Finite Group Databasebeta

← #14219829

Subgroups — #14219829

Order 1792.

2190 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6551C211yesmaximal, characteristic
2C7×#2982C211yesmaximal, characteristic, complemented
3C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
4#11737365C214yesmaximal, complemented
5C8×Q8×D7C214yesmaximal, complemented
6#11737351C214yesmaximal, complemented
7C8×#1161C214yesmaximal, complemented
8#11736736C216yesmaximal, complemented
9#11736721C216yesmaximal, complemented
10C2×#6551C411yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12#90503C411yescharacteristic
13#90265C414yes—
14C4×#1161C2214yes—
15#90263C414yes—
16C4×Q8×D7C2214yes—
17#90195C416yes—
18C4×#1157C2216yes—
19C7×C8×Q8C2214yes—
20C7×C8×Q8C2214yes—
21C7×#431C2216yes—
22Q8×C7⋊C8C2214yes—
23Q8×C7⋊C8C2214yes—
24C2×C4⋊(C7⋊C8)C2213yes—
25C2×C4×C7⋊C8C2213yes—
26#89606C2216yes—
27#89606C2216yes—
28#89605C22112yes—
29D7×C4⋊C8C22112yes—
30#89602C2216yes—
31#89602C2216yes—
32(C22⋊D7)⋊C8C2214yes—
33(C22⋊D7)⋊C8C22112yes—
34C8×D28C22112yes—
35#89457C22112yes—
36C4×C8×D7C22112yes—
37C8×Dic14C22112yes—
38C8×2−1+4—71nomaximal, Sylow
39C22×C7×Q8C2×C411yescharacteristic
40#6551C8116yescomplemented
41C2×#1161C2×C414yes—
42C2×Q8×D7C2×C414yes—
43C2×#1157C2×C416yes—
44C4×C7×Q8C2×C414yes—
45C4×C7×Q8C2314yes—
46C2×C7×C4⋊C4C2×C413yes—
47C2×C42×C7C2313yes—
48Q8×Dic7C2×C414yes—
49Q8×Dic7C2×C414yes—
50C4×C22⋊D7C2×C4112yes—
51C4×C22⋊D7C2314yes—
52#6489C2×C416yes—
53C22×C7⋊C8C2311yescharacteristic
54C22×C7⋊C8C2313yes—
55C2×C8×D7C2314yes—
56C2×C8×D7C23112yes—
57#6456C2×C4112yes—
58#6455C2×C4112yes—
59D7×C4⋊C4C2×C4112yes—
60#6450C2×C4112yes—
61C4×D28C2×C416yes—
62C4×D28C2316yes—
63#6435C2×C4112yes—
64C42×D7C23112yes—
65C4×Dic14C2×C416yes—
66C4×Dic14C2316yes—
67C7×C4⋊C8C2316yes—
68C7×C4⋊C8C2316yes—
69C7×C22⋊C8C2314yes—
70C4×C7×C8C23112yes—
71(C2×D7)⋊C8C23112yes—
72(C2×D7)⋊C8C2314yes—
73Dic7⋊C8C2314yes—
74Dic7⋊C8C23112yes—
75C8×Dic7C2314yes—
76C8×Dic7C23112yes—
77C4⋊(C7⋊C8)C23112yes—
78C4×C7⋊C8C2316yes—
79C4×C7⋊C8C2316yes—
80C4×2−1+4—71no—
81#2988—74no—
82#2988—76no—
83#2982—74no—
84#2982D711yescharacteristic, complemented
85C8×#41—74no—
86C8×#41—76no—
87C2×C8×Q8—74no—
88C2×C8×Q8—71no—
89C2×C7×Q8C2×C818yescomplemented
90C2×C7×Q8C22×C414yes—
91C22×C4×C7C22×C413yes—
92C22×C4×C7C22×C413yes—
93C22×C4×C7C2411yescharacteristic
94C2×C22⋊D7C22×C414yes—
95#1161C2×C8132yescomplemented
96Q8×D7C2×C8132yescomplemented
97#1157C2×C8148yescomplemented
98C2×D28C22×C416yes—
99C2×C4×D7C22×C4112yes—
100C2×C4×D7C22×C4112yes—
101C2×C4×D7C2414yes—
102C2×Dic14C22×C416yes—
