Finite Group Databasebeta

← #33001615

Subgroups — #33001615

Order 18688.

714 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 127 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#22611600C212yesmaximal
2C2×#17988382C212yesmaximal, complemented
3#22610039C212yesmaximal, complemented
4C73×#3510C211yesmaximal, characteristic
5C2×#14556506C2214yes—
6#17988382—22no—
7#17988382—22no—
8#17988382—22no—
9#17988380—22no—
10#17988353—24no—
11#559×C73—22no—
12#556×C73—22no—
13#551×C73—24no—
14#542×C73—22no—
15#536×C73C411yescharacteristic, complemented
16#535×C73C2211yescharacteristic
17#533×C73C411yescharacteristic, complemented
18#529×C73—22no—
19C2×#124×C73C2211yescharacteristic
20#519×C73C411yescharacteristic, complemented
21#518×C73C2211yescharacteristic
22C4×#41×C73C411yescharacteristic, complemented
23C2×C22⋊Dic73—24no—
24#14556506—24no—
25#14556506—24no—
26C2×#41×C73C2×C411yescharacteristic
27C22×Q8×C73C2×C411yescharacteristic
28C22×D4×C73C2311yescharacteristic
29C4⋊Q8×C73—21no—
30C4⋊Q8×C73—21no—
31#127×C73D412yes—
32#126×C73—24no—
33#126×C73—24no—
34#125×C73—21no—
35#124×C73—24no—
36#124×C73—21no—
37#124×C73—22no—
38#124×C73D412yes—
39#124×C73D412yes—
40#124×C73D412yes—
41#123×C73—24no—
42#123×C73—24no—
43#123×C73—22no—
44C22⋊Q8×C73—24no—
45C22⋊Q8×C73—22no—
46C22⋊Q8×C73—22no—
47#121×C73—22no—
48#120×C73—22no—
49C4×Q8×C73—21no—
50C4×Q8×C73—22no—
51C4×Q8×C73—22no—
52C4×Q8×C73—22no—
53C4×D4×C73—24no—
54C4×D4×C73—21no—
55C4×D4×C73—22no—
56C4×D4×C73—22no—
57C4×D4×C73—22no—
58#117×C73—24no—
59#117×C73—22no—
60#117×C73—21no—
61#117×C73C2×C411yescharacteristic
62#117×C73C2×C411yescharacteristic
63#117×C73C2×C411yescharacteristic
64C2×C22⋊C4×C73—22no—
65C2×C22⋊C4×C73D414yes—
66C2×C42×C73—21no—
67C2×C42×C73C2×C411yescharacteristic
68C22×Dic73—44no—
69C22⋊Dic73—44no—
70C22⋊Dic73—44no—
71C24×C73C2×D412yes—
72#41×C73—42no—
73C2×Q8×C73—22no—
74C2×Q8×C73—21no—
75C2×Q8×C73—42no—
76C2×Q8×C73C22⋊C412yes—
77C2×Q8×C73C22⋊C411yescharacteristic
78C2×D4×C73—21no—
79C2×D4×C73—24no—
80C2×D4×C73—42no—
81C2×D4×C73C22⋊C411yescharacteristic
82C2×D4×C73C2×D412yes—
83C2×D4×C73C2×D412yes—
84C22×C4×C73—21no—
85C22×C4×C73—22no—
86C22×C4×C73—21no—
87C22×C4×C73—24no—
88C22×C4×C73—21no—
89C22×C4×C73C22×C411yescharacteristic
90C4⋊C4×C73—22no—
91C4⋊C4×C73—22no—
92C4⋊C4×C73—24no—
93C4⋊C4×C73—24no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73—21no—
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C22⋊C4×C73—22no—
100C22⋊C4×C73—24no—
101C22⋊C4×C73—21no—
102C22⋊C4×C73—24no—
103C22⋊C4×C73—24no—
104C22⋊C4×C73—24no—
105C22⋊C4×C73—24no—
106C22⋊C4×C73C22⋊C412yes—
107C22⋊C4×C73C22⋊C412yes—
108C42×C73—22no—
109C42×C73—24no—
110C42×C73—21no—
111C42×C73—21no—
112C42×C73—21no—
113C42×C73C22⋊C411yescharacteristic
114C42×C73C22⋊C411yescharacteristic
115C42×C73C22⋊C412yes—
116C23×C73—24no—
117C23×C73—21no—
118C23×C73—82no—
119C23×C73C2×C22⋊C411yescharacteristic
120C23×C73#12014yes—
121Q8×C73—42no—
122Q8×C73—41no—
123Q8×C73—42no—
124Q8×C73—82no—
125D4×C73—42no—
126D4×C73—42no—
127D4×C73—41no—
128D4×C73—42no—
129D4×C73—84no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73—21no—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—22no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73—82no—
141C2×C4×C73—84no—
142C2×C4×C73—81no—
143C2×C4×C73C2×C22⋊C411yescharacteristic
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×Dic73—84no—
146C2×Dic73—164no—
147C22×C73—24no—
148C22×C73—41no—
149C22×C73—44no—
150C22×C73—42no—
151C22×C73—41no—
152C22×C73—84no—
153C22×C73#37911yescharacteristic
154C22×C73#41012yes—
