Finite Group Databasebeta

← #34057562

Subgroups — #34057562

Order 19072.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#23750335C211yesmaximal, characteristic
2#23750024C212yesmaximal, complemented
3#23750021C211yesmaximal, characteristic, complemented
4#23749932C212yesmaximal, complemented
5#23749922C212yesmaximal
6#23749920C211yesmaximal, characteristic, complemented
7#23749902C211yesmaximal, characteristic
8Q8×#11924746C211yesmaximal, characteristic, complemented
9C4×#14558723C211yesmaximal, characteristic
10C4×#14558723C212yesmaximal
11#557×C149C211yesmaximal, characteristic, complemented
12C2×#14558723C2211yescharacteristic
13#18011911C2211yescharacteristic
14#18011903C2212yes—
15#18011889—22no—
16#18011887—21no—
17C4×D596C2211yescharacteristic
18C4×D596C2212yes—
19C4⋊C4×D149—21no—
20#18011844C2212yes—
21#18011844C2212yes—
22#18011842C2212yes—
23#18011842C2212yes—
24#18011842C2211yescharacteristic
25C4⋊#11924746C2211yescharacteristic
26C4⋊#11924746C2212yes—
27#18011835C2212yes—
28#18011835C2211yescharacteristic
29#18011831C2212yes—
30#18011831C2211yescharacteristic
31C4×#11924746C2211yescharacteristic
32C4×#11924746C2212yes—
33C4⋊Q8×C149—21no—
34C4⋊Q8×C149C2211yescharacteristic
35#125×C149—22no—
36#125×C149C2212yes—
37C4×Q8×C149—21no—
38C4×Q8×C149C2211yescharacteristic
39C4×Q8×C149C2212yes—
40C4×Q8×C149C2211yescharacteristic
41C2×Dic298—21no—
42C2×D596C2311yescharacteristic
43C2×C4×D149—21no—
44C2×C4×D149—22no—
45#14558723Q814yescomplemented
46C2×#11924746C2311yescharacteristic
47C2×#11924746C2312yes—
48C2×#11924746C2311yescharacteristic
49C4⋊Dic149—21no—
50C4⋊Dic149C2312yes—
51C4⋊Dic149C2311yescharacteristic
52#14558715—22no—
53#14558715—22no—
54#14558713—22no—
55#14558713—21no—
56C2×Q8×C149—21no—
57C2×Q8×C149D411yescharacteristic
58C2×Q8×C149C2311yescharacteristic
59C4⋊C4×C149—22no—
60C4⋊C4×C149—21no—
61C4⋊C4×C149—22no—
62C4⋊C4×C149—21no—
63C4⋊C4×C149—22no—
64C4⋊C4×C149D411yescharacteristic
65C4⋊C4×C149D412yes—
66C4⋊C4×C149C2311yescharacteristic
67C4⋊C4×C149C2312yes—
68C42×C149—22no—
69C42×C149—21no—
70C42×C149C2311yescharacteristic
71C42×C149C2312yes—
72C22×D149—21no—
73Dic298—81no—
74D596—21no—
75D596C2×Q812yes—
76C2×Dic149—22no—
77C2×Dic149—21no—
78C4×D149—42no—
79C4×D149—82no—
80#11924746—21no—
81#11924746—22no—
82#11924746C2×Q812yes—
83Q8×C149—22no—
84Q8×C149—21no—
85Q8×C149—42no—
86Q8×C149C2×Q812yes—
87C2×C4×C149—21no—
88C2×C4×C149—21no—
89C2×C4×C149—22no—
90C2×C4×C149C2×D412yes—
91C2×C4×C149C2×D411yescharacteristic
92C2×C4×C149C2×D412yes—
93C2×C4×C149C2×D411yescharacteristic
94C2×C4×C149C2411yescharacteristic
95C22×C149C22×D411yescharacteristic
96C2×D149—22no—
97C2×D149—41no—
98C2×D149—41no—
99C4×C149—22no—
100C4×C149—22no—
101C4×C149—22no—
102C4×C149—21no—
103C4×C149—42no—
104C4×C149—41no—
105C4×C149#13512yes—
106C4×C149C22×Q811yescharacteristic
107C4×C1492−1+411yescharacteristic
108Dic149—41no—
109Dic149—42no—
110C2×C149D4×Q811yescharacteristic
