Finite Group Databasebeta

← #13251210

Subgroups — #13251210

Order 1792.

570 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 109 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#11736733C211yesmaximal, characteristic
2#11731245C211yesmaximal, characteristic
3#11728534C211yesmaximal, characteristic, complemented
4#11728281C211yesmaximal, characteristic
5#11728262C211yesmaximal, characteristic, complemented
6C7×#1303C211yesmaximal, characteristic, complemented
7#11725426C211yesmaximal, characteristic, complemented
8C2×#6460—22no—
9C2×C4×D28C2211yescharacteristic
10C2×#6432C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13C4×C7×C4⋊C4C2211yescharacteristic
14C2×#6397—21no—
15#89711—21no—
16#89707—21no—
17#89697C2211yescharacteristic
18#89696C2211yescharacteristic
19#89610—22no—
20#89498—22no—
21#89460—22no—
22#89449C414yes—
23#89349—21no—
24#89281—22no—
25#89277—22no—
26#9403—71nomaximal, Sylow
27C22×D28D411yescharacteristic
28C22×C4×D7D412yes—
29C22×Dic14—21no—
30C2×C7×M16—21no—
31C2×C7×C4⋊C4—21no—
32C2×C42×C7—21no—
33C2×C42×C7C2311yescharacteristic
34C2×(C2×D7)⋊C4—24no—
35C2×(C2×D7)⋊C4C2×C412yes—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—24no—
38C2×C4⋊Dic7C2×C412yes—
39C2×C4⋊Dic7D411yescharacteristic
40C2×#1136—21no—
41C2×#1131—24no—
42#6460—48no—
43C4×D28—21no—
44C4×D28—21no—
45C4×D28—21no—
46C4×D28—21no—
47#6432—21no—
48#6432—21no—
49#6432—21no—
50#6432—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#2975—71no—
65#2542—71no—
66#1800—71no—
67#1549—71no—
68#1549—71no—
69#1303—71no—
70#1303D711yescharacteristic, complemented
71C23×D7C22⋊C412yes—
72C22×C4×C7—22no—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—22no—
77C22×Dic7C22⋊C412yes—
78C2×D28—21no—
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28—21no—
83C2×D28—21no—
84C2×C4×D7—22no—
85C2×C4×D7—22no—
86C2×C4×D7—44no—
87C2×Dic14—44no—
88C2×Dic14—41no—
89C2×Dic14—41no—
90C7×M16—41no—
91C7×M16—41no—
92C2×C7×C8—22no—
93C2×C7×C8—41no—
94C7×C4⋊C4—24no—
95C42×C7—41no—
96C42×C7—41no—
97C42×C7C2×D411yescharacteristic
98C42×C7C2×D411yescharacteristic
99C42×C7#4111yescharacteristic
100C42×C7#4111yescharacteristic
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—48no—
104C4⋊Dic7—21no—
105C4⋊Dic7—21no—
106C4⋊Dic7—22no—
107C4⋊Dic7—22no—
108C4⋊Dic7—48no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111#1136—41no—
112#1136—41no—
113C2×C7⋊C8—22no—
114C2×C7⋊C8—41no—
115#1131—84no—
116C2×C4⋊Q8—71no—
117C2×C22⋊Q8—142no—
118C2×C4×D4—71no—
119#436—142no—
120#436—142no—
121#431—71no—
122#431C2×D711yescharacteristic
123#423—71no—
124#423C2×D711yescharacteristic
125C2×#100—141no—
126#407—142no—
127#399—74no—
128#390—141no—
129#384—141no—
130C4×C4⋊C4C2×D711yescharacteristic
131#343—141no—
132#332—142no—
133#332—142no—
134C23×C7C2×C22⋊C411yescharacteristic
135C22×D7—22no—
136C22×D7—22no—
137C22×D7—22no—
138C22×D7—42no—
139C22×D7—42no—
140C2×C4×C7—22no—
141C2×C4×C7—24no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7C4×D411yescharacteristic
146C2×C4×C7C4×D411yescharacteristic
147C2×C4×C7#12011yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12111yescharacteristic
150C2×C4×C7#12311yescharacteristic
151C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×C4×C7#13711yescharacteristic
154C2×C4×C7#13711yescharacteristic
155C2×Dic7—22no—
156C2×Dic7—42no—
157C2×Dic7—44no—
158C2×Dic7—42no—
159D28—41no—
160D28—41no—
161D28—41no—
162D28—41no—
163C4×D7—44no—
164C4×D7—82no—
165Dic14—82no—
166Dic14—82no—
167C7×C8—42no—
168C7×C8—81no—
169C7⋊C8—42no—
170C7⋊C8—81no—
