Finite Group Databasebeta

← #13254792

Subgroups — #13254792

Order 1792.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#11736755C212yesmaximal
2#11735924C211yesmaximal, characteristic
3#11728522C212yesmaximal
4#11727592C211yesmaximal, characteristic
5C7×#1275C211yesmaximal, characteristic
6#90161C2212yes—
7C2×C4×Dic14C2211yescharacteristic
8C7×#431C2212yes—
9C2×C7×#97C2211yescharacteristic
10#89878—22no—
11C2×C4⋊(C7⋊C8)C2211yescharacteristic
12#89622—22no—
13#89621—22no—
14#89511—22no—
15#89494—22no—
16#89474—22no—
17#89474—22no—
18C4×Dic28—24no—
19#89345—22no—
20#89251—24no—
21#10037—71nomaximal, Sylow
22C22×Dic14D411yescharacteristic
23C22×C7×C8D412yes—
24C2×C42×C7C2311yescharacteristic
25#6489—22no—
26C2×C4⋊Dic7D411yescharacteristic
27#6487—22no—
28#6487—22no—
29C2×Dic7⋊C4—21no—
30C2×C4×Dic7—21no—
31C22×C7⋊C8—21no—
32C2×Dic28—24no—
33#6433—22no—
34#6432D412yes—
35#6432D412yes—
36C4×Dic14—22no—
37C4×Dic14—22no—
38C4×Dic14—22no—
39C4×Dic14—22no—
40C7×C4⋊C8—22no—
41C7×C22⋊C8—22no—
42C7×#97—22no—
43C4×C7×C8D414yes—
44#6392—24no—
45#6392—22no—
46#6392—22no—
47#6391—24no—
48#6390—24no—
49#6390—24no—
50C4⋊(C7⋊C8)—22no—
51#3162—71no—
52#3104—72no—
53#1985—71no—
54#1481—72no—
55#1275D711yescharacteristic
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D411yescharacteristic
58C22×C4×C7#4111yescharacteristic
59C22×Dic7—21no—
60C22×Dic7—21no—
61C2×Dic14—22no—
62C2×Dic14—22no—
63C2×Dic14—22no—
64C2×Dic14—42no—
65C2×C7×C8—22no—
66C2×C7×C8—22no—
67C2×C7×C8—24no—
68C2×C7×C8Q1614yescomplemented
69C42×C7C2×D412yes—
70C42×C7C2×D412yes—
71C22⋊Dic7—42no—
72C4⋊Dic7—22no—
73C4⋊Dic7—22no—
74C4⋊Dic7—22no—
75C4⋊Dic7—22no—
76C4⋊Dic7—22no—
77Dic7⋊C4—42no—
78Dic7⋊C4—42no—
79Dic7⋊C4—42no—
80C4×Dic7—41no—
81C4×Dic7—41no—
82C4×Dic7—42no—
83C2×C7⋊C8—41no—
84C2×C7⋊C8—42no—
85Dic28—48no—
86#533—72no—
87C2×C4×Q8—71no—
88#501—142no—
89#501—142no—
90#484—142no—
91#477—142no—
92#462—142no—
93#451—142no—
94C4×Q16—144no—
95#431C2×D712yes—
96#427—142no—
97C2×C4⋊C8—71no—
98C2×#97C2×D711yescharacteristic
99#336—142no—
100Q8⋊C8—144no—
101C23×C7#12111yescharacteristic
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7#12012yes—
105C2×C4×C7#12011yescharacteristic
106C2×C4×C7#12112yes—
107C2×C4×C7#12711yescharacteristic
108C2×C4×C7C2×Q1612yes—
109C2×C4×C7#13712yes—
110C2×Dic7—22no—
111C2×Dic7—41no—
112C2×Dic7—41no—
113C2×Dic7—42no—
114C2×Dic7—41no—
115C2×Dic7—42no—
116C2×Dic7—42no—
117Dic14—24no—
118Dic14—42no—
119Dic14—44no—
120Dic14—44no—
121C7×C8—24no—
122C7×C8—44no—
123C7⋊C8—81no—
124C22×Q8—71no—
125C2×Q16—144no—
126C22×C8—141no—
127C22×C8D2812yes—
128C4⋊Q8—72no—
129C4⋊Q8—72no—
130#125—142no—
131C22⋊Q8—142no—
132C22⋊Q8—142no—
133C4×Q8—142no—
134C4×Q8—142no—
135C4×Q8—142no—
136C4×Q8—142no—
137#117—142no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—141no—
140C2×C42—141no—
141C2×C42C22×D711yescharacteristic
142#107—142no—
143#107—142no—
144#107—144no—
145#106—144no—
146C4⋊C8—22no—
147C4⋊C8—142no—
148#103—144no—
149#103—144no—
150C22⋊C8—22no—
151#97—22no—
152C4×C8D2814yes—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39211yescharacteristic
156C22×C7#46212yes—
157C4×C7—22no—
158C4×C7—22no—
159C4×C7#45112yes—
160C4×C7#49411yescharacteristic
161C4×C7#49711yescharacteristic
162Dic7—42no—
163Dic7—82no—
164Dic7—82no—
165Dic7—81no—
166C2×Q8—142no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—282no—
170C22×C4—141no—
171C22×C4—141no—
172C22×C4C2×D2811yescharacteristic
173C22×C4D4×D711yescharacteristic
174C22×C4#116111yescharacteristic
175Q16—288no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—24no—
179C2×C8—281no—
180C2×C8—282no—
181C2×C8Dic2814yescomplemented
182C4⋊C4—142no—
183C4⋊C4—142no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—142no—
187C4⋊C4—282no—
188C4⋊C4—282no—
189C4⋊C4—282no—
190C22⋊C4—282no—
191C42—281no—
192C42—282no—
193C42—281no—
194C42C2×D2812yes—
195C42D4×D712yes—
196C2×C7—22no—
197C2×C7#201011yescharacteristic
198C2×C7#201611yescharacteristic
199C2×C7#201711yescharacteristic
200C23#645811yescharacteristic
201Q8—144no—
202Q8—282no—
203Q8—284no—
204Q8—284no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—142no—
208C2×C4—281no—
209C2×C4—281no—
210C2×C4—282no—
211C2×C4—281no—
212C2×C4—282no—
213C2×C4—282no—
214C2×C4#643711yescharacteristic
215C2×C4#644512yes—
216C2×C4#644511yescharacteristic
217C2×C4#645812yes—
218C2×C4C2×Dic2812yes—
219C2×C4#647212yes—
220C8—24no—
221C8—44no—
222C8—561no—
223C7#1003711yescharacteristic, complemented, Sylow
224C22—21no—
225C22—21no—
226C22#8944311yescharacteristic
227C22#8962112yes—
228C4—22no—
229C4—22no—
230C4—282no—
231C4—562no—
232C4—561no—
233C4—562no—
234C4#8947111yescharacteristic
235C4#8948711yescharacteristic
236C4#8951112yes—
237C2—22no—
238C2#1172644211yescharacteristic
239C2#1172645711yescharacteristic
240C2#1172648111yescharacteristic
subgroupG/H
centerC2×C4#6437
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1275C2
socleC22×C7#392
prime radicalC23×C7#121
hypercenter#1275D7
hypocenterC7#10037
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
68. C2×C7×C8Q16semidirect
181. C2×C8Dic28semidirect
223. C7#10037semidirect

Maximal subgroups

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

subgroupindex
#117367552
#117359242
#117285222
#117275922
C7×#12752
#100377