Finite Group Databasebeta

← #32974987

Subgroups — #32974987

Order 18688.

424 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 87 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#22611264C211yesmaximal, characteristic
2#22611259C211yesmaximal, characteristic
3#22611030C211yesmaximal, characteristic
4#22607757C211yesmaximal, characteristic
5#22607754C211yesmaximal, characteristic
6#22607754C211yesmaximal, characteristic
7C73×#2805C211yesmaximal, characteristic
8C22×#11924018—21no—
9C2×#14556591C2211yescharacteristic
10C2×#14556524—21no—
11C2×#14556522—21no—
12C2×#14556514—21no—
13#17988349—21no—
14#17988349—21no—
15#17988342C2211yescharacteristic
16#17988342C2211yescharacteristic
17#17988342C2211yescharacteristic
18#400×C73—21no—
19#400×C73—21no—
20#395×C73—21no—
21#391×C73C2211yescharacteristic
22#389×C73—21no—
23#389×C73C2211yescharacteristic
24#389×C73C2211yescharacteristic
25#387×C73—21no—
26#387×C73—21no—
27C22×Dic146—21no—
28C2×#11924018—42no—
29C2×#11924018—42no—
30C2×#11924018—42no—
31C22×#10596831—21no—
32C22×#10596831—21no—
33C22×#10596831—21no—
34#14556591—42no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37#14556524—42no—
38#14556522—42no—
39#14556514—42no—
40C22×Q8×C73D411yescharacteristic
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C42×C73—21no—
51C2×C42×C73C2311yescharacteristic
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61C2×Dic146—22no—
62C2×Dic146—41no—
63C2×Dic146—41no—
64C2×Dic146—41no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C22×Dic73—41no—
68#11924018—48no—
69C2×#10596831—22no—
70C2×#10596831—22no—
71C2×#10596831—22no—
72C2×#10596831—41no—
73C2×#10596831—41no—
74C2×#10596831—41no—
75C2×#10596831—41no—
76C2×#10596831—41no—
77C2×#10596831—41no—
78C4⋊Dic73—22no—
79C4⋊Dic73—22no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82C2×Q8×C73—21no—
83C2×Q8×C73—21no—
84C2×Q8×C73—41no—
85C2×Q8×C73—41no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95C4⋊C4×C73—21no—
96C4⋊C4×C73—21no—
97C4⋊C4×C73—21no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—81no—
106C4⋊C4×C73—81no—
107C4⋊C4×C73Q1612yescomplemented
108C42×C73—41no—
109C42×C73—81no—
110C23×C73#12011yescharacteristic
111Q8×C73—42no—
112Q8×C73—81no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—22no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73C2×Q1611yescharacteristic
138C2×C4×C73#13511yescharacteristic
139C2×C4×C73#13511yescharacteristic
140C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C73#13711yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×Dic73—42no—
144C2×Dic73—81no—
145C2×Dic73—81no—
146Dic146—42no—
147Dic146—42no—
148Dic146—81no—
149Dic146—81no—
150#10596831—42no—
151#10596831—42no—
152#10596831—42no—
153C22×C73#44911yescharacteristic
154C22×C73#45111yescharacteristic
155C22×C73#45211yescharacteristic
156C22×C73#45311yescharacteristic
157C22×C73#45611yescharacteristic
158C22×C73#45611yescharacteristic
159C22×C73B(4,𝔽2)11yescharacteristic
160C4×C73—21no—
161C4×C73—21no—
162C4×C73—21no—
163C4×C73—21no—
164C4×C73—21no—
165C4×C73—21no—
166C4×C73—42no—
167C4×C73—81no—
168C4×C73—81no—
169C4×C73—81no—
170C4×C73—81no—
171C4×C73—81no—
172Dic73—82no—
173#7783—731nomaximal, Sylow
174C2×C73#160011yescharacteristic
175C2×C73#160311yescharacteristic
176C2×C73#160511yescharacteristic
