Finite Group Databasebeta

← #13243105

Subgroups — #13243105

Order 1792.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#11734047C211yesmaximal, characteristic
2C7×#2095C211yesmaximal, characteristic
3#11727326C212yesmaximal
4#11725448C211yesmaximal, characteristic
5#11725283C212yesmaximal
6C2×C7×#107—21no—
7C2×C7×#103—21no—
8C7×#391C2211yescharacteristic
9C7×#384C2211yescharacteristic
10#90002—21no—
11#89758—21no—
12#89700C2212yes—
13#89433—22no—
14#89427—22no—
15C7×#336C2211yescharacteristic
16#89324—22no—
17#89324—22no—
18#89319C2212yes—
19#7831—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×C7×C8—21no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7—21no—
27C2×C4⋊Dic7—21no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×Dic7⋊C4—22no—
31C2×C4×Dic7—21no—
32C22×C7⋊C8—22no—
33C7×#107—42no—
34C7×#103—42no—
35C7×#95—22no—
36C7⋊#95—21no—
37C7⋊#95—22no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40#2809—71no—
41#2100—72no—
42#2095D711yescharacteristic
43#1272—71no—
44#1271—72no—
45C2×C7×Q8—21no—
46C2×C7×Q8—41no—
47C2×C7×Q8—41no—
48C2×C7×Q8Q1612yes—
49C22×C4×C7—21no—
50C22×C4×C7—21no—
51C22×C4×C7—41no—
52C22×C4×C7C2×D411yescharacteristic
53C22×C4×C7C2×D412yes—
54C22×Dic7—22no—
55C22×Dic7—22no—
56C2×C7×C8—22no—
57C2×C7×C8—41no—
58C2×C7×C8—41no—
59C7×C4⋊C4—41no—
60C7×C4⋊C4—41no—
61C7×C4⋊C4—41no—
62C7×C4⋊C4—44no—
63C42×C7—41no—
64C4⋊Dic7—22no—
65C4⋊Dic7—22no—
66C4⋊Dic7—42no—
67C4⋊Dic7—81no—
68C4⋊Dic7—81no—
69Dic7⋊C4—42no—
70Dic7⋊C4—42no—
71C4×Dic7—81no—
72C2×C7⋊C8—42no—
73C2×C7⋊C8—42no—
74C2×C7⋊C8—42no—
75C2×#107—21no—
76C2×#103—21no—
77#400—141no—
78#398—142no—
79#391—141no—
80#391C2×D711yescharacteristic
81#389—72no—
82#387—142no—
83#384C2×D711yescharacteristic
84#340—142no—
85#340—142no—
86#336—72no—
87#336C2×D711yescharacteristic
88C23×C7#12011yescharacteristic
89C7×Q8—42no—
90C7×Q8—81no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—41no—
97C2×C4×C7—41no—
98C2×C4×C7—41no—
99C2×C4×C7—42no—
100C2×C4×C7—81no—
101C2×C4×C7—82no—
102C2×C4×C7C2×Q1611yescharacteristic
103C2×C4×C7#13512yes—
104C2×C4×C7#13712yes—
105C2×C4×C7#13711yescharacteristic
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—42no—
112C7×C8—42no—
113C7⋊C8—82no—
114C22×Q8C22⋊D711yescharacteristic
115C22×C8—21no—
116C22×C8—142no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—142no—
122C2×C4⋊C4—141no—
123C2×C4⋊C4—141no—
124C2×C4⋊C4C22⋊D711yescharacteristic
125C2×C42—141no—
126C2×C42C22×D711yescharacteristic
127#107—42no—
128#103—42no—
129#95—22no—
130#95—141no—
131#95—142no—
132#95—142no—
133#95—142no—
134C22×C7#45111yescharacteristic
135C22×C7#45212yes—
136C22×C7#45311yescharacteristic
137C22×C7#45412yes—
138C22×C7#45811yescharacteristic
139C4×C7—21no—
140C4×C7—21no—
141C4×C7—22no—
142C4×C7—22no—
143C4×C7—42no—
144C4×C7—81no—
145C4×C7—161no—
146Dic7—82no—
147Dic7—82no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8#114412yes—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—142no—
156C22×C4—142no—
157C22×C4D4×D712yes—
158C22×C4C2×C22⋊D711yescharacteristic
159C2×C8—22no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—282no—
163C2×C8—282no—
164C2×C8—282no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—44no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—282no—
172C4⋊C4—282no—
173C4⋊C4—282no—
174C4⋊C4—561no—
175C4⋊C4—561no—
176C42—41no—
177C42—561no—
178C2×C7#161411yescharacteristic
179C2×C7#161512yes—
180C2×C7#163411yescharacteristic
181C2×C7#163612yes—
182C2×C7#165711yescharacteristic
183C23#650011yescharacteristic
184Q8—42no—
185Q8—81no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—282no—
198C2×C4—282no—
199C2×C4—282no—
200C2×C4—282no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4#648112yes—
204C2×C4#648212yes—
205C2×C4C2×#114411yescharacteristic
206C2×C4#651411yescharacteristic
207C8—42no—
208C8—562no—
209C7#783111yescharacteristic, complemented, Sylow
210C22#8982211yescharacteristic
211C22#8982511yescharacteristic
212C22#8984911yescharacteristic
213C22#8995912yes—
214C22#8997312yes—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—81no—
221C4—161no—
222C4—562no—
223C4—562no—
224C2#1172747011yescharacteristic
225C2#1172891212yes—
226C2#1172893212yes—
227C2#1172905811yescharacteristic
228C2#1172908211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2095C2
socle, prime radicalC23×C7#120
hypercenter#2095D7
hypocenterC7#7831
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
209. C7#7831semidirect

Maximal subgroups

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

subgroupindex
#117340472
C7×#20952
#117273262
#117254482
#117252832
#78317