Finite Group Databasebeta

← #32994870

Subgroups — #32994870

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988352C211yesmaximal, characteristic
2C2×#17988346C212yesmaximal
3#22609628C212yesmaximal
4C2×#17988341C211yesmaximal, characteristic
5C2×#427×C73C211yesmaximal, characteristic
6C22×#11924012C2212yes—
7C23×#10596831C2211yescharacteristic
8C22×C4×Dic73C2211yescharacteristic
9#17988352—22no—
10#17988346—24no—
11#17988342—24no—
12#17988341—22no—
13C23×C8×C73C2211yescharacteristic
14C2×#117×C73C2212yes—
15#427×C73—22no—
16#427×C73—22no—
17C2×#107×C73C411yescharacteristic, complemented
18C2×#107×C73C411yescharacteristic, complemented
19C2×#106×C73C412yescomplemented
20C23×Dic73—21no—
21C2×#11924012—22no—
22C2×#11924012—22no—
23C2×#11924012—22no—
24C2×#11924012—22no—
25C2×#11924012—22no—
26C2×#11924012Q814yes—
27C22×#10596831—21no—
28C22×#10596831—22no—
29C22×#10596831—21no—
30C22×#10596831—22no—
31C22×#10596831—22no—
32C22×#10596831D412yes—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—22no—
35C2×C4×Dic73—22no—
36C2×C4×Dic73D412yes—
37C23×C4×C73C2311yescharacteristic
38C22×C8×C73—22no—
39C22×C8×C73—22no—
40C22×C8×C73C2×C411yescharacteristic
41C22×C8×C73C2×C411yescharacteristic
42C22×C8×C73D412yes—
43C22×C8×C73D412yes—
44#117×C73—22no—
45#117×C73—22no—
46#117×C73—22no—
47#117×C73—22no—
48C2×C4⋊C4×C73C2×C412yes—
49C2×C4⋊C4×C73C2×C412yes—
50C2×C22⋊C4×C73—22no—
51C2×C42×C73—22no—
52#107×C73—22no—
53#107×C73—22no—
54#106×C73—24no—
55C22×Dic73—24no—
56C22×Dic73—22no—
57C22×Dic73—42no—
58C22×Dic73—42no—
59#11924012—24no—
60#11924012—24no—
61#11924012—42no—
62#11924012—42no—
63C2×#10596831—22no—
64C2×#10596831—24no—
65C2×#10596831—24no—
66C2×#10596831—41no—
67C2×#10596831—42no—
68C2×#10596831—41no—
69C2×#10596831—42no—
70C2×#10596831—42no—
71C4×Dic73—24no—
72C4×Dic73—41no—
73C4×Dic73—41no—
74C24×C73#4111yescharacteristic
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73C22×C411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73C2×Q812yes—
83C22×C4×C73#4111yescharacteristic
84C22×C4×C73#4112yes—
85C22×C4×C73#4112yes—
86C22×C4×C73#4112yes—
87C2×C8×C73—22no—
88C2×C8×C73—22no—
89C2×C8×C73—22no—
90C2×C8×C73—22no—
91C2×C8×C73—24no—
92C2×C8×C73C22⋊C412yes—
93C2×C8×C73C22⋊C412yes—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73—22no—
97C4⋊C4×C73—22no—
98C22⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C42×C73—42no—
101C42×C73—42no—
102C23×C73—22no—
103C23×C73—22no—
104C23×C73C4×D411yescharacteristic
105C23×C73#12111yescharacteristic
106C23×C73#12411yescharacteristic
107C23×C73#12512yes—
108C23×C73#12612yes—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—44no—
112C2×C4×C73—44no—
113C2×C4×C73C2×C22⋊C411yescharacteristic
114C2×C4×C73#11712yes—
115C2×C4×C73C4×D411yescharacteristic
116C2×C4×C73C4×Q812yes—
117C2×C4×C73#12111yescharacteristic
118C2×C4×C73C22⋊Q812yes—
119C2×C4×C73C22⋊Q812yes—
120C2×C4×C73C22⋊Q812yes—
121C2×C4×C73#12311yescharacteristic
122C2×C4×C73#12311yescharacteristic
123C2×C4×C73#12312yes—
124C2×C4×C73#12312yes—
125C2×C4×C73#12312yes—
126C2×C4×C73#12411yescharacteristic
127C2×C4×C73#12512yes—
128C2×C4×C73#12612yes—
129C2×C4×C73#12711yescharacteristic
130C2×C4×C73C4⋊Q811yescharacteristic
131C2×Dic73—24no—
132C2×Dic73—48no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C8×C73—22no—
136C8×C73—22no—
137#10596831—42no—
138#10596831—44no—
139C22×C73—24no—
140C22×C73—24no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73#38712yes—
146C22×C73#39011yescharacteristic
147C22×C73#39711yescharacteristic
148C22×C73#39811yescharacteristic
149C22×C73#40012yes—
150C22×C73#44312yes—
151C22×C73#44412yes—
152C4×C73—82no—
153C4×C73—82no—
154C4×C73#38111yescharacteristic
155C4×C73#38712yes—
156C4×C73#39111yescharacteristic
157C4×C73#39711yescharacteristic
158C4×C73#39811yescharacteristic
159C4×C73#39912yes—
160Dic73—44no—
161#11376—731nomaximal, Sylow
162C2×C73—21no—
163C2×C73—21no—
164C2×C73—22no—
165C2×C73#144511yescharacteristic
166C2×C73#192214yes—
167C2×C73#197411yescharacteristic
168C2×C73#197511yescharacteristic
