Finite Group Databasebeta

← #47986408

Subgroups — #47986408

Order 35456.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C2×#22579842C212yesmaximal, complemented
2C2×#22579842C211yesmaximal, characteristic, complemented
3C2×#22579838C211yesmaximal, characteristic, complemented
4#31799928C211yesmaximal, characteristic, complemented
5#31799924C212yesmaximal, complemented
6#31799920C212yesmaximal, complemented
7#31799919C211yesmaximal, characteristic, complemented
8#31799912C211yesmaximal, characteristic
9#31799853C211yesmaximal, characteristic, complemented
10#31799849C212yesmaximal, complemented
11#398×C277C211yesmaximal, characteristic, complemented
12C22×D1108C2211yescharacteristic
13C22×D1108C2212yes—
14C22×C4×D277C2212yes—
15C22×C4×D277C2211yescharacteristic
16#22579842—28no—
17#22579842—24no—
18#22579838—24no—
19C2×C4⋊Dic277C2211yescharacteristic
20C2×#17984542C2212yes—
21C2×#17984540C2212yes—
22C2×#17984540C2212yes—
23C2×#17984540C2211yescharacteristic
24C2×#17984540C2212yes—
25C2×#17984540C2211yescharacteristic
26C2×#17984540C2212yes—
27C2×#17984540C2212yes—
28C2×C4×Dic277C2211yescharacteristic
29C2×C4×Dic277C2212yes—
30C2×C4×Dic277C2211yescharacteristic
31#22579787C2212yes—
32#22579787C2211yescharacteristic
33C2×C4⋊C4×C277C2211yescharacteristic
34C2×C4⋊C4×C277C2212yes—
35C2×C4⋊C4×C277C2211yescharacteristic
36#95×C277C2212yes—
37#95×C277C2211yescharacteristic
38C23×D277C2311yescharacteristic
39C23×D277C2312yes—
40C2×D1108—24no—
41C2×D1108—24no—
42C2×D1108—24no—
43C2×D1108—22no—
44C2×D1108—24no—
45C22×Dic277C2312yes—
46C22×Dic277C2311yescharacteristic
47C22×Dic277C2311yescharacteristic
48C22×Dic277C2311yescharacteristic
49C2×C4×D277—22no—
50C2×C4×D277—22no—
51C2×C4×D277—24no—
52C2×C4×D277—24no—
53C2×C4×D277—24no—
54C2×C4×D277D414yescomplemented
55C4⋊Dic277—22no—
56#17984542—24no—
57#17984540—24no—
58#17984540—24no—
59#17984540—22no—
60#17984540—24no—
61#17984540—22no—
62#17984540—24no—
63#17984540—24no—
64C4×Dic277—22no—
65C4×Dic277—24no—
66C4×Dic277D414yescomplemented
67C22×C4×C277C2311yescharacteristic
68C22×C4×C277C2312yes—
69C22×C4×C277C2311yescharacteristic
70C22×C4×C277C2311yescharacteristic
71C22×C4×C277C2312yes—
72C4⋊C4×C277—22no—
73C4⋊C4×C277—24no—
74C4⋊C4×C277—22no—
75C22×D277—21no—
76C22×D277—22no—
77C22×D277—22no—
78C22×D277—22no—
79C22×D277—21no—
80C22×D277—22no—
81C22×D277—22no—
82C22×D277—22no—
83C22×D277—22no—
84C22×D277—22no—
85C22×D277C2×D412yes—
86C22×D277#4114yes—
87D1108—28no—
88D1108—216no—
89C2×Dic277—22no—
90C2×Dic277—22no—
91C2×Dic277—21no—
92C2×Dic277—22no—
93C2×Dic277—22no—
94C2×Dic277—22no—
95C2×Dic277C2×D412yes—
96C2×Dic277C2×D412yes—
97C2×Dic277C2×D412yes—
98C2×Dic277#4112yes—
99C4×D277—28no—
100C4×D277—48no—
101C23×C277C2411yescharacteristic
102C2×C4×C277—22no—
103C2×C4×C277—21no—
104C2×C4×C277—21no—
105C2×C4×C277—22no—
106C2×C4×C277—22no—
107C2×C4×C277—21no—
108C2×C4×C277—22no—
109C2×C4×C277—22no—
110C2×C4×C277—22no—
111C2×C4×C277—22no—
112C2×C4×C277C2×D412yes—
113C2×C4×C277C2×D412yes—
114C2×C4×C277#4114yes—
115C22×C277C22×D411yescharacteristic
116C22×C277C22×D411yescharacteristic
117C22×C277C2×#4111yescharacteristic
118C22×C277C2×#4112yes—
119C22×C277C2×#4112yes—
120C2×D277—22no—
121C2×D277—24no—
122C2×D277—24no—
123C2×D277—24no—
124C2×D277—24no—
125C2×D277—42no—
126C2×D277—41no—
127C2×D277—42no—
128C2×D277—42no—
129C2×D277—42no—
130C2×D277—41no—
131C2×D277—42no—
132C4×C277—22no—
133C4×C277—22no—
134C4×C277—24no—
135C4×C277—42no—
136C4×C277—41no—
137Dic277—22no—
138Dic277—42no—
139Dic277—41no—
140Dic277#12114yes—
141C2×C277C2×#12111yescharacteristic
142C2×C277#52511yescharacteristic
143C2×C277#52512yes—
