Finite Group Databasebeta

← #13251168

Subgroups — #13251168

Order 1792.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#11740704C211yesmaximal, characteristic
2#11736731C211yesmaximal, characteristic
3C7×#1549C211yesmaximal, characteristic, complemented
4#11729529C211yesmaximal, characteristic, complemented
5#11728281C211yesmaximal, characteristic
6#11725552C211yesmaximal, characteristic, complemented
7#11725521C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×#6509—22no—
10C2×C4×D28C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13C2×#6397—21no—
14#89764C414yes—
15#89760—21no—
16#89748—21no—
17C4×C4⋊Dic7C2211yescharacteristic
18#89697C2211yescharacteristic
19#89696C2211yescharacteristic
20#89606—22no—
21D7×C22⋊C8—22no—
22C8×D28—22no—
23C7×#332—22no—
24#89349—21no—
25#89342—22no—
26#9395—71nomaximal, Sylow
27C22×C7×Q8—21no—
28C22×D28D411yescharacteristic
29C22×C4×D7D412yes—
30C2×C7×M16—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C42×C7C2311yescharacteristic
38#6509—48no—
39C2×(C2×D7)⋊C4—24no—
40C2×(C2×D7)⋊C4C2×C412yes—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7D411yescharacteristic
43C2×Dic7⋊C4—24no—
44C2×C4×Dic7—21no—
45C2×#1136—21no—
46C2×C8×D7—24no—
47C4×D28—21no—
48C4×D28—21no—
49C4×D28—21no—
50C4×D28—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8D412yes—
54C7×C22⋊C8—21no—
55C4×C7×C8—21no—
56C7⋊#95C2×C412yes—
57C22⋊(C7⋊C8)—21no—
58#6397—42no—
59(C2×D7)⋊C8—24no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)D412yes—
63C4×C7⋊C8—21no—
64#2972—71no—
65#2542—71no—
66#1800—71no—
67#1549—71no—
68#1549D711yescharacteristic, complemented
69#1303—71no—
70#1303—71no—
71C23×D7C22⋊C412yes—
72C2×C7×Q8—44no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C22×C4×C7—22no—
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—22no—
80C22×Dic7C22⋊C412yes—
81C2×D28—21no—
82C2×D28—21no—
83C2×D28—21no—
84C2×D28—21no—
85C2×D28SD1612yes—
86C2×D28Q1612yes—
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—44no—
90C7×M16—41no—
91C7×M16—41no—
92C2×C7×C8—22no—
93C2×C7×C8—41no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C42×C7C2×D411yescharacteristic
99C42×C7C2×D411yescharacteristic
100C42×C7#4111yescharacteristic
101C42×C7#4111yescharacteristic
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—48no—
105C4⋊Dic7—24no—
106C4⋊Dic7—21no—
107C4⋊Dic7—21no—
108Dic7⋊C4—48no—
109C4×Dic7—41no—
110C4×Dic7—41no—
111#1136—41no—
112#1136—41no—
113C2×C7⋊C8—22no—
114C2×C7⋊C8—41no—
115C8×D7—84no—
116C2×C4⋊Q8C2×D711yescharacteristic
117C2×C22⋊Q8—142no—
118C2×C4×D4—71no—
119C8×D4—142no—
120C8×D4—142no—
121#431—71no—
122#431C2×D711yescharacteristic
123#423—71no—
124#423C2×D711yescharacteristic
125C2×#100—141no—
126C2×C22⋊C8—142no—
127#399—74no—
128#390—141no—
129#384—141no—
130C4×C4⋊C4—71no—
131#343—141no—
132#332—22no—
133#332—142no—
134C23×C7C2×C22⋊C411yescharacteristic
135C22×D7—22no—
136C22×D7—22no—
137C22×D7—42no—
138C22×D7—42no—
139C22×D7#10314yes—
140C7×Q8—82no—
141C7×Q8—82no—
142C2×C4×C7—22no—
143C2×C4×C7—41no—
144C2×C4×C7—44no—
145C2×C4×C7—42no—
146C2×C4×C7C4×D411yescharacteristic
147C2×C4×C7C4×D411yescharacteristic
148C2×C4×C7#12011yescharacteristic
149C2×C4×C7#12111yescharacteristic
150C2×C4×C7#12111yescharacteristic
151C2×C4×C7#12311yescharacteristic
152C2×C4×C7C2×SD1611yescharacteristic
153C2×C4×C7C2×Q1611yescharacteristic
154C2×C4×C7#13511yescharacteristic
155C2×C4×C7#13511yescharacteristic
156C2×Dic7—24no—
157C2×Dic7—22no—
158C2×Dic7—42no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161D28—22no—
162D28—22no—
163D28—22no—
164D28—22no—
165C4×D7—44no—
166C4×D7—82no—
167C7×C8—42no—
168C7×C8—81no—
169C7⋊C8—42no—
170C7⋊C8—81no—
171C22×Q8—21no—
172C22×D4—71no—
173C23×C4—72no—
174C2×M16—21no—
175C2×M16—141no—
