Finite Group Databasebeta

← #34009939

Subgroups — #34009939

Order 18688.

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

normal subgroups only

This group has 392 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#22626159C211yesmaximal, characteristic, complemented
2#22626156C211yesmaximal, characteristic, complemented
3#22624870C212yesmaximal, complemented
4#22624853C212yesmaximal, complemented
5#22618623C211yesmaximal, characteristic
6C2×#14556666—21no—
7C2×#14556664—21no—
8#17989482—21no—
9#17989482—21no—
10C22⋊#11924031C2212yes—
11C22⋊#11924031C2212yes—
12C22×C8×D73C2211yescharacteristic
13#17989094C411yescharacteristic
14#17988656—21no—
15#17988656—21no—
16C8×C22⋊D73—21no—
17C8×C22⋊D73—21no—
18#17988615C2211yescharacteristic
19#17988554C2211yescharacteristic
20#17988550C411yescharacteristic
21#17988542C411yescharacteristic
22C2×#131×C73C411yescharacteristic
23#14556666—42no—
24#14556664—42no—
25C2×#11924031—22no—
26C2×#11924031—22no—
27C22×C4×D73C2×C411yescharacteristic
28C2×#11924012C2×C411yescharacteristic
29C22×#10596831C2311yescharacteristic
30C2×C8×D73—42no—
31C2×C8×D73D411yescharacteristic
32C2×C8×D73D411yescharacteristic
33C2×C8×D73D411yescharacteristic
34C2×C8×D73D411yescharacteristic
35#14556572—21no—
36#14556567C811yescharacteristic
37#14556567C811yescharacteristic
38#14556562C811yescharacteristic
39#14556562C811yescharacteristic
40#14556559—21no—
41C4×C22⋊D73—21no—
42C4×C22⋊D73—21no—
43#14556550C2×C411yescharacteristic
44#14556520—21no—
45#14556520—21no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48#14556503—21no—
49#14556503—21no—
50#14556501—21no—
51C8×Dic73—21no—
52C2×#41×C73C2×C411yescharacteristic
53#131×C73—42no—
54C2×M16×C73—21no—
55C2×M16×C73C2×C411yescharacteristic
56C22×C8×C73—21no—
57C22×C8×C73C2×C411yescharacteristic
58#11924031—42no—
59#11924031—42no—
60C23×D73M1611yescharacteristic
61C2×C22⋊D73—21no—
62C2×C22⋊D73—21no—
63C22×Dic73M1611yescharacteristic
64C2×C4×D73—41no—
65C2×C4×D73—41no—
66C2×C4×D73C22⋊C411yescharacteristic
67C2×C4×D73C22⋊C411yescharacteristic
68C2×C4×D73C2×C811yescharacteristic
69C2×C4×D73C2×C811yescharacteristic
70#11924012—41no—
71C2×#10596831—21no—
72C2×#10596831—41no—
73C2×#10596831C2×D411yescharacteristic
74C2×#10596831C2×D411yescharacteristic
75C8×D73—22no—
76C8×D73—22no—
77C8×D73—41no—
78C8×D73—41no—
79C4⋊Dic73C2×C811yescharacteristic
80C4⋊Dic73C2×C811yescharacteristic
81#11924007—21no—
82#11924007—21no—
83C22⋊Dic73—21no—
84C22⋊Dic73—21no—
85C22⋊Dic73—21no—
86#11924005—21no—
87#11924005—21no—
88C4×Dic73—21no—
89#41×C73—41no—
90#41×C73—41no—
91C2×Q8×C73C2×C811yescharacteristic
92C2×D4×C73—21no—
93C2×D4×C73C2×C811yescharacteristic
94C22×C4×C73—21no—
95C22×C4×C73C22×C411yescharacteristic
96M16×C73—42no—
97M16×C73—41no—
98C2×C8×C73—22no—
99C2×C8×C73—21no—
100C2×C8×C73—42no—
101C2×C8×C73C22⋊C411yescharacteristic
102C2×C8×C73C22⋊C411yescharacteristic
103C23×C73—21no—
104C23×C73C2×M1611yescharacteristic
105C22×D73—22no—
106C22×D73—41no—
107C22×D73—41no—
108C22×D73C22⋊C811yescharacteristic
109C22×D73C22⋊C811yescharacteristic
110Q8×C73—41no—
111D4×C73—41no—
112D4×C73—42no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73C2×C22⋊C411yescharacteristic
119C2×C4×C73C22×C811yescharacteristic
120C22⋊D73—42no—
121C22⋊D73—42no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—41no—
125C2×Dic73C22⋊C811yescharacteristic
126C2×Dic73C22⋊C811yescharacteristic
127C4×D73—21no—
128C4×D73—21no—
129C4×D73—21no—
130C4×D73—21no—
131C4×D73—41no—
132C4×D73—41no—
133C8×C73—21no—
134C8×C73—41no—
135C8×C73#9912yes—
136#10596831—21no—
137#10596831—21no—
138#10596831—41no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73C2×C22⋊C811yescharacteristic
146C2×D73—21no—
147C2×D73—21no—
148C2×D73—21no—
149C2×D73—21no—
150C2×D73—21no—
