Finite Group Databasebeta

← #3313×C173 #36351098

Subgroups — #3313×C173 #36351098

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#556×C173C211yesmaximal, characteristic, complemented
2#548×C173C211yesmaximal, characteristic
3#489×C173C211yesmaximal, characteristic, complemented
4#485×C173C211yesmaximal, characteristic, complemented
5#482×C173C211yesmaximal, characteristic, complemented
6#477×C173C211yesmaximal, characteristic, complemented
7#468×C173C211yesmaximal, characteristic, complemented
8#467×C173C211yesmaximal, characteristic, complemented
9#464×C173C211yesmaximal, characteristic, complemented
10#452×C173C211yesmaximal, characteristic, complemented
11#451×C173C211yesmaximal, characteristic, complemented
12#441×C173C211yesmaximal, characteristic, complemented
13#435×C173C211yesmaximal, characteristic, complemented
14#419×C173C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×#41×C173C2211yescharacteristic
17C22×Q8×C173C2211yescharacteristic
18C2×Q16×C173C2211yescharacteristic
19C2×Q16×C173C2211yescharacteristic
20C2×SD16×C173C2211yescharacteristic
21C2×M16×C173C2211yescharacteristic
22C22×C8×C173C2211yescharacteristic
23C4⋊Q8×C173C2211yescharacteristic
24#126×C173—21no—
25#126×C173—21no—
26#124×C173—21no—
27#124×C173—21no—
28#124×C173C2211yescharacteristic
29C22⋊Q8×C173—21no—
30C22⋊Q8×C173C2211yescharacteristic
31C22⋊Q8×C173C2211yescharacteristic
32C22⋊Q8×C173C2211yescharacteristic
33#121×C173—21no—
34#121×C173C2211yescharacteristic
35C4×Q8×C173—21no—
36C4×Q8×C173C2211yescharacteristic
37C4×Q8×C173C2211yescharacteristic
38C4×D4×C173—21no—
39C4×D4×C173C2211yescharacteristic
40#117×C173C2211yescharacteristic
41#117×C173C2211yescharacteristic
42#107×C173C2211yescharacteristic
43#107×C173C2211yescharacteristic
44#106×C173C2211yescharacteristic
45C4⋊C8×C173C2211yescharacteristic
46#103×C173C2211yescharacteristic
47#103×C173C2211yescharacteristic
48#103×C173C2211yescharacteristic
49#103×C173C2211yescharacteristic
50#103×C173C2211yescharacteristic
51#103×C173C2211yescharacteristic
52#103×C173C2211yescharacteristic
53#102×C173C2211yescharacteristic
54#102×C173C2211yescharacteristic
55#102×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#41×C173—22no—
60#41×C173—41no—
61C2×Q8×C173—21no—
62C2×Q8×C173—21no—
63C2×Q8×C173—22no—
64C2×Q8×C173—41no—
65C2×Q8×C173C2311yescharacteristic
66C2×Q8×C173C2311yescharacteristic
67C2×Q8×C173C2311yescharacteristic
68C2×D4×C173—21no—
69C2×D4×C173—21no—
70C2×D4×C173D411yescharacteristic
71C2×D4×C173C2311yescharacteristic
72C22×C4×C173—21no—
73C22×C4×C173—21no—
74C22×C4×C173C2311yescharacteristic
75C22×C4×C173C2311yescharacteristic
76Q16×C173—22no—
77Q16×C173—41no—
78SD16×C173—41no—
79M16×C173—41no—
80C2×C8×C173—41no—
81C2×C8×C173C2311yescharacteristic
82C2×C8×C173C2311yescharacteristic
83C2×C8×C173C2311yescharacteristic
84C2×C8×C173C2311yescharacteristic
85C4⋊C4×C173—21no—
86C4⋊C4×C173—21no—
87C4⋊C4×C173—21no—
88C4⋊C4×C173—21no—
89C4⋊C4×C173—21no—
90C4⋊C4×C173—21no—
91C4⋊C4×C173D411yescharacteristic
92C4⋊C4×C173C2311yescharacteristic
93C4⋊C4×C173C2311yescharacteristic
94C4⋊C4×C173C2311yescharacteristic
95C4⋊C4×C173C2311yescharacteristic
96C22⋊C4×C173—21no—
97C22⋊C4×C173—21no—
98C22⋊C4×C173—21no—
99C22⋊C4×C173—21no—
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C22⋊C4×C173D411yescharacteristic
105C22⋊C4×C173D411yescharacteristic
106C42×C173—21no—
107C42×C173—21no—
108C42×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C23×C173—21no—
112C23×C173C2×D411yescharacteristic
113C23×C173C2×D411yescharacteristic
114Q8×C173—21no—
115Q8×C173—21no—
116Q8×C173—21no—
117Q8×C173—21no—
118Q8×C173—21no—
119Q8×C173—42no—
120Q8×C173—42no—
121Q8×C173#4112yes—
122D4×C173—21no—
123D4×C173—21no—
124D4×C173—41no—
125D4×C173—42no—
126D4×C173—42no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—42no—
139C2×C4×C173—42no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2411yescharacteristic
145C8×C173—21no—
146C8×C173—21no—
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—22no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173C2×#4111yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172C2×C173—41no—
173C2×C173—41no—
174C2×C173—81no—
175C2×C173#54611yescharacteristic
176C2×C173#57511yescharacteristic
177C2×C173#57811yescharacteristic
178C173#331311yescharacteristic, direct product factor, Sylow
179#3313C17311yesmaximal, characteristic, direct product factor, Sylow
180#556C2×C17311yescharacteristic, complemented
181#548C2×C17311yescharacteristic
182#489C2×C17311yescharacteristic, complemented
183#485C2×C17311yescharacteristic, complemented
184#482C2×C17311yescharacteristic, complemented
185#477C2×C17311yescharacteristic, complemented
186#468C2×C17311yescharacteristic, complemented
187#467C2×C17311yescharacteristic, complemented
188#464C2×C17311yescharacteristic, complemented
189#452C2×C17311yescharacteristic, complemented
190#451C2×C17311yescharacteristic, complemented
191#441C2×C17311yescharacteristic, complemented
192#435C2×C17311yescharacteristic, complemented
193#419C2×C17311yescharacteristic, complemented
194#416C2×C17311yescharacteristic, complemented
195C2×#41C22×C17311yescharacteristic
196C22×Q8C22×C17311yescharacteristic
197C2×Q16C22×C17311yescharacteristic
198C2×Q16C22×C17311yescharacteristic
199C2×SD16C22×C17311yescharacteristic
200C2×M16C22×C17311yescharacteristic
201C22×C8C22×C17311yescharacteristic
202C4⋊Q8C22×C17311yescharacteristic
203#126—21no—
204#126—21no—
205#124—21no—
206#124—21no—
207#124C22×C17311yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8C22×C17311yescharacteristic
210C22⋊Q8C22×C17311yescharacteristic
211C22⋊Q8C22×C17311yescharacteristic
212#121—21no—
213#121C22×C17311yescharacteristic
214C4×Q8—21no—
215C4×Q8C22×C17311yescharacteristic
216C4×Q8C22×C17311yescharacteristic
217C4×D4—21no—
218C4×D4C22×C17311yescharacteristic
219#117C22×C17311yescharacteristic
220#117C22×C17311yescharacteristic
221#107C22×C17311yescharacteristic
222#107C22×C17311yescharacteristic
223#106C22×C17311yescharacteristic
224C4⋊C8C22×C17311yescharacteristic
225#103C22×C17311yescharacteristic
226#103C22×C17311yescharacteristic
227#103C22×C17311yescharacteristic
228#103C22×C17311yescharacteristic
229#103C22×C17311yescharacteristic
230#103C22×C17311yescharacteristic
231#103C22×C17311yescharacteristic
232#102C22×C17311yescharacteristic
233#102C22×C17311yescharacteristic
234#102C22×C17311yescharacteristic
235C22⋊C8C22×C17311yescharacteristic
236C22⋊C8C22×C17311yescharacteristic
237#97C22×C17311yescharacteristic
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8C23×C17311yescharacteristic
245C2×Q8C23×C17311yescharacteristic
246C2×Q8C23×C17311yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4D4×C17311yescharacteristic
250C2×D4C23×C17311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4C23×C17311yescharacteristic
254C22×C4C23×C17311yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—41no—
258M16—41no—
259C2×C8—41no—
260C2×C8C23×C17311yescharacteristic
261C2×C8C23×C17311yescharacteristic
262C2×C8C23×C17311yescharacteristic
263C2×C8C23×C17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4D4×C17311yescharacteristic
271C4⋊C4C23×C17311yescharacteristic
272C4⋊C4C23×C17311yescharacteristic
273C4⋊C4C23×C17311yescharacteristic
274C4⋊C4C23×C17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4D4×C17311yescharacteristic
284C22⋊C4D4×C17311yescharacteristic
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42C23×C17311yescharacteristic
290C23—21no—
291C23C2×D4×C17311yescharacteristic
292C23C2×D4×C17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—42no—
300Q8#41×C17312yes—
301D4—21no—
302D4—21no—
303D4—41no—
304D4—42no—
305D4—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4C2×D4×C17311yescharacteristic
320C2×C4C2×D4×C17311yescharacteristic
321C2×C4C2×D4×C17311yescharacteristic
322C2×C4C2×D4×C17311yescharacteristic
323C2×C4C24×C17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×D4×C17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4C2×#41×C17311yescharacteristic
350C4H(5,𝔽2)×C17311yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2#546×C17311yescharacteristic
355C2#575×C17311yescharacteristic
356C2#578×C17311yescharacteristic
subgroupG/H
center, socleC22×C173C22×D4
commutator, Frattini, cosocleC2×C4C24×C173
prime radical#548×C173C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C173 ◁ C4×D4×C173 ◁ G

