Finite Group Databasebeta

← #21437027

Subgroups — #21437027

Order 7552.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic118C211yesmaximal, characteristic
2#16838127C211yesmaximal, characteristic
3#16837809C212yesmaximal
4#16837798C211yesmaximal, characteristic
5#16837711C212yesmaximal
6#16837698C211yesmaximal, characteristic
7#16837686C212yesmaximal
8#16837529C211yesmaximal, characteristic
9#16837506C212yesmaximal
10C4×#11745672C211yesmaximal, characteristic
11C59×#475C211yesmaximal, characteristic
12C2×#11745672C2211yescharacteristic
13#14273727—21no—
14#14273727C2211yescharacteristic
15#14273726—22no—
16#14273726C2212yes—
17#14273725—21no—
18#14273718C2212yes—
19#14273717—22no—
20Q8×Dic59—21no—
21Q8×Dic59C2212yes—
22C4×Dic118—21no—
23C4×Dic118C2211yescharacteristic
24C4×Dic118C2211yescharacteristic
25#14273692C2212yes—
26#14273661C2212yes—
27#14273660C2211yescharacteristic
28#14273658C2212yes—
29#14273658C2212yes—
30Dic59⋊C8C2212yes—
31#14273652C2212yes—
32#14273651C2211yescharacteristic
33#14273650C2211yescharacteristic
34#14273637C2212yes—
35C4×C59⋊C8C2211yescharacteristic
36C4⋊Q8×C59C2211yescharacteristic
37C4×Q8×C59C2211yescharacteristic
38#106×C59C2212yes—
39C4⋊C8×C59C2211yescharacteristic
40#103×C59C2212yes—
41C2×Q8×C59—21no—
42C2×Q8×C59C2311yescharacteristic
43C2×Dic118—21no—
44C2×Dic118—22no—
45C2×Dic118D411yescharacteristic
46C2×Dic118C2311yescharacteristic
47C2×C8×C59C2312yes—
48C4⋊C4×C59—21no—
49C4⋊C4×C59—21no—
50C4⋊C4×C59C2312yes—
51C4⋊C4×C59C2311yescharacteristic
52C42×C59—21no—
53C42×C59C2311yescharacteristic
54#11745672Q814yescomplemented
55C4⋊Dic59—21no—
56C4⋊Dic59—22no—
57C4⋊Dic59—22no—
58C4⋊Dic59D411yescharacteristic
59C4⋊Dic59C2311yescharacteristic
60C4⋊Dic59C2312yes—
61Dic59⋊C4—21no—
62Dic59⋊C4—22no—
63Dic59⋊C4—22no—
64Dic59⋊C4—22no—
65Dic59⋊C4D412yes—
66C4×Dic59—21no—
67C4×Dic59—22no—
68C4×Dic59—22no—
69C4×Dic59C2312yes—
70C2×C59⋊C8C2311yescharacteristic
71C2×C59⋊C8C2311yescharacteristic
72Q8×C59—21no—
73Q8×C59—42no—
74Q8×C59C2×Q812yes—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—21no—
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2411yescharacteristic
81C2×Dic59—21no—
82C2×Dic59—21no—
83C2×Dic59—22no—
84C2×Dic59C2×D412yes—
85C2×Dic59C2×D412yes—
86Dic118—22no—
87Dic118—21no—
88Dic118—82no—
89Dic118—81no—
90Dic118C2×Q812yes—
91C8×C59—22no—
92C59⋊C8—21no—
93C59⋊C8C2×Q812yes—
94C22×C59C22×D411yescharacteristic
95C4×C59—22no—
96C4×C59—21no—
97C4×C59—42no—
98C4×C59—41no—
99C4×C59#13712yes—
100C4×C59C22×Q811yescharacteristic
101C4×C592−1+411yescharacteristic
102Dic59—22no—
103Dic59—22no—
104Dic59—22no—
105Dic59—42no—
106Dic59—41no—
107#3387—591nomaximal, Sylow
108C2×C59D4×Q811yescharacteristic
109C2×C59C2×#13711yescharacteristic
110C2×C59#57711yescharacteristic
111Q82—591no—
112#557—591no—
113#501—592no—
114#498—591no—
115#479—592no—
116#477—591no—
117#475—592no—
118#475D5911yescharacteristic
119#446—591no—
120#441—592no—
121C4×Q16—591no—
122C59#338711yescharacteristic, complemented, Sylow
123C2×Q16—591no—
124C4⋊Q8—592no—
125C4⋊Q8—591no—
126C4⋊Q8—1181no—
127C4⋊Q8—1182no—
128C4⋊Q8—1181no—
129C4⋊Q8C2×D5911yescharacteristic
130#125—592no—
131#125—1182no—
132C4×Q8—591no—
133C4×Q8—592no—
134C4×Q8—592no—
135C4×Q8—591no—
136C4×Q8—1181no—
137C4×Q8—1181no—
138C4×Q8C2×D5911yescharacteristic
139#107—592no—
140#107—591no—
141#106—592no—
142#106—592no—
143#106C2×D5912yes—
144C4⋊C8—592no—
145C4⋊C8C2×D5911yescharacteristic
146#103—591no—
147#103—592no—
148#103—591no—
149#103C2×D5912yes—
150#97—592no—
151C4×C8—591no—
152C2×Q8—21no—
153C2×Q8—591no—
154C2×Q8—591no—
155C2×Q8—1182no—
156C2×Q8—1181no—
157C2×Q8C22×D5911yescharacteristic
158Q16—594no—
159C2×C8—591no—
160C2×C8—591no—
161C2×C8C22×D5912yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—591no—
165C4⋊C4—592no—
166C4⋊C4—592no—
167C4⋊C4—591no—
168C4⋊C4—1182no—
169C4⋊C4—1182no—
170C4⋊C4—1182no—
171C4⋊C4—1182no—
172C4⋊C4—1182no—
173C4⋊C4—1181no—
174C4⋊C4—1181no—
175C4⋊C4C22×D5911yescharacteristic
176C4⋊C4C22×D5912yes—
177C42—21no—
178C42—592no—
179C42—1182no—
180C42—1182no—
181C42—1181no—
182C42C22×D5911yescharacteristic
183Q8—21no—
184Q8—42no—
185Q8—592no—
186Q8—1182no—
187Q8—1181no—
188Q8—4721no—
189Q8—4722no—
190Q8Q8×D5912yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—592no—
195C2×C4—592no—
196C2×C4—1181no—
197C2×C4—1181no—
198C2×C4—1182no—
199C2×C4D4×D5911yescharacteristic
200C2×C4D4×D5911yescharacteristic
201C2×C4C23×D5911yescharacteristic
202C8—22no—
203C8—592no—
204C8—1181no—
205C22C2×D4×D5911yescharacteristic
206C4—21no—
207C4—22no—
208C4—42no—
209C4—41no—
210C4—1182no—
211C4—1182no—
212C4—1182no—
213C4—2362no—
214C4—2361no—
215C4#1427375612yes—
216C4C2×Q8×D5911yescharacteristic
217C4#1427377411yescharacteristic
218C2#1683806511yescharacteristic
219C2C2×#1427375611yescharacteristic
220C2#1683821111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#475C2
socleC22×C59C22×D4
hypercenter#475D59
hypocenterC59#3387
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
54. #11745672Q8semidirect
122. C59#3387semidirect