Finite Group Databasebeta

← #34010022

Subgroups — #34010022

Order 18688.

360 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 88 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#22626293C211yesmaximal, characteristic, complemented
2#22626289C211yesmaximal, characteristic, complemented
3#22624854C212yesmaximal, complemented
4#22624853C212yesmaximal, complemented
5#22618641C211yesmaximal, characteristic
6#17989508—21no—
7#17989508—21no—
8C22⋊#11924031C2212yes—
9C22⋊#11924031C2212yes—
10#17989476—21no—
11#17989473—21no—
12C2×#14556597C411yescharacteristic
13#17989133C411yescharacteristic
14#17988663—21no—
15#17988663—21no—
16C8×C22⋊D73—21no—
17C8×C22⋊D73—21no—
18#17988610C411yescharacteristic
19#17988555C2211yescharacteristic
20#17988554C2211yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22C2×#131×C73C411yescharacteristic
23C2×#11924031—22no—
24C2×#11924031—22no—
25C2×#11924028C2×C411yescharacteristic
26#14556597—42no—
27C2×#11924012C2×C411yescharacteristic
28C2×#11924011—21no—
29C22×#10596831C2311yescharacteristic
30C2×C8×D73—21no—
31#14556589C811yescharacteristic
32#14556587C811yescharacteristic
33#14556580C811yescharacteristic
34#14556578C811yescharacteristic
35#14556563—21no—
36#14556563—21no—
37C4×C22⋊D73—21no—
38C4×C22⋊D73—21no—
39C2×C4×Dic73C2×C411yescharacteristic
40#14556520—21no—
41#14556520—21no—
42C22⋊#10596831—21no—
43C22⋊#10596831—21no—
44#14556503—21no—
45#14556503—21no—
46C8×Dic73D411yescharacteristic
47C8×Dic73D411yescharacteristic
48C8×Dic73D411yescharacteristic
49C8×Dic73D411yescharacteristic
50C2×#41×C73C2×C411yescharacteristic
51#131×C73—42no—
52C2×M16×C73—21no—
53C2×M16×C73C2×C411yescharacteristic
54C22×C8×C73—21no—
55C22×C8×C73C2×C411yescharacteristic
56#11924031—42no—
57#11924031—42no—
58#11924028—41no—
59#11924028—41no—
60C2×Dic146C2×C811yescharacteristic
61C2×D292C2×C811yescharacteristic
62C2×C22⋊D73—21no—
63C2×C22⋊D73—21no—
64C2×C22⋊D73—21no—
65C22×Dic73M1611yescharacteristic
66C22×Dic73M1611yescharacteristic
67C2×C4×D73—21no—
68#11924012—41no—
69#11924011—42no—
70C2×#10596831—21no—
71C2×#10596831—41no—
72C2×#10596831C2×D411yescharacteristic
73C2×#10596831C2×D411yescharacteristic
74C8×D73—42no—
75#11924007—21no—
76#11924007—21no—
77C22⋊Dic73—21no—
78C22⋊Dic73—21no—
79#11924005—21no—
80#11924005—21no—
81C4×Dic73C22⋊C411yescharacteristic
82C4×Dic73C22⋊C411yescharacteristic
83C4×Dic73C2×C811yescharacteristic
84C4×Dic73C2×C811yescharacteristic
85#41×C73—41no—
86#41×C73—41no—
87C2×Q8×C73C2×C811yescharacteristic
88C2×D4×C73—21no—
89C2×D4×C73C2×C811yescharacteristic
90C22×C4×C73—21no—
91C22×C4×C73C22×C411yescharacteristic
92M16×C73—42no—
93M16×C73—41no—
94C2×C8×C73—22no—
95C2×C8×C73—21no—
96C2×C8×C73—42no—
97C2×C8×C73C22⋊C411yescharacteristic
98C2×C8×C73C22⋊C411yescharacteristic
99C23×C73—21no—
100C23×C73C2×M1611yescharacteristic
101C22×D73—21no—
102Q8×C73—41no—
103D4×C73—41no—
104D4×C73—42no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73C2×C22⋊C411yescharacteristic
111C2×C4×C73C22×C811yescharacteristic
112C22⋊D73—42no—
113C22⋊D73—42no—
114C22⋊D73—42no—
115C2×Dic73—22no—
116C2×Dic73—21no—
117C2×Dic73—41no—
118C2×Dic73C22⋊C811yescharacteristic
119C2×Dic73C22⋊C811yescharacteristic
120C2×Dic73C22⋊C811yescharacteristic
121C2×Dic73C22⋊C811yescharacteristic
122D292—41no—
123C4×D73—41no—
124C4×D73—41no—
125Dic146—41no—
126C8×C73—21no—
127C8×C73—41no—
128C8×C73#10012yes—
129#10596831—21no—
130#10596831—21no—
131#10596831—41no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—41no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73C2×C22⋊C811yescharacteristic
139C2×D73—41no—
140C2×D73—41no—
141C2×D73—41no—
142C4×C73—21no—
