Finite Group Databasebeta

← #13263231

Subgroups — #13263231

Order 1792.

284 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 69 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#11736484C211yesmaximal, characteristic
2#11726868C212yesmaximal, complemented
3C7×#1327C211yesmaximal, characteristic, complemented
4#11725523C211yesmaximal, characteristic, complemented
5#11725336C212yesmaximal
6C2×#6435C411yescharacteristic
7C7×#423C2211yescharacteristic
8#89898C411yescharacteristic
9#89898C2211yescharacteristic
10M16×Dic7C411yescharacteristic
11M16×Dic7C2211yescharacteristic
12#89696C411yescharacteristic
13#89612—22no—
14#89612—22no—
15C7×#349C2212yes—
16#89348C2212yes—
17#89307—22no—
18#89306—22no—
19#11522—71nomaximal, Sylow
20C22×C4×D7C2×C411yescharacteristic
21C2×C7×M16C2×C411yescharacteristic
22C2×C7×M16C2311yescharacteristic
23C2×C42×C7C2×C411yescharacteristic
24C2×(C2×D7)⋊C4—21no—
25C2×Dic7⋊C4—21no—
26C2×C4×Dic7C2×C411yescharacteristic
27C2×#1136C2×C411yescharacteristic
28C2×#1136C2×C411yescharacteristic
29#6435—22no—
30#6435—21no—
31#6435—21no—
32C7×M32—22no—
33C7×C4⋊C8—22no—
34(C2×D7)⋊C8—22no—
35(C2×D7)⋊C8—22no—
36#6389—21no—
37#6389D412yes—
38C8×Dic7—21no—
39C8×Dic7D412yes—
40#6386—22no—
41C4⋊(C7⋊C8)—22no—
42#2921—71no—
43#2165—72no—
44#1327—71no—
45#1327D711yescharacteristic, complemented
46#1320—72no—
47C23×D7M1611yescharacteristic
48C22×C4×C7—21no—
49C22×C4×C7C22×C411yescharacteristic
50C22×Dic7—21no—
51C22×Dic7M1611yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7—21no—
54C2×C4×D7—41no—
55C2×C4×D7—41no—
56C7×M16—41no—
57C7×M16—41no—
58C2×C7×C8—21no—
59C2×C7×C8C2×D412yes—
60C42×C7—21no—
61C42×C7M1612yes—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66C4×Dic7—21no—
67C4×Dic7C22⋊C411yescharacteristic
68C4×Dic7C22⋊C411yescharacteristic
69#1136—41no—
70#1136—41no—
71C2×C7⋊C8—21no—
72C2×C7⋊C8C22⋊C412yes—
73C7×C16—42no—
74C7⋊C16—42no—
75C2×#117—71no—
76#433—142no—
77#433—142no—
78#423—71no—
79#423C2×D711yescharacteristic
80#407—71no—
81#407—71no—
82C4×M16—71no—
83C4×M16—71no—
84#364—142no—
85#349—72no—
86#349C2×D712yes—
87#347—142no—
88C23×C7C2×M1611yescharacteristic
89C22×D7—21no—
90C22×D7—41no—
91C22×D7—81no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—81no—
96C2×C4×C7C2×C22⋊C411yescharacteristic
97C2×C4×C7C2×M1611yescharacteristic
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—41no—
102C2×Dic7—81no—
103C4×D7—42no—
104C4×D7—41no—
105C4×D7—41no—
106C7×C8—22no—
107C7×C8—41no—
108C7⋊C8—22no—
109C7⋊C8—41no—
110C23×C4—71no—
111C2×M16—71no—
112C2×M16—71no—
113C2×M16C4×D711yescharacteristic
114C2×M16C22×D711yescharacteristic
115#117—141no—
116#117—141no—
117#117—142no—
118C2×C4⋊C4—141no—
119C2×C22⋊C4—141no—
120C2×C42—71no—
121C2×C42C4×D711yescharacteristic
122M32—22no—
123M32—142no—
124C4⋊C8—22no—
125C4⋊C8—142no—
126C22⋊C8—142no—
127C22⋊C8—142no—
128#97—72no—
129#97—141no—
130C4×C8—72no—
131C4×C8—141no—
132C22×C7—21no—
133C22×C7—41no—
134C22×C7#40711yescharacteristic
135C2×D7—42no—
136C2×D7—41no—
137C2×D7—82no—
138C4×C7—21no—
139C4×C7—42no—
140C4×C7#41011yescharacteristic
141C4×C7#41311yescharacteristic
142Dic7—41no—
143Dic7—41no—
144Dic7—41no—
145C24—71no—
146C22×C4—21no—
147C22×C4—71no—
148C22×C4—141no—
149C22×C4—141no—
150C22×C4—141no—
151C22×C4—281no—
152C22×C4—281no—
153C22×C4C2×C4×D711yescharacteristic
154M16—41no—
155M16—41no—
156M16—281no—
157M16—281no—
158C2×C8—21no—
159C2×C8—72no—
160C2×C8—141no—
161C2×C8D4×D712yes—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C22⋊C4—142no—
165C22⋊C4—142no—
166C42—21no—
167C42—71no—
168C42—71no—
169C42—141no—
170C42#113112yes—
171C16—42no—
172C16—282no—
173C2×C7—21no—
174C2×C7—41no—
175C2×C7#146411yescharacteristic
176D7—81no—
177C23—141no—
178C23—281no—
179C23—561no—
180C23D7×M1611yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—81no—
185C2×C4—141no—
186C2×C4—141no—
187C2×C4—141no—
188C2×C4—282no—
189C2×C4—281no—
190C2×C4—281no—
191C2×C4—281no—
192C2×C4—561no—
193C2×C4D7×C22⋊C411yescharacteristic
194C2×C4C2×#113111yescharacteristic
195C8—22no—
196C8—41no—
197C8—142no—
198C8—281no—
199C7#1152211yescharacteristic, complemented, Sylow
200C22—21no—
201C22—41no—
202C22—282no—
203C22—281no—
204C22—562no—
205C22#8949811yescharacteristic
206C4—21no—
207C4—42no—
208C4—281no—
209C4—281no—
210C4—281no—
211C4#8951211yescharacteristic
212C4#8952311yescharacteristic
213C2—21no—
214C2—41no—
215C2—561no—
216C2#1172588011yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×M16C22×D7
FittingC7×#1327C2
socleC2×C7#1464
prime radicalC23×D7M16
cosocleC2×C7×M16C23
hypercenter#1327D7
hypocenterC7#11522
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #11726868C2semidirect
3. C7×#1327C2semidirect
4. #11725523C2semidirect
45. #1327D7semidirect
199. C7#11522semidirect

Maximal subgroups

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

subgroupindex
#117364842
#117268682
C7×#13272
#117255232
#117253362
#115227