17988445—22no—
15#17988433C411yescharacteristic, complemented
16#17988380—22no—
17#17988353—22no—
18C2×#127×C73C2211yescharacteristic
19C2×#99×C73C2212yes—
20#353×C73—22no—
21C23×Dic73—21no—
22C23×Dic73D412yes—
23D4×Dic73—42no—
24D4×Dic73—42no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—42no—
28C2×C22⋊Dic73—21no—
29C2×C22⋊Dic73—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73D412yes—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35#14556506—22no—
36#14556506—22no—
37#14556498—21no—
38#14556498—21no—
39#14556498—22no—
40#14556498—42no—
41C22×D4×C73—22no—
42C22×D4×C73C2×C411yescharacteristic
43C22×D4×C73C2311yescharacteristic
44#127×C73—22no—
45#127×C73D412yes—
46#127×C73Q812yes—
47C2×C22⋊C4×C73—22no—
48C2×C42×C73C2×C411yescharacteristic
49#99×C73—22no—
50#99×C73—22no—
51C22×Dic73—22no—
52C22×Dic73—22no—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73—42no—
56C22×Dic73—42no—
57C22×Dic73—82no—
58C22×Dic73—82no—
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C22⋊C412yes—
61C22×Dic73C22⋊C411yescharacteristic
62C4⋊Dic73—41no—
63C4⋊Dic73—41no—
64#11924007—84no—
65C22⋊Dic73—42no—
66C22⋊Dic73—42no—
67C22⋊Dic73—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—82no—
70C22⋊Dic73—82no—
71C4×Dic73—22no—
72C4×Dic73—41no—
73C24×C73—21no—
74C24×C73C2×D412yes—
75C2×D4×C73—24no—
76C2×D4×C73—24no—
77C2×D4×C73—21no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73—41no—
82C2×D4×C73—82no—
83C2×D4×C73—82no—
84C2×D4×C73C4⋊C412yescomplemented
85C2×D4×C73C2×D411yescharacteristic
86C2×D4×C73C2×Q811yescharacteristic
87C2×D4×C73#4112yes—
88C22×C4×C73—21no—
89C22×C4×C73—42no—
90C22×C4×C73C22×C411yescharacteristic
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C42×C73—21no—
94C42×C73C4⋊C412yescomplemented
95C23×C73—22no—
96C23×C73—22no—
97C23×C73—22no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—42no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73#12112yes—
106C23×C73C22⋊Q812yes—
107D4×C73—41no—
108D4×C73—41no—
109D4×C73—42no—
110D4×C73—42no—
111D4×C73—42no—
112D4×C73—84no—
113D4×C73—84no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—22no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—82no—
120C2×C4×C73—82no—
121C2×C4×C73—82no—
122C2×C4×C73C2×C4⋊C411yescharacteristic
123C2×C4×C73#11711yescharacteristic
124C2×Dic73—21no—
125C2×Dic73—22no—
126C2×Dic73—41no—
127C2×Dic73—44no—
128C2×Dic73—42no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—42no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—82no—
136C2×Dic73—84no—
137C2×Dic73—82no—
138C2×Dic73—82no—
139C2×Dic73—82no—
140C2×Dic73#9912yes—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—44no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—42no—
151C22×C73—82no—
152C22×C73—82no—
153C22×C73—81no—
154C22×C73—81no—
155C22×C73#38011yescharacteristic
156C22×C73C2×#9911yescharacteristic
157C22×C73#41011yescharacteristic
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—42no—
161C4×C73—162no—
162Dic73—22no—
163Dic73—42no—
164Dic73—41no—
165Dic73—81no—
166Dic73—81no—
167Dic73—162no—
168#12799—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—42no—
172C2×C73—42no—
173C2×C73—82no—
174C2×C73#179911yescharacteristic
175C2×C73#212711yescharacteristic
176C2×C73#212811yescharacteristic
177#2328—731no—
178C2×#353—731no—
179C2×#353D7311yescharacteristic
