Finite Group Databasebeta

← #32988256

Subgroups — #32988256

Order 18688.

462 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 101 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#22620879C211yesmaximal, characteristic
2#22620453C211yesmaximal, characteristic
3#22610905C211yesmaximal, characteristic
4#22608496C211yesmaximal, characteristic
5#22608495C211yesmaximal, characteristic
6#22607775C211yesmaximal, characteristic
7C73×#3230C211yesmaximal, characteristic
8#17989144C2211yescharacteristic
9#17988997—21no—
10#17988997—21no—
11#17988993—21no—
12#17988993—21no—
13#17988966—21no—
14#17988937—21no—
15#17988794—22no—
16#17988743—22no—
17#17988640C2211yescharacteristic
18#17988623—22no—
19C2×#14556502C2211yescharacteristic
20#17988453C2211yescharacteristic
21#17988349—22no—
22#17988325—22no—
23#17988324—22no—
24#533×C73C2211yescharacteristic
25C2×C4⋊Q8×C73C2211yescharacteristic
26#500×C73—21no—
27#487×C73—21no—
28#484×C73—22no—
29#477×C73—22no—
30#431×C73C2211yescharacteristic
31C2×#11924018—24no—
32C22×#10596831D412yes—
33#14556591D411yescharacteristic
34#14556591D411yescharacteristic
35#14556583—21no—
36#14556570—21no—
37#14556570—21no—
38C4×Dic146—21no—
39C4×Dic146—21no—
40#14556550—21no—
41C2×C4⋊Dic73—21no—
42#14556524—22no—
43#14556524—22no—
44#14556522—22no—
45#14556522—22no—
46C4⋊#10596831—21no—
47#14556515—24no—
48C22⋊#10596831D412yes—
49#14556502—21no—
50#14556502—21no—
51C4×#10596831—21no—
52#14556498—21no—
53#14556498—21no—
54C22×Q8×C73—21no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73D412yes—
59C4⋊Q8×C73D411yescharacteristic
60C4⋊Q8×C73D411yescharacteristic
61#125×C73—21no—
62C22⋊Q8×C73—21no—
63C22⋊Q8×C73—21no—
64C4×Q8×C73—21no—
65C4×Q8×C73—21no—
66#117×C73—21no—
67C2×C4⋊C4×C73—21no—
68C2×C4⋊C4×C73—21no—
69C2×C42×C73C2311yescharacteristic
70#107×C73—24no—
71C4⋊C8×C73—21no—
72#103×C73—24no—
73C22⋊C8×C73D412yes—
74C4×C8×C73—21no—
75C2×Dic146—21no—
76C2×Dic146—21no—
77C22×Dic73—41no—
78#11924018—84no—
79C2×#10596831—22no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831Q1614yescomplemented
83C4⋊Dic73—21no—
84C4⋊Dic73—21no—
85C4⋊Dic73—21no—
86C4⋊Dic73—21no—
87C4⋊Dic73—42no—
88#11924007—41no—
89#11924007—41no—
90C22⋊Dic73—41no—
91C4×Dic73—41no—
92C2×Q8×C73—21no—
93C2×Q8×C73—22no—
94C2×Q8×C73—21no—
95C2×Q8×C73—41no—
96C2×Q8×C73—82no—
97C22×C4×C73—41no—
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D411yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C2×C8×C73—22no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—21no—
104C4⋊C4×C73—21no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—21no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—41no—
112C22⋊C4×C73—41no—
113C42×C73—41no—
114C42×C73C2×D411yescharacteristic
115C42×C73C2×D411yescharacteristic
116C42×C73C2×D411yescharacteristic
117C42×C73#4111yescharacteristic
118C23×C73#12711yescharacteristic
119Q8×C73—42no—
120Q8×C73—44no—
121Q8×C73—81no—
122Q8×C73—82no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—42no—
128C2×C4×C73—82no—
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73#12011yescharacteristic
131C2×C4×C73#12011yescharacteristic
132C2×C4×C73#12111yescharacteristic
133C2×C4×C73#12111yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73C2×Q1612yes—
136C2×C4×C73#13712yes—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—81no—
140C2×Dic73—81no—
141C2×Dic73—81no—
142Dic146—42no—
143Dic146—81no—
144C8×C73—42no—
145#10596831—24no—
146#10596831—42no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73#39211yescharacteristic
150C22×C73#49411yescharacteristic
151C22×C73#49711yescharacteristic
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—21no—
156C4×C73—82no—