103C2×C7×C8C2414yes—
104C2×C7×C8C2414yes—
105C7×C4⋊C4C22×C4112yes—
106C42×C7C22×C416yes—
107C42×C7C2416yes—
108C22⋊Dic7C22×C414yes—
109(C2×D7)⋊C4C22×C414yes—
110(C2×D7)⋊C4C22×C4112yes—
111C4⋊Dic7C22×C416yes—
112C4⋊Dic7C22×C416yes—
113Dic7⋊C4C22×C414yes—
114Dic7⋊C4C22×C4112yes—
115C4×Dic7C22×C4112yes—
116C4×Dic7C22×C4112yes—
117C4×Dic7C2414yes—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—43no—
120C2×C7⋊C8C2416yes—
121C2×C7⋊C8C2411yescharacteristic
122C2×C7⋊C8C2411yescharacteristic
123C8×D7—28no—
124C8×D7—412no—
125C2×2−1+4—71no—
126#520—74no—
127#520—76no—
128#519—71no—
129#519—74no—
130C4×#41—74no—
131C4×#41—76no—
132C2×C4×Q8—74no—
133C2×C4×Q8C2×D711yescharacteristic
134C8×Q8—74no—
135C8×Q8—74no—
136C8×Q8—76no—
137C8×Q8—76no—
138C8×Q8—712no—
139C8×Q8C2×D714yes—
140C8×Q8C2×D714yes—
141C8×D4—76no—
142C8×D4—76no—
143C8×D4—74no—
144C8×D4—712no—
145C8×D4—712no—
146#431—712no—
147#431—712no—
148#431C2×D716yes—
149C2×C4⋊C8—73no—
150C2×C4⋊C8—712no—
151C2×C4×C8—73no—
152C2×C4×C8—712no—
153C23×C7C23×C411yescharacteristic
154C22×D7C23×C414yes—
155C7×Q8C22×C8116yes—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—43no—
159C2×C4×C7C22×C8112yes—
160C2×C4×C7C23×C416yes—
161C2×C4×C7C23×C411yescharacteristic
162C2×C4×C7C23×C416yes—
163C2×C4×C7C2511yescharacteristic
164C22⋊D7C22×C8116yes—
165C2×Dic7C23×C414yes—
166C2×Dic7C23×C414yes—
167C2×Dic7C23×C414yes—
168D28C22×C8124yes—
169C4×D7—24no—
170C4×D7—24no—
171C4×D7—412no—
172C4×D7C22×C8148yes—
173Dic14C22×C8124yes—
174C7×C8—24no—
175C7×C8—24no—
176C7⋊C8—21no—
177C7⋊C8—26no—
178C7⋊C82−1+412yescomplemented
1792−1+4—716no—
180C2×#41—74no—
181C2×#41—76no—
182C22×Q8—74no—
183C22×Q8C4×D711yescharacteristic
184C22×C8—71no—
185C22×C8—73no—
186C22×C8—74no—
187C22×C8—712no—
188C4×Q8—74no—
189C4×Q8—74no—
190C4×Q8—712no—
191C4×Q8—76no—
192C4×Q8—76no—
193C4×Q8C4×D714yes—
194C4×Q8C22×D714yes—
195C4×D4—712no—
196C4×D4—74no—
197C4×D4—712no—
198C4×D4—76no—
199C4×D4—76no—
200#117—76no—
201#117—712no—
202#117—712no—
203C2×C4⋊C4—712no—
204C2×C4⋊C4C4×D713yes—
205C2×C42—712no—
206C2×C42C22×D713yes—
207C4⋊C8—712no—
208C4⋊C8—74no—
209C4⋊C8—712no—
210C4⋊C8C22×D716yes—
211C4⋊C8C22×D716yes—
212C22⋊C8—74no—
213C22⋊C8—712no—
214C22⋊C8C22×D714yes—
215C4×C8—74no—
216C4×C8—712no—
217C4×C8—76no—
218C4×C8—76no—
219C4×C8C22×D7112yes—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7C23×C812yes—
223C22×C7C24×C411yescharacteristic
224C2×D7—24no—
225C2×D7—24no—
226C2×D7C23×C818yes—
227C4×C7—21no—
228C4×C7—26no—
229C4×C7C23×C8112yes—
230C4×C7C2×2−1+411yescharacteristic
231C4×C7#58511yescharacteristic