176C22×C8—144no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C22⋊Q8—288no—
182C4×D4—141no—
183C4×D4—141no—
184C4×D4—141no—
185C4×D4—141no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—141no—
189C2×C4⋊C4—144no—
190C2×C4⋊C4C4×D712yes—
191C2×C22⋊C4—72no—
192C2×C22⋊C4—144no—
193C2×C42—141no—
194C2×C42C22×D711yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—72no—
198C4⋊C8—141no—
199C4⋊C8—141no—
200C4⋊C8C22⋊D712yes—
201#100—282no—
202C22⋊C8—21no—
203C22⋊C8—144no—
204C22⋊C8—141no—
205C4×C8—21no—
206C4×C8—141no—
207#95—72no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7#38611yescharacteristic
211C22×C7C2×#10311yescharacteristic
212C22×C7#41611yescharacteristic
213C2×D7—24no—
214C2×D7—42no—
215C2×D7—42no—
216C2×D7—44no—
217C2×D7—82no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—82no—
222C4×C7#45312yes—
223C4×C7#46511yescharacteristic
224C4×C7#46511yescharacteristic
225C4×C7#46711yescharacteristic
226C4×C7#46711yescharacteristic
227Dic7—44no—
228Dic7—42no—
229C24—72no—
230C2×Q8—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×D4—72no—
234C2×D4—72no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C22×C4—22no—
240C22×C4—72no—
241C22×C4—142no—
242C22×C4—142no—
243C22×C4—142no—
244C22×C4—284no—
245C22×C4C2×C4×D711yescharacteristic
246C22×C4D4×D711yescharacteristic
247C22×C4D4×D711yescharacteristic
248M16—41no—
249M16—41no—
250M16—281no—
251M16—281no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—142no—
255C2×C8—281no—
256C2×C8—564no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—144no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—288no—
265C22⋊C4—142no—
266C22⋊C4—142no—
267C22⋊C4—288no—
268C42—281no—
269C42—281no—
270C42#115711yescharacteristic
271C42D4×D711yescharacteristic
272C42#115911yescharacteristic
273C42C2×C22⋊D711yescharacteristic
274C2×C7—21no—
275C2×C7—21no—
276C2×C7#188711yescharacteristic
277C2×C7#189211yescharacteristic
278C2×C7#191511yescharacteristic
279D7—44no—
280C23—74no—
281C23—142no—
282C23—142no—
283C23—282no—
284C23—282no—
285C23D7×C22⋊C411yescharacteristic
286Q8—82no—
287Q8—82no—
288D4—142no—
289D4—142no—
290D4—142no—
291D4—142no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—144no—
297C2×C4—142no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—284no—
301C2×C4—282no—
302C2×C4—562no—
303C2×C4(C22⋊D7)⋊C411yescharacteristic
304C2×C4#644611yescharacteristic
305C2×C4#644711yescharacteristic
306C2×C4D7×SD1611yescharacteristic
307C2×C4#647911yescharacteristic
308C2×C4D7×Q1611yescharacteristic
309C2×C4#648211yescharacteristic
310C2×C4C4×C22⋊D711yescharacteristic
311C2×C4#650011yescharacteristic
312C2×C4#650111yescharacteristic
313C8—42no—
314C8—81no—
315C8—282no—
316C8—561no—
317C7#939511yescharacteristic, complemented, Sylow
318C22—21no—
319C22—21no—
320C22—144no—
321C22—284no—
322C22—282no—
323C22—282no—
324C22—562no—
325C22D7×#10311yescharacteristic
326C22#8957611yescharacteristic
327C22#8972811yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—284no—
333C4—282no—
334C4#8957911yescharacteristic
335C4#8958511yescharacteristic
336C4#8994311yescharacteristic
337C4#8996011yescharacteristic
338C4#8997012yes—
339C2—21no—
340C2—21no—
341C2—284no—
342C2#1172787811yescharacteristic
343C2#1172802411yescharacteristic
344C2#1172807711yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1549C2
socleC22×C7#386
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1549D7
hypocenterC7#9395
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C7×#1549C2semidirect
4. #11729529C2semidirect
6. #11725552C2semidirect
7. #11725521C2semidirect
68. #1549D7semidirect
317. C7#9395semidirect

Maximal subgroups

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

subgroupindex
#117407042
#117367312
C7×#15492
#117295292
#117282812
#117255522
#117255212
#93957