Finite Group Databasebeta

← #14536742

Subgroups — #14536742

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762743C211yesmaximal, characteristic
2#11762743C211yesmaximal, characteristic
3#11762742C211yesmaximal, characteristic
4#11762741C211yesmaximal, characteristic
5#11762730C211yesmaximal, characteristic
6#11762730C211yesmaximal, characteristic
7#11762715C211yesmaximal, characteristic
8#11762713C211yesmaximal, characteristic
9#11762713C211yesmaximal, characteristic
10#11762713C211yesmaximal, characteristic
11#11762708C211yesmaximal, characteristic
12#11762707C211yesmaximal, characteristic
13#11762705C211yesmaximal, characteristic
14#11762695C211yesmaximal, characteristic
15C17×#387C211yesmaximal, characteristic
16C22×Dic34C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×C4×Dic17C2211yescharacteristic
34C2×C4×Dic17C2211yescharacteristic
35C17×#95C2211yescharacteristic
36C17×#95C2211yescharacteristic
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C22×C4×C17C2311yescharacteristic
52C22×C4×C17C2311yescharacteristic
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C2×Dic34—22no—
67C2×Dic34—41no—
68C2×Dic34—41no—
69C2×Dic34—41no—
70C2×Dic34—41no—
71C2×Dic34Q814yescomplemented
72C4⋊C4×C17—41no—
73C4⋊C4×C17Q814yescomplemented
74C42×C17—41no—
75C4⋊Dic17—22no—
76C4⋊Dic17—22no—
77C4⋊Dic17—41no—
78C4⋊Dic17—41no—
79C4⋊Dic17—41no—
80C4⋊Dic17—41no—
81#63225—41no—
82#63225—41no—
83#63225—41no—
84#63225—41no—
85#63225—41no—
86#63225—41no—
87C4×Dic17—41no—
88C4×Dic17—41no—
89C23×C17C2411yescharacteristic
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17C2×Q812yes—
108C2×C4×C17C2×Q812yes—
109C2×C4×C17C2×Q812yes—
110C2×C4×C17C2×Q812yes—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17C2×Q812yes—
133C2×Dic17C2×Q812yes—
134C2×Dic17#4112yes—
135Dic34—44no—
136Dic34—44no—
137#2806—171nomaximal, Sylow
138C22×C17C22×Q811yescharacteristic
139C22×C17C22×Q811yescharacteristic
140C22×C17C2×#4111yescharacteristic
141C22×C17H(5,𝔽2)11yescharacteristic
142C22×C17H(5,𝔽2)11yescharacteristic
143C22×C17H(5,𝔽2)11yescharacteristic
144C22×C172−1+411yescharacteristic
145C4×C17—22no—
146C4×C17—22no—
147C4×C17—22no—
148C4×C17—22no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17—41no—
152Dic17—22no—
153Dic17—22no—
154Dic17—22no—
155Dic17—41no—
156Dic17—41no—
157Dic17—41no—
158Dic17—41no—
159Dic17—41no—
160#398—171no—
161#398—171no—
162#398—171no—
163#398—171no—
164#395—171no—
165#395—171no—
166#391—171no—
167#391—171no—
168#391—171no—
169#391—171no—
170#389—171no—
171#389—171no—
172#389—171no—
173#387—171no—
174#387D1711yescharacteristic
175C2×C17#53811yescharacteristic
176C2×C17#54311yescharacteristic
177C2×C17#54411yescharacteristic
178C2×C17#55411yescharacteristic
179C2×C17#55711yescharacteristic
180C2×C17Q8211yescharacteristic
181C2×C17#56211yescharacteristic
182C22×Q8—171no—
183C22×Q8—171no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—171no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4C2×D1711yescharacteristic
197C2×C4⋊C4C2×D1711yescharacteristic
198C2×C42—171no—
199C2×C42—171no—
200C2×C42C2×D1711yescharacteristic
201#95—171no—
202#95—171no—
203#95—171no—
204#95—171no—
205#95—171no—
206#95—171no—
207#95—171no—
208#95—171no—
209#95—171no—
210#95—171no—
211#95—171no—
212#95—171no—
213#95C2×D1711yescharacteristic
214#95C2×D1711yescharacteristic
215#95C2×D1711yescharacteristic
216#95C2×D1711yescharacteristic
217C17#280611yescharacteristic, complemented, Sylow
218C2×Q8—174no—
219C2×Q8—342no—
220C2×Q8—681no—
221C2×Q8—681no—
222C2×Q8—681no—
223C2×Q8—681no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4C22×D1711yescharacteristic
233C22×C4C22×D1711yescharacteristic
234C22×C4C22×D1711yescharacteristic
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1711yescharacteristic
237C22×C4C22×D1711yescharacteristic
238C22×C4C22×D1711yescharacteristic
239C4⋊C4—41no—
240C4⋊C4—342no—
241C4⋊C4—342no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C4⋊C4—681no—
249C4⋊C4—681no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4Dic3414yescomplemented
253C42—41no—
254C42—681no—
255C42—681no—
256C23C23×D1711yescharacteristic
257Q8—684no—
258Q8—684no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—172no—
277C2×C4—172no—
278C2×C4—172no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4C2×Dic3412yes—
301C2×C4C2×Dic3412yes—
302C2×C4C2×Dic3412yes—
303C2×C4Q8×D1712yes—
304C22C22×Dic3411yescharacteristic
305C22#1058769111yescharacteristic
306C22C2×Q8×D1711yescharacteristic
307C22#41×D1711yescharacteristic
308C22#1058769611yescharacteristic
309C22#1058769611yescharacteristic
310C22#1058769711yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—342no—
319C4—342no—
320C4—342no—
321C4—681no—
322C4—681no—
323C4—681no—
324C4—681no—
325C4—681no—
326C2#1176339811yescharacteristic
327C2#1176343411yescharacteristic
328C2#1176343711yescharacteristic
329C2#1176351811yescharacteristic
330C2#1176353611yescharacteristic
331C2Q8×Dic3411yescharacteristic
332C2#1176355611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#387C2
hypercenter#387D17
hypocenterC17#2806
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
71. C2×Dic34Q8semidirect
73. C4⋊C4×C17Q8semidirect
217. C17#2806semidirect
252. C4⋊C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627432
#117627422
#117627412
#117627302
#117627152
#117627132
#117627082
#117627072
#117627052
#117626952
C17×#3872
#280617