143C4×C73—41no—
144C4×C73#41011yescharacteristic
145C4×C73C2×#10011yescharacteristic
146Dic73—21no—
147Dic73—21no—
148Dic73—21no—
149Dic73—41no—
150Dic73#35012yes—
151#11492—731nomaximal, Sylow
152C2×C73—41no—
153C2×C73—81no—
154C2×C73#146011yescharacteristic
155C2×C73C2×#35011yescharacteristic
156C2×C73#211711yescharacteristic
157D73—81no—
158#2931—731no—
159#2163—731no—
160#2162—731no—
161#1315—732no—
162#1315—732no—
163C73#1149211yescharacteristic, complemented, Sylow
164C2×#131—731no—
165C2×#131#6440811yescharacteristic
166#532—731no—
167#436—1461no—
168#436—1461no—
169C8×D4—1461no—
170C8×D4—1461no—
171#423—731no—
172#408—731no—
173#408—731no—
174C2×C4×C8—731no—
175C4⋊C16—1461no—
176C4⋊C16—1461no—
177C22⋊C16—732no—
178C22⋊C16—732no—
179#346—1461no—
180C4×C16—1461no—
181C2×#41—731no—
182C2×#41C2×#6440811yescharacteristic
183#131—42no—
184#131—2922no—
185C2×M16—21no—
186C2×M16—731no—
187C2×M16—1461no—
188C2×M16C2×#6440811yescharacteristic
189C22×C8—21no—
190C22×C8—731no—
191C22×C8—1461no—
192C22×C8C2×#6440811yescharacteristic
193C4⋊Q8—731no—
194#127—731no—
195#124—731no—
196#124—731no—
197#121—1461no—
198#121—1461no—
199C4×D4—1461no—
200C4×D4—1461no—
201C2×C42—731no—
202C2×C16—1462no—
203C2×C16—1462no—
204C4⋊C8—1461no—
205C4⋊C8—1461no—
206C22⋊C8—1461no—
207C22⋊C8—1461no—
208C22⋊C8—1461no—
209C22⋊C8—1461no—
210C4×C8—731no—
211C4×C8—731no—
212C4×C8—731no—
213C4×C8—731no—
214#41—41no—
215#41—41no—
216#41—2921no—
217#41—2921no—
218C2×Q8—731no—
219C2×Q8#1192403611yescharacteristic
220C2×D4—21no—
221C2×D4—731no—
222C2×D4—1461no—
223C2×D4—1461no—
224C2×D4—1461no—
225C2×D4#1192403611yescharacteristic
226C22×C4—21no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—1461no—
230C22×C4C22×#6440811yescharacteristic
231M16—42no—
232M16—41no—
233M16—2922no—
234M16—2921no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—2922no—
239C2×C8—2921no—
240C2×C8—731no—
241C2×C8—731no—
242C2×C8—1461no—
243C2×C8C22⋊#6440811yescharacteristic
244C2×C8C22⋊#6440811yescharacteristic
245C4⋊C4—1461no—
246C4⋊C4—1461no—
247C22⋊C4—1461no—
248C22⋊C4—1461no—
249C22⋊C4—1461no—
250C22⋊C4—1461no—
251C42—731no—
252C42—731no—
253C42—731no—
254C42—731no—
255C16—2922no—
256C16—2922no—
257C23—21no—
258C23—1461no—
259C23D73⋊M1611yescharacteristic
260Q8—41no—
261Q8—2921no—
262D4—41no—
263D4—42no—
264D4—2921no—
265D4—2922no—
266D4—2922no—
267D4—2922no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—2921no—
274C2×C4—2921no—
275C2×C4—2921no—
276C2×C4—731no—
277C2×C4—731no—
278C2×C4—731no—
279C2×C4—731no—
280C2×C4—1461no—
281C2×C4—1462no—
282C2×C4C2×C22⋊#6440811yescharacteristic
283C2×C4C2×#1192403611yescharacteristic
284C8—21no—
285C8—41no—
286C8—2921no—
287C8—1461no—
288C8—1461no—
289C8#1455665112yes—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—2921no—
297C22—2921no—
298C22—2921no—
299C22#1798955511yescharacteristic
300C4—21no—
301C4—41no—
302C4—2921no—
303C4—732no—
304C4—1461no—
305C4—1461no—
306C4—1461no—
307C4#1798956611yescharacteristic
308C4C2×#1455665111yescharacteristic
309C2—41no—
310C2—81no—
311C2—5841no—
312C2#2262517011yescharacteristic
313C2#2262617111yescharacteristic
314C2#2262617411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#131×C73C4
socleC22×C73C2×C22⋊C8
prime radical#17989133C4
cosocleC22×#10596831C23
hypercenterC2×#131#64408
hypocenterC73#11492
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #22626293C2semidirect
2. #22626289C2semidirect
3. #22624854C2semidirect
4. #22624853C2semidirect
163. C73#11492semidirect

Maximal subgroups

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

subgroupindex
#226262932
#226262892
#226248542
#226248532
#226186412
#1149273