Complements

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

NH ≅ G/Ntype
1. #556×C173C2semidirect
3. #489×C173C2semidirect
4. #485×C173C2semidirect
5. #482×C173C2semidirect
6. #477×C173C2semidirect
7. #468×C173C2semidirect
8. #467×C173C2semidirect
9. #464×C173C2semidirect
10. #452×C173C2semidirect
11. #451×C173C2semidirect
12. #441×C173C2semidirect
13. #435×C173C2semidirect
14. #419×C173C2semidirect
15. #416×C173C2semidirect
178. C173#3313direct (N × H)
179. #3313C173direct (N × H)
180. #556C2×C173semidirect
182. #489C2×C173semidirect
183. #485C2×C173semidirect
184. #482C2×C173semidirect
185. #477C2×C173semidirect
186. #468C2×C173semidirect
187. #467C2×C173semidirect
188. #464C2×C173semidirect
189. #452C2×C173semidirect
190. #451C2×C173semidirect
191. #441C2×C173semidirect
192. #435C2×C173semidirect
193. #419C2×C173semidirect
194. #416C2×C173semidirect

Maximal subgroups

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

subgroupindex
#556×C1732
#548×C1732
#489×C1732
#485×C1732
#482×C1732
#477×C1732
#468×C1732
#467×C1732
#464×C1732
#452×C1732
#451×C1732
#441×C1732
#435×C1732
#419×C1732
#416×C1732
#3313173