232Dic7—24no—
233Dic7—24no—
234Dic7C23×C818yes—
235#41—732no—
236#41—748no—
237C2×Q8—732no—
238C2×Q8—76no—
239C2×Q8C8×D718yescomplemented
240C2×Q8C2×C4×D714yes—
241C2×D4—74no—
242C2×D4—76no—
243C22×C4—712no—
244C22×C4—74no—
245C22×C4—712no—
246C22×C4C2×C4×D713yes—
247C22×C4C2×C4×D713yes—
248C22×C4C23×D711yescharacteristic
249C2×C8—71no—
250C2×C8—71no—
251C2×C8—76no—
252C2×C8—148no—
253C2×C8—142no—
254C2×C8—2812no—
255C2×C8—283no—
256C2×C8C23×D714yes—
257C2×C8C23×D714yes—
258C4⋊C4—712no—
259C4⋊C4—74no—
260C4⋊C4—76no—
261C4⋊C4—76no—
262C4⋊C4C2×C4×D7112yes—
263C22⋊C4—74no—
264C22⋊C4—712no—
265C22⋊C4—74no—
266C42—712no—
267C42—712no—
268C42—74no—
269C42C2×C4×D716yes—
270C42C23×D716yes—
271C2×C7—22no—
272C2×C7C4×2−1+411yescharacteristic
273C2×C7C24×C811yescharacteristic
274C2×C7#357311yescharacteristic
275D7—28no—
276C23—74no—
277C23C22×C4×D711yescharacteristic
278Q8—724no—
279Q8C2×C8×D7116yes—
280D4—716no—
281D4—724no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—43no—
285C2×C4—74no—
286C2×C4—748no—
287C2×C4—74no—
288C2×C4—74no—
289C2×C4—144no—
290C2×C4—144no—
291C2×C4—2812no—
292C2×C4C2×C8×D7112yes—
293C2×C4C22×C4×D716yes—
294C2×C4C22×C4×D711yescharacteristic
295C2×C4C22×C4×D716yes—
296C2×C4C24×D711yescharacteristic
297C8—24no—
298C8—24no—
299C8—72no—
300C8—141no—
301C8—146no—
302C7C8×2−1+411yescharacteristic, complemented, Sylow
303C22—21no—
304C22—21no—
305C22—78no—
306C22—144no—
307C22—144no—
308C22C22×C8×D712yes—
309C22C23×C4×D711yescharacteristic
310C4—21no—
311C4—26no—
312C4—78no—
313C4—144no—
314C4—144no—
315C4C22×C8×D7112yes—
316C4D7×2−1+411yescharacteristic
317C4#9062011yescharacteristic
318C2—22no—
319C2—148no—
320C2#1174294111yescharacteristic
321C2C23×C8×D711yescharacteristic
322C2#1174423411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutatorC2×C7C24×C8
FittingC7×#2982C2
socleC22×C7C24×C4
prime radicalC2×#6551C4
cosocleC2×C4×C7C25
hypercenter#2982D7
hypocenterC7C8×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C7×#2982C2semidirect
3. C2×Q8×C7⋊C8C2semidirect
4. #11737365C2semidirect
5. C8×Q8×D7C2semidirect
6. #11737351C2semidirect
7. C8×#1161C2semidirect
8. #11736736C2semidirect
9. #11736721C2semidirect
40. #6551C8semidirect
84. #2982D7semidirect
89. C2×C7×Q8C2×C8semidirect
95. #1161C2×C8semidirect
96. Q8×D7C2×C8semidirect
97. #1157C2×C8semidirect
178. C7⋊C82−1+4semidirect
239. C2×Q8C8×D7semidirect
302. C7C8×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C4×#65512
C7×#29822
C2×Q8×C7⋊C82
#117373652
C8×Q8×D72
#117373512
C8×#11612
#117367362
#117367212
C8×2−1+47