157C4×C73—81no—
158C4×C73—81no—
159C4×C73#45112yes—
160C4×C73#46212yes—
161Dic73—81no—
162Dic73—81no—
163Dic73—161no—
164#10224—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73#200811yescharacteristic
168C2×C73#200911yescharacteristic
169C2×C73#201611yescharacteristic
170#3230D7311yescharacteristic
171#3209—731no—
172#3162—731no—
173#1993—731no—
174#1518—731no—
175#1518—731no—
176#1281—731no—
177C73#1022411yescharacteristic, complemented, Sylow
178#533—731no—
179#533C2×D7311yescharacteristic
180C2×C4⋊Q8C2×D7311yescharacteristic
181#501—1461no—
182#501—1461no—
183#501—1461no—
184#501—1461no—
185#500—21no—
186#494—1461no—
187#487—21no—
188#487—1461no—
189#484—22no—
190#477—22no—
191#462—1462no—
192#451—1462no—
193#431—731no—
194#431C2×D7311yescharacteristic
195#427—1462no—
196C2×#97—731no—
197#382—731no—
198#340—1462no—
199#328—1462no—
200#328—1462no—
201C22×Q8—21no—
202C2×Q16—1464no—
203C22×C8—732no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—731no—
208C4⋊Q8—731no—
209C4⋊Q8C22⋊D7312yes—
210C4⋊Q8C22⋊D7311yescharacteristic
211C4⋊Q8C22⋊D7311yescharacteristic
212#125—21no—
213#125—1461no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—1461no—
217C22⋊Q8—1461no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—1461no—
221C4×Q8—1461no—
222#117—21no—
223#117—1461no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1461no—
227C2×C42C22×D7311yescharacteristic
228#107—24no—
229#107—1462no—
230#107—1462no—
231#106—1462no—
232#106—1462no—
233C4⋊C8—21no—
234C4⋊C8—1461no—
235#103—24no—
236#103—1464no—
237C22⋊C8—732no—
238C22⋊C8C22⋊D7312yes—
239#97—1461no—
240#97—1461no—
241C4×C8—21no—
242C4×C8—1461no—
243#95—1461no—
244#95—1461no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—1461no—
251C2×Q8—1461no—
252C22×C4—41no—
253C22×C4—2921no—
254C22×C4D4×D7311yescharacteristic
255C22×C4D4×D7311yescharacteristic
256C22×C4C2×C22⋊D7311yescharacteristic
257Q16—5844no—
258C2×C8—22no—
259C2×C8—734no—
260C2×C8—1462no—
261C2×C8—1462no—
262C2×C8—1462no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—2921no—
274C4⋊C4—2921no—
275C4⋊C4—2922no—
276C4⋊C4—1461no—
277C4⋊C4—1461no—
278C4⋊C4—1461no—
279C4⋊C4—1461no—
280C22⋊C4—41no—
281C22⋊C4—2921no—
282C42—41no—
283C42—2921no—
284C42D4×D7311yescharacteristic
285C42C2×C22⋊D7311yescharacteristic
286C42C2×C22⋊D7311yescharacteristic
287C42#1192402711yescharacteristic
288C23#1455658711yescharacteristic
289Q8—44no—
290Q8—42no—
291Q8—81no—
292Q8—82no—
293Q8—2922no—
294Q8—5841no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—5841no—
304C2×C4—5841no—
305C2×C4—5841no—
306C2×C4#1455655911yescharacteristic
307C2×C4#1455655911yescharacteristic
308C2×C4#1455656011yescharacteristic
309C2×C4#1455656211yescharacteristic
310C2×C4#1455656311yescharacteristic
311C2×C4#1455656311yescharacteristic
312C2×C4Q16×D7312yes—
313C2×C4#1455662312yes—
314C8—42no—
315C8—2922no—
316C8—1464no—
317C22—21no—
318C22—21no—
319C22#1798849511yescharacteristic
320C22#1798896511yescharacteristic
321C22#1798898011yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—5841no—
330C4—5841no—
331C4—11681no—
332C4#1798874112yes—
333C4#1798879512yes—
334C2—21no—
335C2—21no—
336C2#2261098211yescharacteristic
337C2#2261099111yescharacteristic
338C2#2261102911yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#3230C2
socleC22×C73#392
prime radicalC23×C73#127
hypercenter#3230D73
hypocenterC73#10224
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
82. C2×#10596831Q16semidirect
177. C73#10224semidirect

Maximal subgroups

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

subgroupindex
#226208792
#226204532
#226109052
#226084962
#226084952
#226077752
C73×#32302
#1022473