Finite Group Databasebeta

Subgroups — #14271861

Order 1856.

238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#11745292	C₂	1	1	yes	maximal, characteristic, complemented
3	#11745292	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊D₂₉	C₂	1	2	yes	maximal, complemented
6	#11745260	C₂	1	2	yes	maximal, complemented
7	#11745257	C₂	1	1	yes	maximal, characteristic, complemented
8	#11745256	C₂	1	2	yes	maximal, complemented
9	#11745255	C₂	1	1	yes	maximal, characteristic, complemented
10	#11745254	C₂	1	1	yes	maximal, characteristic, complemented
11	#11745252	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₂₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₂₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₂₉	C₂²	1	2	yes	—
16	C₂×D₁₁₆	C₂²	1	1	yes	characteristic
17	C₂×D₁₁₆	C₂²	1	2	yes	—
18	C₂×C₄×D₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₂₉	C₂²	1	2	yes	—
20	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₉	C₂²	1	1	yes	characteristic
24	#90781	C₂²	1	1	yes	characteristic
25	#90781	C₂²	1	2	yes	—
26	#90781	C₂²	1	2	yes	—
27	#90781	C₂²	1	2	yes	—
28	#90781	C₂²	1	2	yes	—
29	C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₂₉	C₂²	1	2	yes	—
31	#90779	C₂²	1	2	yes	—
32	#90779	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₉	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₉	C₂²	1	2	yes	—
35	C₂³×C₂₉	C₂³	1	1	yes	characteristic
36	C₂²×D₂₉	C₂³	1	1	yes	characteristic
37	C₂²×D₂₉	C₂³	1	2	yes	—
38	C₂×C₄×C₂₉	—	2	2	no	—
39	C₂×C₄×C₂₉	—	4	2	no	—
40	C₂×C₄×C₂₉	C₂³	1	2	yes	—
41	C₂×C₄×C₂₉	C₂³	1	2	yes	—
42	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
44	C₂²⋊D₂₉	—	2	2	no	—
45	C₂²⋊D₂₉	—	2	4	no	—
46	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₉	C₂³	1	2	yes	—
50	D₁₁₆	—	2	2	no	—
51	D₁₁₆	—	4	2	no	—
52	C₄×D₂₉	—	2	2	no	—
53	C₄×D₂₉	—	4	2	no	—
54	C₂²×C₂₉	—	2	1	no	—
55	C₂²×C₂₉	—	2	1	no	—
56	C₂²×C₂₉	#41	1	2	yes	—
57	C₂²×C₂₉	C₂⁴	1	1	yes	characteristic
58	C₂×D₂₉	—	2	1	no	—
59	C₂×D₂₉	—	2	2	no	—
60	C₂×D₂₉	—	2	1	no	—
61	C₂×D₂₉	—	2	2	no	—
62	C₂×D₂₉	—	2	2	no	—
63	C₂×D₂₉	—	2	1	no	—
64	C₄×C₂₉	—	2	1	no	—
65	C₄×C₂₉	—	2	2	no	—
66	C₄×C₂₉	—	2	2	no	—
67	C₄×C₂₉	#41	1	2	yes	—
68	Dic₂₉	—	2	1	no	—
69	Dic₂₉	—	2	1	no	—
70	Dic₂₉	—	2	2	no	—
71	Dic₂₉	—	2	1	no	—
72	#553	—	29	1	no	maximal, Sylow
73	C₂×C₂₉	—	2	2	no	—
74	C₂×C₂₉	C₂×#41	1	1	yes	characteristic
75	C₂×C₂₉	C₂×#41	1	1	yes	characteristic
76	C₂×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
77	D₂₉	—	4	1	no	—
78	D₂₉	—	4	2	no	—
79	#126	—	29	2	no	—
80	#125	—	29	1	no	—
81	#123	—	29	2	no	—
82	#121	—	29	2	no	—
83	#121	—	29	1	no	—