169C2×#427D7311yescharacteristic
170C2×#341—731no—
171C2×#339—732no—
172#1729—732no—
173C2×#336—731no—
174C73#1137611yescharacteristic, complemented, Sylow
175C22×M16—732no—
176C23×C8—731no—
177C23×C8C2×D7311yescharacteristic
178C2×#117C2×D7312yes—
179C22×C42—731no—
180#427—22no—
181#427—22no—
182C2×#107Dic7311yescharacteristic, complemented
183C2×#107Dic7311yescharacteristic, complemented
184C2×#106Dic7312yescomplemented
185#341—1462no—
186#339—1464no—
187#336—1464no—
188#336—1462no—
189C23×C4—1461no—
190C23×C4C22×D7311yescharacteristic
191C2×M16—734no—
192C2×M16—1462no—
193C2×M16—1462no—
194C2×M16—1462no—
195C2×M16—1462no—
196C2×M16—1462no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—732no—
200C22×C8—1462no—
201C22×C8—1461no—
202C22×C8—1461no—
203C22×C8—1462no—
204C22×C8—1462no—
205C22×C8C2×Dic7311yescharacteristic
206C22×C8C2×Dic7311yescharacteristic
207C22×C8C22⋊D7312yes—
208C22×C8C22⋊D7312yes—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213C2×C4⋊C4C2×Dic7312yes—
214C2×C4⋊C4C2×Dic7312yes—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—732no—
218C2×C42—1461no—
219C2×C42—1462no—
220C2×C42—1462no—
221#107—22no—
222#107—22no—
223#106—24no—
224C24#1192402711yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—2922no—
228C22×C4—2922no—
229C22×C4—1464no—
230C22×C4—1462no—
231C22×C4C22×Dic7311yescharacteristic
232C22×C4D4×D7311yescharacteristic
233C22×C4D4×D7311yescharacteristic
234C22×C4C2×C22⋊D7311yescharacteristic
235C22×C4C2×C22⋊D7311yescharacteristic
236C22×C4Q8×D7312yes—
237C22×C4#1192402711yescharacteristic
238C22×C4#1192402812yes—
239C22×C4#1192402812yes—
240C22×C4#1192402912yes—
241M16—2922no—
242M16—2922no—
243M16—1464no—
244M16—1464no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—2921no—
251C2×C8—2922no—
252C2×C8—2921no—
253C2×C8—2922no—
254C2×C8—2922no—
255C2×C8—1462no—
256C2×C8—1464no—
257C2×C8—1464no—
258C2×C8C22⋊Dic7312yes—
259C2×C8C22⋊Dic7312yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—42no—
267C42—42no—
268C42—2921no—
269C42—2921no—
270C42—1464no—
271C23—22no—
272C23—22no—
273C23D4×Dic7311yescharacteristic
274C23#1455656211yescharacteristic
275C23#1455657811yescharacteristic
276C23#1455658112yes—
277C23#1455658612yes—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—2928no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—1464no—
286C2×C4C2×C22⋊Dic7311yescharacteristic
287C2×C4#1455655012yes—
288C2×C4D4×Dic7311yescharacteristic
289C2×C4Q8×Dic7312yes—
290C2×C4#1455656211yescharacteristic
291C2×C4#1455656712yes—
292C2×C4#1455656912yes—
293C2×C4#1455656912yes—
294C2×C4#1455657411yescharacteristic
295C2×C4#1455657411yescharacteristic
296C2×C4#1455657412yes—
297C2×C4#1455657612yes—
298C2×C4#1455657612yes—
299C2×C4#1455658011yescharacteristic
300C2×C4#1455658112yes—
301C2×C4#1455658612yes—
302C2×C4#1455658711yescharacteristic
303C2×C4#1455658911yescharacteristic
304C8—22no—
305C8—22no—
306C8—2922no—
307C8—2924no—
308C22—24no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22#1798847312yes—
315C22#1798849011yescharacteristic
316C22#1798851311yescharacteristic
317C22#1798851811yescharacteristic
318C22#1798852512yes—
319C22#1798870412yes—
320C22#1798870912yes—
321C4—82no—
322C4—82no—
323C4—2924no—
324C4#1798845011yescharacteristic
325C4#1798847312yes—
326C4#1798849411yescharacteristic
327C4#1798851311yescharacteristic
328C4#1798851811yescharacteristic
329C4#1798852212yes—
330C2—21no—
331C2—21no—
332C2—22no—
333C2#2260819911yescharacteristic
334C2#2261053214yes—
335C2#2261082011yescharacteristic
336C2#2261082511yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#427×C73C2
socleC23×C73C4×D4
prime radicalC24×C73#41
cosocleC23×C4×C73C23
hypercenterC2×#427D73
hypocenterC73#11376
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
17. C2×#107×C73C4semidirect
18. C2×#107×C73C4semidirect
19. C2×#106×C73C4semidirect
174. C73#11376semidirect
182. C2×#107Dic73semidirect
183. C2×#107Dic73semidirect
184. C2×#106Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179883522
C2×#179883462
#226096282
C2×#179883412
C2×#427×C732
#1137673