Finite Group Databasebeta

← C23×#10386 #19158012

Subgroups — C23×#10386 #19158012

Order 5888.

602 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C23×#2607C211yesmaximal, characteristic, complemented
2C23×#1789C212yesmaximal
3C23×#1782C211yesmaximal, characteristic
4C23×#1306C211yesmaximal, characteristic, complemented
5C23×#1284C212yesmaximal, complemented
6C22×#41×C23C2211yescharacteristic
7C2×C23×#104—22no—
8C2×C23×#104—22no—
9C2×C23×#99—21no—
10C2×C23×#99—21no—
11C23×#407C2212yes—
12C23×#399—21no—
13C23×#393—21no—
14C23×#391—22no—
15C23×#387—22no—
16C23×#381C2211yescharacteristic
17C23×#381C2211yescharacteristic
18C4×C22⋊C4×C23C2212yes—
19C2×#41×C23—22no—
20C2×#41×C23—22no—
21C2×#41×C23—42no—
22C2×#41×C23—42no—
23C22×Q8×C23—21no—
24C22×Q8×C23Q811yescharacteristic, complemented
25C22×D4×C23—21no—
26C22×D4×C23Q811yescharacteristic, complemented
27C23×C4×C23—21no—
28C23×C4×C23C2311yescharacteristic
29C2×M16×C23—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—21no—
32C2×C22⋊C4×C23—22no—
33C2×C22⋊C4×C23—21no—
34C2×C22⋊C4×C23D411yescharacteristic
35C2×C22⋊C4×C23D411yescharacteristic
36C2×C42×C23D412yes—
37C2×C42×C23D412yes—
38C23×#104—44no—
39C23×#104—44no—
40C23×#99—42no—
41C23×#99—42no—
42C23×C22⋊C8—22no—
43C23×C22⋊C8—22no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95—21no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—22no—
50C24×C23—21no—
51C24×C23C2×D411yescharacteristic
52#41×C23—24no—
53#41×C23—24no—
54#41×C23—42no—
55#41×C23—42no—
56#41×C23—44no—
57#41×C23—44no—
58C2×Q8×C23—22no—
59C2×Q8×C23—42no—
60C2×Q8×C23—44no—
61C2×Q8×C23—81no—
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C2×D4×C23—22no—
65C2×D4×C23—42no—
66C2×D4×C23—41no—
67C2×D4×C23—41no—
68C2×D4×C23—41no—
69C2×D4×C23—41no—
70C22×C4×C23—22no—
71C22×C4×C23—21no—
72C22×C4×C23—22no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23—41no—
81C22×C4×C23C2×D412yes—
82C22×C4×C23C2×Q811yescharacteristic
83C22×C4×C23#4112yes—
84C22×C4×C23#4111yescharacteristic
85M16×C23—42no—
86M16×C23—42no—
87C2×C8×C23—42no—
88C4⋊C4×C23—44no—
89C4⋊C4×C23—81no—
90C22⋊C4×C23—44no—
91C22⋊C4×C23—41no—
92C22⋊C4×C23—41no—
93C22⋊C4×C23—41no—
94C22⋊C4×C23—41no—
95C42×C23—42no—
96C42×C23—42no—
97#10386C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—22no—
101C23×C23—42no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23—82no—
105C23×C23#12011yescharacteristic
106C23×C23C22⋊Q811yescharacteristic
107C23×C23#12411yescharacteristic
108Q8×C23—24no—
109Q8×C23—42no—
110Q8×C23—84no—
111D4×C23—24no—
112D4×C23—42no—
113D4×C23—42no—
114D4×C23—42no—
115D4×C23—42no—
116D4×C23—42no—
117C2×C4×C23—22no—
118C2×C4×C23—22no—
119C2×C4×C23—24no—
120C2×C4×C23—22no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—42no—
124C2×C4×C23—44no—
125C2×C4×C23—44no—
126C2×C4×C23—41no—
127C2×C4×C23—42no—
128C2×C4×C23—41no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—81no—
138C2×C4×C23—82no—
139C2×C4×C23C22⋊Q812yes—
140C2×C4×C23#12412yes—
141C8×C23—82no—
142#2607C2×C2311yescharacteristic, complemented
143#1789C2×C2312yes—
144#1782C2×C2311yescharacteristic
145#1306C2×C2311yescharacteristic, complemented