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—42no—
158C4×C73—41no—
159C4×C73—42no—
160C4×C73—41no—
161C4×C73—42no—
162C4×C73—41no—
163C4×C73—42no—
164Dic73—164no—
165#12728—731nomaximal, Sylow
166C2×C73—21no—
167C2×C73—22no—
168C2×C73—44no—
169C2×C73—81no—
170C2×C73#178211yescharacteristic
171#3510D7311yescharacteristic
172#2127—732no—
173C2×#354—732no—
174#1828—732no—
175C73#1272811yescharacteristic, complemented, Sylow
176#559—22no—
177#556—22no—
178#551—24no—
179#542—22no—
180#536Dic7311yescharacteristic, complemented
181#535C2×D7311yescharacteristic
182#533Dic7311yescharacteristic, complemented
183#529—22no—
184C2×#124C2×D7311yescharacteristic
185#519Dic7311yescharacteristic, complemented
186#518C2×D7311yescharacteristic
187C4×#41Dic7311yescharacteristic, complemented
188C2×#99—734no—
189#354—1462no—
190#354—1462no—
191#354—1462no—
192#353—1462no—
193#342—1464no—
194C2×#41C2×Dic7311yescharacteristic
195C22×Q8C2×Dic7311yescharacteristic
196C22×D4C22×D7311yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199#127C22⋊D7312yes—
200#126—24no—
201#126—24no—
202#125—21no—
203#124—24no—
204#124—21no—
205#124—22no—
206#124C22⋊D7312yes—
207#124C22⋊D7312yes—
208#124C22⋊D7312yes—
209#123—24no—
210#123—24no—
211#123—22no—
212C22⋊Q8—24no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215#121—22no—
216#120—22no—
217C4×Q8—21no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×D4—24no—
222C4×D4—21no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226#117—24no—
227#117—22no—
228#117—21no—
229#117C2×Dic7311yescharacteristic
230#117C2×Dic7311yescharacteristic
231#117C2×Dic7311yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1464no—
234C2×C22⋊C4C22⋊D7314yes—
235C2×C42—21no—
236C2×C42C2×Dic7311yescharacteristic
237#99—1464no—
238#99—1464no—
239C24C2×C22⋊D7312yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic7312yes—
245C2×Q8C22⋊Dic7311yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊Dic7311yescharacteristic
250C2×D4C2×C22⋊D7312yes—
251C2×D4C2×C22⋊D7312yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—2924no—
258C22×C4C22×Dic7311yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2924no—
276C22⋊C4—2924no—
277C22⋊C4C22⋊Dic7312yes—
278C22⋊C4C22⋊Dic7312yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊Dic7311yescharacteristic
285C42C22⋊Dic7311yescharacteristic
286C42C22⋊Dic7312yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊Dic7311yescharacteristic
291C23#1455656014yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—5844no—
315C2×C4—11684no—
316C2×C4C2×C22⋊Dic7311yescharacteristic
317C2×C4C2×C22⋊Dic7311yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22#1798844311yescharacteristic
325C22#1798856112yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—11684no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2#2260984911yescharacteristic
subgroupG/H
centerC2#22609849
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#3510C2
socleC2×C73#1782
prime radical#535×C73C22
cosocleC22×D4×C73C23
hypercenter#3510D73
hypocenterC73#12728
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C23 ◁ #518 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. C2×#17988382C2semidirect
3. #22610039C2semidirect
15. #536×C73C4semidirect
17. #533×C73C4semidirect
20. #519×C73C4semidirect
22. C4×#41×C73C4semidirect
175. C73#12728semidirect
180. #536Dic73semidirect
182. #533Dic73semidirect
185. #519Dic73semidirect
187. C4×#41Dic73semidirect

Maximal subgroups

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

subgroupindex
#226116002
C2×#179883822
#226100392
C73×#35102
#1272873