177C2×C73#163211yescharacteristic
178C2×C73#164811yescharacteristic
179C2×C73#165211yescharacteristic
180C2×C73#166011yescharacteristic
181#2805D7311yescharacteristic
182#2058—731no—
183#2057—731no—
184#2016—731no—
185#1272—731no—
186#1271—731no—
187#1271—731no—
188C73#778311yescharacteristic, complemented, Sylow
189C22×Q16—1461no—
190C2×C4⋊Q8—731no—
191C2×#107—1461no—
192C2×#106—1461no—
193C2×#103—1461no—
194#400—21no—
195#400—21no—
196#395—21no—
197#391C2×D7311yescharacteristic
198#389—21no—
199#389C2×D7311yescharacteristic
200#389C2×D7311yescharacteristic
201#387—21no—
202#387—21no—
203#340—1461no—
204#340—1461no—
205#336—731no—
206#336—731no—
207#336—731no—
208C22×Q8—1461no—
209C22×Q8C22⋊D7311yescharacteristic
210C2×Q16—2922no—
211C2×Q16—2922no—
212C2×Q16—2922no—
213C22×C8—1461no—
214C22×C8—1461no—
215C22×C8—1461no—
216C4⋊Q8—2922no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—1461no—
222C2×C4⋊C4—1461no—
223C2×C4⋊C4C22⋊D7311yescharacteristic
224C2×C4⋊C4C22⋊D7311yescharacteristic
225C2×C4⋊C4C22⋊D7311yescharacteristic
226C2×C4⋊C4C22⋊D7311yescharacteristic
227C2×C4⋊C4C22⋊D7311yescharacteristic
228C2×C42—21no—
229C2×C42C22×D7311yescharacteristic
230#107—2922no—
231#106—2922no—
232#103—2922no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—2921no—
247C2×Q8—2921no—
248C2×Q8—2921no—
249C2×Q8—2921no—
250C2×Q8—2921no—
251C2×Q8—1462no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—2921no—
259C22×C4C2×C22⋊D7311yescharacteristic
260C22×C4C2×C22⋊D7311yescharacteristic
261C22×C4C2×C22⋊D7311yescharacteristic
262Q16—2928no—
263C2×C8—2921no—
264C2×C8—2921no—
265C2×C8—2921no—
266C2×C8—2921no—
267C2×C8—2921no—
268C2×C8—2921no—
269C2×C8—1462no—
270C2×C8—1462no—
271C2×C8—1462no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—2921no—
285C4⋊C4—2921no—
286C4⋊C4—1462no—
287C4⋊C4—1462no—
288C4⋊C4#1192401812yescomplemented
289C42—41no—
290C42—81no—
291C23#1455656011yescharacteristic
292Q8—42no—
293Q8—81no—
294Q8—2922no—
295Q8—2922no—
296Q8—5841no—
297Q8—5841no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—2922no—
323C2×C4—5841no—
324C2×C4—5841no—
325C2×C4C2×#1192401811yescharacteristic
326C2×C4#1455661111yescharacteristic
327C2×C4#1455661111yescharacteristic
328C2×C4#1455661611yescharacteristic
329C2×C4#1455662111yescharacteristic
330C2×C4#1455662211yescharacteristic
331C8—2922no—
332C8—2922no—
333C8—2922no—
334C22#1798873211yescharacteristic
335C22#1798874311yescharacteristic
336C22#1798874911yescharacteristic
337C22#1798875511yescharacteristic
338C22#1798877011yescharacteristic
339C22#1798877011yescharacteristic
340C22#1798877411yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—5842no—
354C2#2260887711yescharacteristic
355C2#2260889411yescharacteristic
356C2#2260890511yescharacteristic
357C2#2260906511yescharacteristic
358C2#2260917311yescharacteristic
359C2#2260920111yescharacteristic
360C2#2260924211yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2805C2
socle, prime radicalC23×C73#120
hypercenter#2805D73
hypocenterC73#7783
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
107. C4⋊C4×C73Q16semidirect
188. C73#7783semidirect
288. C4⋊C4#11924018semidirect

Maximal subgroups

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

subgroupindex
#226112642
#226112592
#226110302
#226077572
#226077542
C73×#28052
#778373