Finite Group Databasebeta

← #13267672

Subgroups — #13267672

Order 1792.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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#11737359C211yesmaximal, characteristic
2C7×#1674C211yesmaximal, characteristic
3#11729143C211yesmaximal, characteristic
4#11729136C211yesmaximal, characteristic
5Q8⋊(C7⋊C16)C211yesmaximal, characteristic
6#11725280C211yesmaximal, characteristic
7#11725278C211yesmaximal, characteristic
8#90261—21no—
9C7×#500C2211yescharacteristic
10C7×#462—21no—
11C7×C8×Q8C2211yescharacteristic
12#89621—22no—
13#89621—21no—
14#89471—21no—
15#89471C2211yescharacteristic
16C4×Dic28—21no—
17C4×Dic28C2211yescharacteristic
18C7×#335C2211yescharacteristic
19C7×#332—21no—
20#89287—22no—
21#89287—21no—
22#89285—21no—
23C4⋊(C7⋊C16)C2211yescharacteristic
24C4×C7⋊C16C2211yescharacteristic
25#12382—71nomaximal, Sylow
26C2×C7×Q16—41no—
27C7×C4⋊Q8—21no—
28C4×C7×Q8D411yescharacteristic
29Q8×Dic7—41no—
30C2×Dic28—21no—
31C2×Dic28—22no—
32C2×Dic28D411yescharacteristic
33C2×Dic28D412yes—
34#6453—42no—
35#6453—41no—
36#6432—22no—
37#6432—21no—
38C4×Dic14—22no—
39C4×Dic14—21no—
40C7×#107—21no—
41C7×#107—21no—
42C7×C4⋊C8—21no—
43C7×C4⋊C8—21no—
44C7×C4⋊C8D411yescharacteristic
45C7×#103—41no—
46C4×C7×C8—21no—
47C4×C7×C8C2311yescharacteristic
48#6392D411yescharacteristic
49#6390—22no—
50#6390—21no—
51C2×C7⋊C16—21no—
52C2×C7⋊C16—21no—
53#3395—71no—
54#2241—71no—
55#2225—71no—
56#1674D711yescharacteristic
57#1362—71no—
58Q8⋊C16—71no—
59#1332—71no—
60C2×C7×Q8—41no—
61C2×C7×Q8D811yescharacteristic
62C2×Dic14—21no—
63C2×Dic14—22no—
64C7×Q16—82no—
65C2×C7×C8—21no—
66C2×C7×C8—41no—
67C2×C7×C8C2×D411yescharacteristic
68C2×C7×C8C2×D411yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4D811yescharacteristic
73C42×C7—21no—
74C42×C7C2×D411yescharacteristic
75C4⋊Dic7—21no—
76C4⋊Dic7—42no—
77C4⋊Dic7—42no—
78C4⋊Dic7—41no—
79C4⋊Dic7—41no—
80Dic7⋊C4—42no—
81Dic7⋊C4—41no—
82C4×Dic7—42no—
83C4×Dic7—41no—
84Dic28—21no—
85Dic28—21no—
86Dic28—42no—
87Dic28—41no—
88Dic28—41no—
89Dic28—82no—
90C7⋊C16—41no—
91C7⋊C16—41no—
92#557—141no—
93#500C2×D711yescharacteristic
94#494—71no—
95#494—141no—
96#462—21no—
97#462—142no—
98#462—141no—
99C8×Q8C2×D711yescharacteristic
100C4×Q16—71no—
101C4×Q16—141no—
102#367—141no—
103C4⋊C16—71no—
104#358—141no—
105#358—142no—
106C4×C16—71no—
107#335C2×D711yescharacteristic
108#332—21no—
109C7×Q8—21no—
110C7×Q8—81no—
111C7×Q8—81no—
112C7×Q8SD3212yes—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—81no—
116C2×C4×C7#12011yescharacteristic
117C2×C4×C7C2×D811yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×Dic7—41no—
120C2×Dic7—42no—
121C2×Dic7—41no—
122Dic14—41no—
123Dic14—42no—
124Dic14—42no—
125Dic14—41no—
126C7×C8—21no—
127C7×C8—41no—
