Finite Group Databasebeta

← #32996915

Subgroups — #32996915

Order 18688.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#22615311C211yesmaximal, characteristic
2C2×#17988313C211yesmaximal, characteristic
3#22607960C212yesmaximal, complemented
4#22607936C212yesmaximal, complemented
5C73×#1821C211yesmaximal, characteristic
6C2×#14556570C411yescharacteristic
7C2×C22⋊#10596831—21no—
8#17988517C411yescharacteristic
9#17988517C2211yescharacteristic
10#17988507C411yescharacteristic
11#17988507C2211yescharacteristic
12#17988482—21no—
13#17988482—21no—
14#17988460—21no—
15#17988460—21no—
16C4×C22⋊Dic73C411yescharacteristic
17#17988313—21no—
18#17988313C2212yes—
19C22×C4⋊C4×C73C2211yescharacteristic
20#407×C73—21no—
21#343×C73—21no—
22#343×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C22×#10596831—41no—
25#14556570—41no—
26#14556570—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×#11924007—21no—
32C2×#11924007—21no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C22⋊#10596831—22no—
37C22⋊#10596831—41no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C2×M16×C73—22no—
48C2×C4⋊C4×C73—41no—
49C2×C4⋊C4×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C4⋊C4×C73D411yescharacteristic
53C2×C4⋊C4×C73C2311yescharacteristic
54C22⋊C8×C73—41no—
55C22⋊C8×C73—41no—
56C2×Dic146—41no—
57C2×Dic146—81no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C2×#10596831—42no—
63C2×#10596831—82no—
64C4⋊Dic73—41no—
65#11924007—42no—
66#11924007—42no—
67#11924007—42no—
68#11924007—81no—
69#11924007—81no—
70C22⋊Dic73—24no—
71C22⋊Dic73—41no—
72C22⋊Dic73—41no—
73C4×Dic73—81no—
74C24×C73C22⋊C411yescharacteristic
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—22no—
82C22×C4×C73—41no—
83C22×C4×C73—81no—
84C22×C4×C73—81no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91M16×C73—42no—
92M16×C73—42no—
93C2×C8×C73—42no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C23×C73—22no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—41no—
102C23×C73#9911yescharacteristic
103C23×C73#9911yescharacteristic
104C23×C73C2×C22⋊C411yescharacteristic
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73—81no—
124C2×C4×C73#13512yes—
125C2×Dic73—22no—
126C2×Dic73—22no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137Dic146—82no—
138C8×C73—82no—
139#10596831—82no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73#35612yes—
150C22×C73C2×#9911yescharacteristic
151C22×C73#41611yescharacteristic
152C22×C73#42111yescharacteristic
153C4×C73—22no—
154C4×C73—41no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159Dic73—42no—
160Dic73—42no—
161Dic73—81no—
162Dic73—81no—
163#11854—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—22no—
167C2×C73—41no—
168C2×C73#151611yescharacteristic
169C2×C73#212111yescharacteristic
170C2×C73C2×#35611yescharacteristic
171#2583—731no—
172#1821D7311yescharacteristic
173C2×#324—731no—
174#1354—732no—
175#1346—732no—
176C73#1185411yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—731no—
178C22×C4⋊C4C2×D7311yescharacteristic
179#407—21no—
180C2×C22⋊C8—1461no—
181#398—731no—
182#398—731no—
183#395—731no—
184#395—731no—
185#389—1461no—
186#389—1461no—
187#385—1461no—
188#385—1461no—
189C4×C22⋊C4—731no—
190#343—21no—
191#343C2×D7312yes—
192#324—732no—
193#324—1461no—
194C22×Q8—731no—
195C23×C4C4×D7311yescharacteristic
196C23×C4D29211yescharacteristic
197C23×C4C22⋊D7311yescharacteristic
198C2×M16—22no—
199C22×C8—2921no—
200C22⋊Q8—2921no—
201C22⋊Q8—2921no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4C4×D7311yescharacteristic
211C2×C4⋊C4D29211yescharacteristic
212C2×C4⋊C4C22⋊D7311yescharacteristic
213C2×C4⋊C4C22×D7311yescharacteristic
214C2×C22⋊C4—1461no—
215C2×C22⋊C4—1461no—
216C2×C42—1461no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—2921no—
220C22⋊C8—1462no—
221#95—731no—
222#95—731no—
223#95—1461no—
224#95—1461no—
225#95—1461no—
226#95—1461no—
227C24#1192400511yescharacteristic
228C2×Q8—2921no—
229C2×Q8—5841no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4—1461no—
244C22×C4#1192400511yescharacteristic
245C22×C4#1192400511yescharacteristic
246C22×C4#1192400511yescharacteristic
247C22×C4C2×C4×D7311yescharacteristic
248C22×C4C2×C22⋊D7311yescharacteristic
249C22×C4C2×D29211yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—42no—
253C2×C8—2922no—
254C2×C8—5842no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—2921no—
260C4⋊C4—2922no—
261C4⋊C4—2922no—
262C4⋊C4—2922no—
263C4⋊C4—5841no—
264C4⋊C4—5841no—
265C22⋊C4—2921no—
266C22⋊C4—2921no—
267C22⋊C4—1464no—
268C42—5841no—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23#1455650511yescharacteristic
274C23#1455650511yescharacteristic
275C23C2×#1192400511yescharacteristic
276Q8—5842no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—1462no—
307C2×C4—1462no—
308C2×C4#1455661112yes—
309C8—82no—
310C8—5842no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1798838512yes—
321C22C2×#1455650511yescharacteristic
322C22#1798858111yescharacteristic
323C22#1798860511yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—2922no—
331C4—2922no—
332C4—5841no—
333C4—5841no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2260848511yescharacteristic
339C2#2261157811yescharacteristic
340C2C2×#1798838511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1821C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1821D73
hypocenterC73#11854
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #22607960C2semidirect
4. #22607936C2semidirect
176. C73#11854semidirect

Maximal subgroups

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

subgroupindex
#226153112
C2×#179883132
#226079602
#226079362
C73×#18212
#1185473