144C2×C277#52912yes—
145C2×C277#53211yescharacteristic
146D277—42no—
147D277—44no—
148C277#239211yescharacteristic, complemented, Sylow
149#2392—2771nomaximal, Sylow
150C2×C4×D4—2771no—
151C2×C4×D4—2772no—
152C2×#117—2771no—
153#398D27711yescharacteristic, complemented
154#397—2771no—
155#396—2772no—
156#394—2771no—
157#394—2772no—
158#392—2771no—
159C4×C4⋊C4—2771no—
160C4×C22⋊C4—2772no—
161C22×D4—2771no—
162C22×D4—2772no—
163C23×C4—2771no—
164C23×C4—2772no—
165C4×D4—5544no—
166C4×D4—5548no—
167#117—5544no—
168C2×C4⋊C4—2771no—
169C2×C4⋊C4—2772no—
170C2×C4⋊C4C2×D27711yescharacteristic
171C2×C4⋊C4C2×D27712yes—
172C2×C4⋊C4C2×D27711yescharacteristic
173C2×C22⋊C4—2771no—
174C2×C22⋊C4—2771no—
175C2×C22⋊C4—2772no—
176C2×C22⋊C4—2772no—
177C2×C22⋊C4—2772no—
178C2×C22⋊C4—2772no—
179C2×C22⋊C4—2772no—
180C2×C42—2771no—
181C2×C42—2771no—
182C2×C42—2772no—
183#95—2772no—
184#95—2771no—
185#95C2×D27712yes—
186#95C2×D27711yescharacteristic
187C24—2771no—
188C24—2772no—
189C2×D4—5544no—
190C2×D4—5544no—
191C2×D4—5544no—
192C2×D4—5542no—
193C2×D4—5544no—
194C22×C4—2771no—
195C22×C4—2771no—
196C22×C4—2774no—
197C22×C4—2772no—
198C22×C4—2771no—
199C22×C4—5542no—
200C22×C4—5542no—
201C22×C4—5544no—
202C22×C4—5544no—
203C22×C4—5544no—
204C22×C4C22×D27712yes—
205C22×C4C22×D27711yescharacteristic
206C22×C4C22×D27711yescharacteristic
207C22×C4C22×D27711yescharacteristic
208C22×C4C22×D27712yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—5542no—
213C4⋊C4—5544no—
214C22⋊C4—5542no—
215C22⋊C4—5544no—
216C22⋊C4—5544no—
217C22⋊C4—5544no—
218C22⋊C4—5542no—
219C22⋊C4—5544no—
220C22⋊C4—5544no—
221C42—2774no—
222C42—5542no—
223C42—5544no—
224C23—2772no—
225C23—2774no—
226C23—5541no—
227C23—5542no—
228C23—5542no—
229C23—5541no—
230C23—5542no—
231C23—5542no—
232C23—5542no—
233C23—5542no—
234C23—5542no—
235C23—5542no—
236C23C23×D27711yescharacteristic
237D4—5548no—
238D4—55416no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—2772no—
250C2×C4—2772no—
251C2×C4—2772no—
252C2×C4—2772no—
253C2×C4—5548no—
254C2×C4—5542no—
255C2×C4—5542no—
256C2×C4—5541no—
257C2×C4—5542no—
258C2×C4—5542no—
259C2×C4—5542no—
260C2×C4—11088no—
261C2×C4D4×D27712yes—
262C2×C4D4×D27712yes—
263C2×C4#1798456414yes—
264C22—5542no—
265C22—5544no—
266C22—5544no—
267C22—5544no—
268C22—5544no—
269C22—11082no—
270C22—11081no—
271C22—11082no—
272C22—11082no—
273C22—11082no—
274C22—11081no—
275C22—11082no—
276C22C2×D4×D27711yescharacteristic
277C22C2×D4×D27711yescharacteristic
278C22#41×D27711yescharacteristic
279C22#41×D27712yes—
280C22C2×#1798456412yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—2774no—
287C4—5542no—
288C4—11082no—
289C4—11081no—
290C2—11082no—
291C2—11084no—
292C2#121×D27711yescharacteristic
293C2#3180051011yescharacteristic
294C2#3180051512yes—
295C2#3180053712yes—
296C2#3180055311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D277
commutator, socle, cosocleC23×C277C24
Fitting#398×C277C2
prime radical#31799912C2
hypercenter#398D277
hypocenterC277#2392
LCSG ▷ C23×C277 ▷ C277
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#22579842C2semidirect
2. C2×#22579842C2semidirect
3. C2×#22579838C2semidirect
4. #31799928C2semidirect
5. #31799924C2semidirect
6. #31799920C2semidirect
7. #31799919C2semidirect
9. #31799853C2semidirect
10. #31799849C2semidirect
11. #398×C277C2semidirect
54. C2×C4×D277D4semidirect
66. C4×Dic277D4semidirect
148. C277#2392semidirect
153. #398D277semidirect

Maximal subgroups

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

subgroupindex
C2×#225798422
C2×#225798382
#317999282
#317999242
#317999202
#317999192
#317999122
#317998532
#317998492
#398×C2772
#2392277