151C2×D73—44no—
152C2×D73#35012yes—
153C4×C73—21no—
154C4×C73—41no—
155C4×C73C2×#9911yescharacteristic
156C4×C73#41311yescharacteristic
157Dic73—21no—
158Dic73—21no—
159Dic73—41no—
160Dic73—41no—
161#11470—731nomaximal, Sylow
162C2×C73—41no—
163C2×C73—81no—
164C2×C73#145911yescharacteristic
165C2×C73C2×#35011yescharacteristic
166C2×C73#211711yescharacteristic
167D73—22no—
168D73—41no—
169#2929—731no—
170#2112—731no—
171C2×C22⋊C16—731no—
172#1323—732no—
173#1315—732no—
174C73#1147011yescharacteristic, complemented, Sylow
175C2×#131#6440811yescharacteristic
176C23×C8—731no—
177#525—731no—
178C2×M32—1461no—
179C22×C16—1461no—
180#435—1461no—
181#435—1461no—
182C8×D4—1461no—
183C8×D4—1461no—
184#424—731no—
185#408—731no—
186#407—731no—
187#405—731no—
188C22⋊C16—732no—
189C22⋊C16—732no—
190C22⋊C16—1461no—
191C22⋊C16—1461no—
192C2×#41C2×#6440811yescharacteristic
193C23×C4—731no—
194#131—42no—
195C2×M16—21no—
196C2×M16—731no—
197C2×M16C2×#6440811yescharacteristic
198C22×C8—21no—
199C22×C8—2922no—
200C22×C8—731no—
201C22×C8—731no—
202C22×C8—731no—
203C22×C8—731no—
204C22×C8—731no—
205C22×C8C2×#6440811yescharacteristic
206#123—1461no—
207C22⋊Q8—731no—
208C22⋊Q8—731no—
209#121—731no—
210#121—731no—
211#120—1461no—
212C4×D4—1461no—
213C4×D4—1461no—
214#117—731no—
215M32—2922no—
216C2×C16—2922no—
217C2×C16—1462no—
218C2×C16—1462no—
219C4⋊C8—1461no—
220C4⋊C8—1461no—
221C22⋊C8—1461no—
222C22⋊C8—1461no—
223C22⋊C8—1461no—
224C22⋊C8—1461no—
225#97—1461no—
226C4×C8—1461no—
227C24—731no—
228#41—41no—
229#41—41no—
230C2×Q8#1192403611yescharacteristic
231C2×D4—21no—
232C2×D4—1461no—
233C2×D4—1461no—
234C2×D4#1192403611yescharacteristic
235C22×C4—21no—
236C22×C4—2921no—
237C22×C4—2921no—
238C22×C4—731no—
239C22×C4—731no—
240C22×C4—731no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4C22×#6440811yescharacteristic
244M16—42no—
245M16—41no—
246M16—2921no—
247C2×C8—22no—
248C2×C8—21no—
249C2×C8—42no—
250C2×C8—2921no—
251C2×C8—2921no—
252C2×C8—2921no—
253C2×C8—731no—
254C2×C8—731no—
255C2×C8—1461no—
256C2×C8—1462no—
257C2×C8—1462no—
258C2×C8C22⋊#6440811yescharacteristic
259C2×C8C22⋊#6440811yescharacteristic
260C4⋊C4—731no—
261C4⋊C4—731no—
262C4⋊C4—1461no—
263C4⋊C4—1461no—
264C22⋊C4—1461no—
265C22⋊C4—1461no—
266C22⋊C4—1461no—
267C22⋊C4—1461no—
268C22⋊C4—1461no—
269C42—1461no—
270C16—2922no—
271C16—2922no—
272C23—21no—
273C23—2921no—
274C23—2921no—
275C23—731no—
276C23—731no—
277C23—1462no—
278C23D73⋊M1611yescharacteristic
279Q8—41no—
280D4—41no—
281D4—42no—
282D4—2922no—
283D4—2922no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—731no—
293C2×C4—731no—
294C2×C4—1461no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4C2×C22⋊#6440811yescharacteristic
301C2×C4C2×#1192403611yescharacteristic
302C8—21no—
303C8—41no—
304C8—2921no—
305C8—1461no—
306C8—1461no—
307C8#1455664912yes—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—2924no—
315C22—732no—
316C22—1461no—
317C22—1461no—
318C22—1461no—
319C22—1461no—
320C22—1461no—
321C22#1798955511yescharacteristic
322C4—21no—
323C4—41no—
324C4—2921no—
325C4—2921no—
326C4—1461no—
327C4—1461no—
328C4C2×#1455664911yescharacteristic
329C4#1798957311yescharacteristic
330C2—41no—
331C2—81no—
332C2—2921no—
333C2—1462no—
334C2#2262516611yescharacteristic
335C2#2262617111yescharacteristic
336C2#2262617411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#131×C73C4
socleC22×C73C2×C22⋊C8
prime radical#17989094C4
cosocleC22×#10596831C23
hypercenterC2×#131#64408
hypocenterC73#11470
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. #22626159C2semidirect
2. #22626156C2semidirect
3. #22624870C2semidirect
4. #22624853C2semidirect
174. C73#11470semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226261562
#226248702
#226248532
#226186232
#1147073