180#1799—732no—
181#1393—732no—
182C73#1279911yescharacteristic, complemented, Sylow
183C2×#127C2×D7311yescharacteristic
184C2×C4×D4—731no—
185C2×C4×D4—731no—
186C2×#99—732no—
187C2×#99C2×D7312yes—
188#390—731no—
189#390—731no—
190#386—1462no—
191#386—1462no—
192#382—731no—
193#380—1462no—
194#376—731no—
195#353—22no—
196#353—1462no—
197#342—1462no—
198C22×D4—22no—
199C22×D4C4×D7311yescharacteristic
200C22×D4C22×D7311yescharacteristic
201C23×C4—732no—
202C23×C4—1461no—
203#127—22no—
204#127Dic14612yes—
205#127C22⋊D7312yes—
206C4×D4—2922no—
207C4×D4—2922no—
208C2×C4⋊C4—2922no—
209C2×C4⋊C4—731no—
210C2×C4⋊C4—731no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—732no—
213C2×C22⋊C4—1462no—
214C2×C22⋊C4—1461no—
215C2×C22⋊C4—1462no—
216C2×C22⋊C4—1462no—
217C2×C42—731no—
218C2×C42—731no—
219C2×C42C4×D7311yescharacteristic
220#99—22no—
221#99—22no—
222#99—1462no—
223#99—1462no—
224#95—2922no—
225#95—1461no—
226#95—1461no—
227#95—1462no—
228C24—21no—
229C24D4×D7312yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4#1192400712yescomplemented
240C2×D4C2×C22⋊D7311yescharacteristic
241C2×D4C2×Dic14611yescharacteristic
242C2×D4#1192402712yes—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—2922no—
246C22×C4—2922no—
247C22×C4—5842no—
248C22×C4—5842no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4—732no—
252C22×C4—1461no—
253C22×C4—1461no—
254C22×C4—1462no—
255C22×C4—1462no—
256C22×C4C2×C4×D7311yescharacteristic
257C4⋊C4—2921no—
258C4⋊C4—2921no—
259C4⋊C4—5844no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—2922no—
263C22⋊C4—2922no—
264C22⋊C4—2922no—
265C22⋊C4—2922no—
266C22⋊C4—5842no—
267C22⋊C4—5842no—
268C42—21no—
269C42—2921no—
270C42—1462no—
271C42#1192400712yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23C22⋊C4×D7311yescharacteristic
282C23#1455656312yes—
283C23#1455656812yes—
284D4—41no—
285D4—41no—
286D4—42no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—2924no—
300C2×C4—2922no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2922no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—5844no—
309C2×C4—5842no—
310C2×C4—5842no—
311C2×C4—5842no—
312C2×C4—5842no—
313C2×C4—732no—
314C2×C4—1461no—
315C2×C4—1462no—
316C2×C4C2×#1192400711yescharacteristic
317C2×C4#1455654711yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22#1798844511yescharacteristic
333C22#99×D7311yescharacteristic
334C22#1798855911yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—162no—
339C4—2922no—
340C4—2921no—
341C4—5841no—
342C4—5841no—
343C4—11682no—
344C4—1462no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#2260991711yescharacteristic
351C2#2261159711yescharacteristic
352C2#2261160111yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC2×#353×C73C2
socleC22×C73C2×#99
prime radicalC2×#127×C73C22
cosocleC22×D4×C73C23
hypercenterC2×#353D73
hypocenterC73#12799
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #22609917C2semidirect
4. #22608055C2semidirect
6. C2×D4×Dic73C4semidirect
9. #17988490C4semidirect
13. #17988453C4semidirect
15. #17988433C4semidirect
84. C2×D4×C73C4⋊C4semidirect
94. C42×C73C4⋊C4semidirect
182. C73#12799semidirect
239. C2×D4#11924007semidirect
271. C42#11924007semidirect

Maximal subgroups

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

subgroupindex
#226126572
C2×#179883802
#226099172
#226080552
C2×#353×C732
#1279973