128C7×C8—81no—
129C7×C8#13511yescharacteristic
130C7×C8#13711yescharacteristic
131C2×Q16—41no—
132C2×Q16—71no—
133C2×Q16—72no—
134C2×Q16—142no—
135C2×Q16—141no—
136C4⋊Q8—21no—
137C4⋊Q8—142no—
138C4⋊Q8—141no—
139#125—282no—
140#125—281no—
141C4×Q8—142no—
142C4×Q8—141no—
143C4×Q8—281no—
144C4×Q8C22⋊D711yescharacteristic
145C2×C16—141no—
146C2×C16—141no—
147#107—21no—
148#107—21no—
149#107—71no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8C22⋊D711yescharacteristic
153#103—41no—
154#103—142no—
155#103—141no—
156C4×C8—21no—
157C4×C8C22×D711yescharacteristic
158C22×C7#44711yescharacteristic
159C4×C7—22no—
160C4×C7—21no—
161C4×C7—41no—
162C4×C7—161no—
163C4×C7#45211yescharacteristic
164C4×C7#45511yescharacteristic
165C4×C7C2×SD3211yescharacteristic
166C4×C7#51011yescharacteristic
167Dic7—81no—
168Dic7—82no—
169Dic7—81no—
170C2×Q8—41no—
171C2×Q8—141no—
172C2×Q8—142no—
173C2×Q8#114111yescharacteristic
174Q16—82no—
175Q16—141no—
176Q16—141no—
177Q16—282no—
178Q16—281no—
179Q16—281no—
180Q16—562no—
181C2×C8—21no—
182C2×C8—41no—
183C2×C8D4×D711yescharacteristic
184C2×C8D4×D711yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—141no—
189C4⋊C4—282no—
190C4⋊C4—282no—
191C4⋊C4—282no—
192C4⋊C4—281no—
193C4⋊C4—281no—
194C4⋊C4—281no—
195C4⋊C4#114111yescharacteristic
196C42—21no—
197C42—282no—
198C42—281no—
199C42C2×C22⋊D711yescharacteristic
200C16—281no—
201C16—281no—
202C2×C7#164111yescharacteristic
203C2×C7#218511yescharacteristic
204C2×C7#219511yescharacteristic
205Q8—21no—
206Q8—81no—
207Q8—81no—
208Q8—281no—
209Q8—282no—
210Q8—282no—
211Q8—281no—
212Q8#640112yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—81no—
216C2×C4—281no—
217C2×C4—282no—
218C2×C4—281no—
219C2×C4C2×#114111yescharacteristic
220C2×C4#650011yescharacteristic
221C2×C4#651211yescharacteristic
222C8—21no—
223C8—41no—
224C8—81no—
225C8#648111yescharacteristic
226C8#648211yescharacteristic
227C7#1238211yescharacteristic, complemented, Sylow
228C22#8980811yescharacteristic
229C4—22no—
230C4—21no—
231C4—41no—
232C4—161no—
233C4—561no—
234C4—562no—
235C4—561no—
236C4C2×#640111yescharacteristic
237C4#8995911yescharacteristic
238C4#8996711yescharacteristic
239C4#8997411yescharacteristic
240C2#1172881511yescharacteristic
241C2#1172883111yescharacteristic
242C2#1172885311yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1674C2
socle, prime radicalC22×C7#447
hypercenter#1674D7
hypocenterC7#12382
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1674

Complements

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

NH ≅ G/Ntype
227. C7#12382semidirect

Maximal subgroups

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

subgroupindex
#117373592
C7×#16742
#117291432
#117291362
Q8⋊(C7⋊C16)2
#117252802
#117252782
#123827