146#1284C2×C2312yescomplemented
147C22×C23—22no—
148C22×C23—21no—
149C22×C23—21no—
150C22×C23—22no—
151C22×C23—22no—
152C22×C23—42no—
153C22×C23—81no—
154C22×C23—81no—
155C22×C23—82no—
156C22×C23—82no—
157C22×C23—82no—
158C22×C23#39311yescharacteristic
159C22×C23B(4,𝔽2)11yescharacteristic
160C22×C23B(4,𝔽2)11yescharacteristic
161C4×C23—22no—
162C4×C23—22no—
163C4×C23—44no—
164C4×C23—42no—
165C4×C23—82no—
166C4×C23—81no—
167C4×C23—81no—
168C4×C23—161no—
169C22×#41C22×C2311yescharacteristic
170C2×#104—22no—
171C2×#104—22no—
172C2×#99—21no—
173C2×#99—21no—
174#407C22×C2312yes—
175#399—21no—
176#393—21no—
177#391—22no—
178#387—22no—
179#381C22×C2311yescharacteristic
180#381C22×C2311yescharacteristic
181C4×C22⋊C4C22×C2312yes—
182C2×C23—21no—
183C2×C23—21no—
184C2×C23—41no—
185C2×C23—41no—
186C2×C23—82no—
187C2×C23#202911yescharacteristic
188C2×C23#203112yes—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—42no—
192C2×#41—42no—
193C22×Q8—21no—
194C22×Q8Q8×C2311yescharacteristic, complemented
195C22×D4—21no—
196C22×D4Q8×C2311yescharacteristic, complemented
197C23×C4—21no—
198C23×C4C23×C2311yescharacteristic
199C2×M16—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4D4×C2311yescharacteristic
205C2×C22⋊C4D4×C2311yescharacteristic
206C2×C42D4×C2312yes—
207C2×C42D4×C2312yes—
208#104—44no—
209#104—44no—
210#99—42no—
211#99—42no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220C23#1038611yescharacteristic, direct product factor, Sylow
221C24—21no—
222C24C2×D4×C2311yescharacteristic
223#41—24no—
224#41—24no—
225#41—42no—
226#41—42no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—81no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4C2×D4×C2312yes—
253C22×C4C2×Q8×C2311yescharacteristic
254C22×C4#41×C2312yes—
255C22×C4#41×C2311yescharacteristic
256M16—42no—
257M16—42no—
258C2×C8—42no—
259C4⋊C4—44no—
260C4⋊C4—81no—
261C22⋊C4—44no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—42no—
267C42—42no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23C23×#12011yescharacteristic
276C23C23×C22⋊Q811yescharacteristic
277C23C23×#12411yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—84no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4C23×C22⋊Q812yes—
310C2×C4C23×#12412yes—
311C8—82no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C23×#39311yescharacteristic
324C22C23×B(4,𝔽2)11yescharacteristic
325C22C23×B(4,𝔽2)11yescharacteristic
326C4—22no—
327C4—22no—
328C4—44no—
329C4—42no—
330C4—81no—
331C4—82no—
332C4—81no—
333C4—161no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2C23×#202911yescharacteristic
340C2C23×#203112yes—
subgroupG/H
center, socleC22×C23#393
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2607C2semidirect
4. C23×#1306C2semidirect
5. C23×#1284C2semidirect
24. C22×Q8×C23Q8semidirect
26. C22×D4×C23Q8semidirect
97. #10386C23direct (N × H)
142. #2607C2×C23semidirect
145. #1306C2×C23semidirect
146. #1284C2×C23semidirect
194. C22×Q8Q8×C23semidirect
196. C22×D4Q8×C23semidirect
220. C23#10386direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26072
C23×#17892
C23×#17822
C23×#13062
C23×#12842
#1038623