111C2×C149C2×#13511yescharacteristic
112C2×C149#57711yescharacteristic
113D149—42no—
114C149#338111yescharacteristic, complemented, Sylow
115#3381—1491nomaximal, Sylow
116#557D14911yescharacteristic, complemented
117#554—1491no—
118#499—1492no—
119#498—1491no—
120#479—1492no—
121#478—1492no—
122#477—1491no—
123#474—1491no—
124C8×Q8—1491no—
125C4×SD16—1491no—
126C4×SD16—1492no—
127C2×SD16—1491no—
128C4⋊Q8—21no—
129C4⋊Q8—1491no—
130C4⋊Q8C2×D14911yescharacteristic
131#125—22no—
132#125—1492no—
133#125C2×D14912yes—
134C22⋊Q8—2981no—
135C22⋊Q8—2982no—
136C4×Q8—21no—
137C4×Q8C2×D14911yescharacteristic
138C4×Q8C2×D14912yes—
139C4×Q8C2×D14911yescharacteristic
140C4×D4—1491no—
141C4×D4—1492no—
142C2×C4⋊C4—2981no—
143#107—1492no—
144#107—1492no—
145#106—1492no—
146#106—1491no—
147#106—1492no—
148C4⋊C8—1491no—
149C4⋊C8—1492no—
150#103—1492no—
151#103—1491no—
152#102—1492no—
153#102—1491no—
154C4×C8—1491no—
155C4×C8—1492no—
156C2×Q8—21no—
157C2×Q8—2981no—
158C2×Q8C22⋊D14911yescharacteristic
159C2×Q8C22×D14911yescharacteristic
160C2×D4—1491no—
161C22×C4—2981no—
162C22×C4—2982no—
163SD16—1494no—
164C2×C8—1491no—
165C2×C8—1492no—
166C2×C8—1491no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—2982no—
173C4⋊C4—2981no—
174C4⋊C4—2982no—
175C4⋊C4—1492no—
176C4⋊C4—1491no—
177C4⋊C4C22⋊D14911yescharacteristic
178C4⋊C4C22⋊D14912yes—
179C4⋊C4C22×D14912yes—
180C4⋊C4C22×D14911yescharacteristic
181C22⋊C4—2981no—
182C22⋊C4—2982no—
183C42—22no—
184C42—21no—
185C42C22×D14912yes—
186C42C22×D14911yescharacteristic
187C23—2981no—
188Q8—22no—
189Q8—21no—
190Q8—42no—
191Q8—11921no—
192Q8Q8×D14912yes—
193D4—2981no—
194D4—1492no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—2981no—
199C2×C4—2982no—
200C2×C4—5962no—
201C2×C4—11922no—
202C2×C4C2×C22⋊D14912yes—
203C2×C4C2×C22⋊D14912yes—
204C2×C4C2×C22⋊D14911yescharacteristic
205C2×C4C2×C22⋊D14911yescharacteristic
206C2×C4C23×D14911yescharacteristic
207C8—2981no—
208C8—2982no—
209C8—1492no—
210C22—2982no—
211C22—5961no—
212C22—5961no—
213C22C22×C22⋊D14911yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—22no—
218C4—42no—
219C4—41no—
220C4—5961no—
221C4—5962no—
222C4#1801193112yes—
223C4C2×Q8×D14911yescharacteristic
224C4#1801195911yescharacteristic
225C2—5962no—
226C2Q8×C22⋊D14911yescharacteristic
227C2C2×#1801193111yescharacteristic
228C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#557×C149C2
socleC22×C149C22×D4
prime radicalC2×D596C23
hypercenter#557D149
hypocenterC149#3381
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #23750024C2semidirect
3. #23750021C2semidirect
4. #23749932C2semidirect
6. #23749920C2semidirect
8. Q8×#11924746C2semidirect
11. #557×C149C2semidirect
45. #14558723Q8semidirect
114. C149#3381semidirect
116. #557D149semidirect

Maximal subgroups

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

subgroupindex
#237503352
#237500242
#237500212
#237499322
#237499222
#237499202
#237499022
Q8×#119247462
C4×#145587232
#557×C1492
#3381149