Finite Group Databasebeta

← #13273463

Subgroups — #13273463

Order 1792.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C7×#2683C211yesmaximal, characteristic
2#11727396C211yesmaximal, characteristic
3#11727351C212yesmaximal
4#11725447C211yesmaximal, characteristic
5#11725443C212yesmaximal
6C2×C7×C22⋊Q8—22no—
7#90489C2211yescharacteristic
8C7×#396—22no—
9C7×#395—21no—
10C7×#390C2211yescharacteristic
11C7×#389—21no—
12C7×#386C2212yes—
13C2×#6408C2212yes—
14C2×#6407C2211yescharacteristic
15#89982—22no—
16#89736—21no—
17#89322—22no—
18#89318—21no—
19#13461—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4C2311yescharacteristic
22C23×C4×C7D412yes—
23C7×C22⋊Q8—48no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4D412yes—
32C2×C42×C7D411yescharacteristic
33C2×C22⋊Dic7—22no—
34C2×C22⋊Dic7—22no—
35D4×Dic7—41no—
36D4×Dic7—41no—
37#6489—21no—
38C2×Dic7⋊C4—42no—
39C2×#1136—21no—
40C7×#95—21no—
41C7×#95—22no—
42#6408—22no—
43#6408Q814yescomplemented
44#6407—21no—
45#6407Q812yescomplemented
46C7⋊#95—42no—
47#6382—41no—
48#6381—41no—
49#2683D711yescharacteristic
50#1867—72no—
51#1856—71no—
52#1393—71no—
53#1390—72no—
54C24×C7C2×D412yes—
55C2×C7×Q8—44no—
56C2×C7×Q8—81no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4C2×Q811yescharacteristic
60C2×C7×D4C2×Q812yes—
61C2×C7×D4#4111yescharacteristic
62C22×C4×C7—22no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—44no—
67C22×C4×C7—42no—
68C22×C4×C7C2×D411yescharacteristic
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—42no—
72C7×C4⋊C4—24no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—44no—
75C7×C4⋊C4—44no—
76C7×C4⋊C4—81no—
77C7×C22⋊C4—44no—
78C7×C22⋊C4—44no—
79C7×C22⋊C4—42no—
80C42×C7—41no—
81C22⋊Dic7—24no—
82C22⋊Dic7—42no—
83C22⋊Dic7—42no—
84C22⋊Dic7—41no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87Dic7⋊C4—84no—
88C4×Dic7—41no—
89#1136—22no—
90#1136—41no—
91C2×C7⋊C8—41no—
92C2×C22⋊Q8—22no—
93#518—71no—
94#428—141no—
95C2×#100—71no—
96C2×#99—72no—
97#396—22no—
98#395—21no—
99#390C2×D711yescharacteristic
100#389—21no—
101#386C2×D712yes—
102#385—142no—
103#342—142no—
104#339—141no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—42no—
108C23×C7—42no—
109C23×C7#12011yescharacteristic
110C23×C7C22⋊Q812yes—
111C23×C7C4⋊Q812yes—
112C7×Q8—84no—
113C7×D4—41no—
114C7×D4—42no—
115C7×D4—41no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—24no—
120C2×C4×C7—42no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—41no—
124C2×C4×C7—44no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—82no—
131C2×C4×C7C22⋊Q811yescharacteristic
132C2×C4×C7C4⋊Q811yescharacteristic
133C2×Dic7—44no—
134C2×Dic7—41no—
135C2×Dic7—41no—
136C2×Dic7—82no—
137C2×Dic7—82no—
138C2×Dic7—82no—
139C2×Dic7—82no—
140C7⋊C8—42no—
141C22×Q8—21no—
142C22×D4C22×D711yescharacteristic
143C23×C4C22⋊D712yes—
144C2×M16—141no—
145C22⋊Q8—48no—
146C4×D4—281no—
147C4×D4—281no—
148#117—141no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—282no—
154C2×C4⋊C4C22⋊D711yescharacteristic
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—142no—
158C2×C22⋊C4—142no—
159C2×C22⋊C4C22⋊D712yes—
160C2×C42C22⋊D711yescharacteristic
161#107—281no—
162#106—281no—
163#100—72no—
164#100—141no—
165#99—74no—
166#99—142no—
167#95—21no—
168#95—22no—
169#95—282no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—42no—
175C22×C7—42no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—82no—
179C22×C7#39511yescharacteristic
180C22×C7B(4,𝔽2)11yescharacteristic
181C22×C7#45811yescharacteristic
182C4×C7—21no—
183C4×C7—21no—
184C4×C7—42no—
185C4×C7—44no—
186C4×C7—82no—
187C4×C7—81no—
188Dic7—84no—
189Dic7—81no—
190Dic7—81no—
191C24C2×C22⋊D712yes—
192C2×Q8—44no—
193C2×Q8—81no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4Q8×D712yes—
197C2×D4Q8×D711yescharacteristic
198C2×D4#116111yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—142no—
206C22×C4—142no—
207C22×C4—282no—
208C22×C4C2×C22⋊D711yescharacteristic
209M16—142no—
210M16—281no—
211C2×C8—281no—
212C4⋊C4—24no—
213C4⋊C4—41no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—81no—
217C4⋊C4—141no—
218C4⋊C4—141no—
219C4⋊C4—564no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—144no—
224C22⋊C4—282no—
225C22⋊C4—282no—
226C22⋊C4—281no—
227C42—41no—
228C42—281no—
229C2×C7—21no—
230C2×C7—21no—
231C2×C7—42no—
232C2×C7—42no—
233C2×C7#206211yescharacteristic
234C2×C7#219611yescharacteristic
235C2×C7#220111yescharacteristic
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23#650712yes—
241C23#650912yes—
242C23#651611yescharacteristic
243Q8—84no—
244D4—41no—
245D4—42no—
246D4—41no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—284no—
263C2×C4—281no—
264C2×C4—281no—
265C2×C4—562no—
266C2×C4—562no—
267C2×C4—562no—
268C2×C4—562no—
269C2×C4#650711yescharacteristic
270C2×C4#650911yescharacteristic
271C8—282no—
272C7#1346111yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#9000111yescharacteristic
283C22#9001511yescharacteristic
284C22#9001611yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—44no—
289C4—82no—
290C4—81no—
291C4—564no—
292C4—561no—
293C4—561no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#1172964911yescharacteristic
299C2#1172966311yescharacteristic
300C2#1172971211yescharacteristic
subgroupG/H
centerC22#90016
commutator, prime radical, cosocleC22×C7×D4C23
FrattiniC22×D4C22×D7
FittingC7×#2683C2
socleC22×C7B(4,𝔽2)
hypercenter#2683D7
hypocenterC7#13461
LCSG ▷ C22×C7×D4 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
43. #6408Q8semidirect
45. #6407Q8semidirect
272. C7#13461semidirect

Maximal subgroups

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

subgroupindex
C7×#26832
#117273962
#117273512
#117254472
#117254432
#134617