84	#121	—	29	1	no	—
85	C₄×D₄	—	29	2	no	—
86	C₄×D₄	—	29	1	no	—
87	C₄×D₄	—	29	1	no	—
88	#117	—	29	1	no	—
89	C₂×C₄⋊C₄	D₂₉	1	1	yes	characteristic, complemented
90	C₂₉	#553	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₄	—	29	2	no	—
92	C₂×D₄	—	29	1	no	—
93	C₂×D₄	—	29	1	no	—
94	C₂×D₄	—	29	2	no	—
95	C₂²×C₄	—	29	1	no	—
96	C₂²×C₄	—	29	2	no	—
97	C₂²×C₄	C₂×D₂₉	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₂₉	1	2	yes	—
99	C₄⋊C₄	—	29	1	no	—
100	C₄⋊C₄	—	29	2	no	—
101	C₄⋊C₄	—	29	1	no	—
102	C₄⋊C₄	—	29	2	no	—
103	C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
104	C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
105	C₄⋊C₄	C₂×D₂₉	1	2	yes	—
106	C₂²⋊C₄	—	29	1	no	—
107	C₂²⋊C₄	—	29	2	no	—
108	C₂²⋊C₄	—	29	2	no	—
109	C₂²⋊C₄	—	29	2	no	—
110	C₂²⋊C₄	—	29	1	no	—
111	C₂²⋊C₄	—	29	2	no	—
112	C₄²	—	29	2	no	—
113	C₄²	—	29	1	no	—
114	C₂³	—	29	1	no	—
115	C₂³	—	29	2	no	—
116	C₂³	C₂²×D₂₉	1	1	yes	characteristic
117	D₄	—	58	4	no	—
118	D₄	—	58	2	no	—
119	D₄	—	58	2	no	—
120	D₄	—	116	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	29	1	no	—
124	C₂×C₄	—	29	2	no	—
125	C₂×C₄	—	29	1	no	—
126	C₂×C₄	—	29	1	no	—
127	C₂×C₄	—	58	2	no	—
128	C₂×C₄	—	116	2	no	—
129	C₂×C₄	C₂²×D₂₉	1	2	yes	—
130	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₂₉	1	2	yes	—
132	C₂×C₄	C₂²×D₂₉	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	58	1	no	—
136	C₂²	—	58	2	no	—
137	C₂²	—	58	2	no	—
138	C₂²	—	58	1	no	—
139	C₂²	—	58	1	no	—
140	C₂²	—	58	2	no	—
141	C₂²	#90809	1	2	yes	—
142	C₂²	C₂³×D₂₉	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	58	1	no	—
147	C₄	—	58	1	no	—
148	C₄	—	58	2	no	—
149	C₄	—	58	1	no	—
150	C₄	#90805	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	116	1	no	—
153	C₂	—	116	2	no	—
154	C₂	C₂×#90805	1	1	yes	characteristic
155	C₂	#11745359	1	1	yes	characteristic
156	C₂	C₂×#90809	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₉
commutator, socle, cosocle	C₂²×C₂₉	C₂⁴
Fitting	C₂×C₄⋊C₄×C₂₉	C₂
hypercenter	C₂×C₄⋊C₄	D₂₉
hypocenter	C₂₉	#553
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×C₂₉	C₂	semidirect
2. #11745292	C₂	semidirect
3. #11745292	C₂	semidirect
4. C₄×C₂²⋊D₂₉	C₂	semidirect
5. C₄×C₂²⋊D₂₉	C₂	semidirect
6. #11745260	C₂	semidirect
7. #11745257	C₂	semidirect
8. #11745256	C₂	semidirect
9. #11745255	C₂	semidirect
10. #11745254	C₂	semidirect
11. #11745252	C₂	semidirect
89. C₂×C₄⋊C₄	D₂₉	semidirect
90. C₂₉	#553	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×C₂₉	2
#11745292	2
C₄×C₂²⋊D₂₉	2
#11745260	2
#11745257	2
#11745256	2
#11745255	2
#11745254	2
#11745252	2
#553	29