Finite Group Databasebeta

← #13274157

Subgroups — #13274157

Order 1792.

312 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 59 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#11739184C211yesmaximal, characteristic
2D7×#356C212yesmaximal
3C7×#1413C211yesmaximal, characteristic, complemented
4#11725592C211yesmaximal, characteristic, complemented
5#11725357C212yesmaximal, complemented
6D7×C4⋊Q8C2211yescharacteristic
7#90379C411yescharacteristic
8C7×#491C2211yescharacteristic
9#89854C2211yescharacteristic
10#89841C411yescharacteristic
11#89574—22no—
12#89564—22no—
13D7×#101—22no—
14#89479C411yescharacteristic
15#89475C411yescharacteristic
16C7×#356C2212yes—
17#89343C2212yes—
18#89273—22no—
19#13599—71nomaximal, Sylow
20C2×Q8×D7D412yes—
21C7×C4⋊Q8C2311yescharacteristic
22C7×#125C2×C411yescharacteristic
23#6507D412yes—
24D7×M16—42no—
25C2×#1131—21no—
26#6459—41no—
27#6455—21no—
28D7×C4⋊C4—41no—
29#6451—21no—
30#6451—41no—
31#6450—41no—
32C42×D7C2×C411yescharacteristic
33#6433C2×C411yescharacteristic
34#6432C2×C411yescharacteristic
35C7×#103—22no—
36C7×#101—22no—
37C7×#97C2×C411yescharacteristic
38#6410—22no—
39#6409—22no—
40#6398—42no—
41#6390—22no—
42C8×Dic7—21no—
43#6384—22no—
44#6377C2×C411yescharacteristic
45#3139—71no—
46C2×#356—72no—
47#1413—71no—
48#1413—72no—
49#1413D711yescharacteristic, complemented
50C2×C7×Q8C2×D412yes—
51Q8×D7—42no—
52Q8×D7—42no—
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7C22⋊C411yescharacteristic
56C2×Dic14—22no—
57C2×Dic14—41no—
58C2×Dic14C22⋊C412yes—
59C7×M16—42no—
60C2×C7×C8—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—41no—
63C7×C4⋊C4—41no—
64C42×C7C22×C411yescharacteristic
65(C2×D7)⋊C4—41no—
66C4⋊Dic7—21no—
67C4⋊Dic7—41no—
68Dic7⋊C4—42no—
69Dic7⋊C4—41no—
70Dic7⋊C4—41no—
71C4×Dic7—21no—
72C4×Dic7—41no—
73C4×Dic7C22⋊C411yescharacteristic
74#1136—42no—
75C2×C7⋊C8—21no—
76#1131—42no—
77#1131—82no—
78C8×D7—82no—
79#533—71no—
80C2×C4⋊Q8—71no—
81#491—71no—
82#491—71no—
83#491—71no—
84#491C2×D711yescharacteristic
85#487—142no—
86#419—142no—
87C2×#101—142no—
88C4×M16—71no—
89#356—72no—
90#356—142no—
91#356C2×D712yes—
92C22×D7#9911yescharacteristic
93C7×Q8—42no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—41no—
97C2×C4×C7C2×C22⋊C411yescharacteristic
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—41no—
101C2×Dic7#9911yescharacteristic
102C4×D7—22no—
103C4×D7—42no—
104C4×D7—42no—
105C4×D7—42no—
106Dic14—42no—
107Dic14—82no—
108Dic14—82no—
109C7×C8—41no—
110C7×C8—82no—
111C7⋊C8—41no—
112C7⋊C8—82no—
113C22×Q8—72no—
114C2×M16—141no—
115C2×M16—282no—
116C4⋊Q8—72no—
117C4⋊Q8—71no—
118C4⋊Q8—141no—
119C4⋊Q8—281no—
120C4⋊Q8C22×D711yescharacteristic
121#125—71no—
122#125C4×D711yescharacteristic
123C22⋊Q8—281no—
124C4×Q8—281no—
125#117—141no—
126C2×C4⋊C4—281no—
127C2×C42—71no—
128#103—22no—
129#103—142no—
130#103—142no—
131#103—142no—
132#101—22no—
133#101—142no—
134#101—282no—
135#97—71no—
136#97C4×D711yescharacteristic
137C4×C8—141no—
138C22×C7C2×#9911yescharacteristic
139C2×D7—41no—
140C2×D7#35412yes—
141C4×C7—21no—
142C4×C7—41no—
143C4×C7—42no—
144C4×C7—81no—
145Dic7—22no—
146Dic7—21no—
147Dic7—42no—
148Dic7—81no—
149C2×Q8—72no—
150C2×Q8—142no—
151C2×Q8—281no—
152C2×Q8—282no—
153C2×Q8—282no—
154C2×Q8D4×D712yes—
155C22×C4—71no—
156C22×C4—141no—
157C22×C4—142no—
158M16—42no—
159M16—282no—
160M16—282no—
161M16—562no—
162C2×C8—21no—
163C2×C8—141no—
164C2×C8—562no—
165C4⋊C4—21no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—141no—
169C4⋊C4—282no—
170C4⋊C4—281no—
171C4⋊C4—281no—
172C4⋊C4—281no—
173C22⋊C4—281no—
174C42—71no—
175C42—141no—
176C42—281no—
177C42C2×C4×D711yescharacteristic
178C2×C7—21no—
179C2×C7C2×#35411yescharacteristic
180D7—22no—
181C23—71no—
182Q8—42no—
183Q8—282no—
184Q8—562no—
185Q8—562no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—71no—
190C2×C4—142no—
191C2×C4—141no—
192C2×C4—142no—
193C2×C4—282no—
194C2×C4—282no—
195C2×C4—282no—
196C2×C4—281no—
197C2×C4D7×C22⋊C411yescharacteristic
198C8—41no—
199C8—82no—
200C8—281no—
201C8—562no—
202C7#1359911yescharacteristic, complemented, Sylow
203C22—72no—
204C22—281no—
205C22D7×#9911yescharacteristic
206C4—21no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—142no—
211C4—141no—
212C4—282no—
213C4—561no—
214C2—21no—
215C2—142no—
216C2D7×#35411yescharacteristic
subgroupG/H
centerC2D7×#354
commutatorC42×C7C22×C4
FrattiniC4⋊Q8C22×D7
FittingC7×#1413C2
socleC2×C7C2×#354
prime radicalC22×D7#99
cosocleC7×C4⋊Q8C23
hypercenter#1413D7
hypocenterC7#13599
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. C7×#1413C2semidirect
4. #11725592C2semidirect
5. #11725357C2semidirect
49. #1413D7semidirect
202. C7#13599semidirect

Maximal subgroups

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

subgroupindex
#117391842
D7×#3562
C7×#14132
#117255922
#117253572
#135997