171C22×Q8—141no—
172C22×D4—71no—
173C23×C4—72no—
174C2×M16—21no—
175C2×M16—144no—
176C2×M16—141no—
177C4⋊Q8—141no—
178C4⋊Q8—141no—
179C4⋊Q8—141no—
180C4⋊Q8—141no—
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—72no—
189C2×C4⋊C4—144no—
190C2×C4⋊C4—141no—
191C2×C22⋊C4—72no—
192C2×C22⋊C4—144no—
193C2×C42—21no—
194C2×C42C22×D711yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—72no—
198C4⋊C8—141no—
199C4⋊C8—141no—
200C4⋊C8D2812yes—
201#100—282no—
202C22⋊C8—21no—
203C22⋊C8—144no—
204C22⋊C8—141no—
205C4×C8—21no—
206C4×C8—141no—
207#95C4×D712yes—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7#38611yescharacteristic
211C22×C7#41711yescharacteristic
212C22×C7#41911yescharacteristic
213C2×D7—44no—
214C2×D7—42no—
215C2×D7—42no—
216C2×D7—42no—
217C2×D7—82no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—44no—
222C4×C7#45312yes—
223C4×C7#46811yescharacteristic
224C4×C7#46811yescharacteristic
225C4×C7#47011yescharacteristic
226C4×C7#47011yescharacteristic
227Dic7—42no—
228Dic7—82no—
229C24—72no—
230C2×Q8—284no—
231C2×Q8—281no—
232C2×Q8—281no—
233C2×D4—141no—
234C2×D4—141no—
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—
252M16—564no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—142no—
256C2×C8—281no—
257C4⋊C4—24no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—288no—
263C4⋊C4—281no—
264C4⋊C4—281no—
265C22⋊C4—142no—
266C22⋊C4—142no—
267C22⋊C4—288no—
268C42—41no—
269C42—41no—
270C42C2×D2811yescharacteristic
271C42#115711yescharacteristic
272C42D4×D711yescharacteristic
273C42#116111yescharacteristic
274C2×C7—21no—
275C2×C7—21no—
276C2×C7#188711yescharacteristic
277C2×C7#189411yescharacteristic
278C2×C7#191511yescharacteristic
279D7—82no—
280C23—142no—
281C23—142no—
282C23—142no—
283C23—282no—
284C23—282no—
285C23D7×C22⋊C411yescharacteristic
286Q8—562no—
287Q8—562no—
288D4—281no—
289D4—281no—
290D4—281no—
291D4—281no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—142no—
298C2×C4—284no—
299C2×C4—282no—
300C2×C4—284no—
301C2×C4—282no—
302C2×C4—562no—
303C2×C4C4×D2811yescharacteristic
304C2×C4#644511yescharacteristic
305C2×C4#644711yescharacteristic
306C2×C4#645611yescharacteristic
307C2×C4#645711yescharacteristic
308C2×C4#645811yescharacteristic
309C2×C4#647411yescharacteristic
310C2×C4#647811yescharacteristic
311C2×C4#647811yescharacteristic
312C2×C4#648111yescharacteristic
313C8—42no—
314C8—81no—
315C8—282no—
316C8—561no—
317C7#940311yescharacteristic, complemented, Sylow
318C22—21no—
319C22—21no—
320C22—284no—
321C22—282no—
322C22—282no—
323C22—282no—
324C22—562no—
325C22#8943711yescharacteristic
326C22#8954311yescharacteristic
327C22#8957411yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—282no—
333C4—562no—
334C4#8955211yescharacteristic
335C4#8958811yescharacteristic
336C4#8959412yes—
337C4#8964011yescharacteristic
338C4#8965811yescharacteristic
339C2—21no—
340C2—21no—
341C2—562no—
342C2#1172651411yescharacteristic
343C2#1172663811yescharacteristic
344C2#1172679511yescharacteristic
subgroupG/H
centerC22#89437
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1303C2
socleC22×C7#386
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1303D7
hypocenterC7#9403
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #11728534C2semidirect
5. #11728262C2semidirect
6. C7×#1303C2semidirect
7. #11725426C2semidirect
70. #1303D7semidirect
317. C7#9403semidirect

Maximal subgroups

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

subgroupindex
#117367332
#117312452
#117285342
#117282812
#117282622
C7×#13032
#117254262
#94037