Subgroups — #16822115

Order 3712.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745288	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₂₉	C₂	1	1	yes	maximal, characteristic
3	#14271062	C₂	1	4	yes	maximal, complemented
4	#14271038	C₂	1	4	yes	maximal
5	#14271033	C₂	1	2	yes	maximal, complemented
6	#14271029	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₉×#117	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₉	C₂²	1	2	yes	—
10	C₂₉×#117	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₂₉	C₂²	1	2	yes	—
15	#11745288	—	2	4	no	—
16	C₂×C₄⋊Dic₂₉	—	2	4	no	—
17	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₂₉	C₂²	1	8	yes	—
22	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
23	#11745179	C₂²	1	8	yes	—
24	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₉	D₄	1	4	yes	—
27	C₂²×C₄×C₂₉	Q₈	1	4	yes	—
28	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₉	—	2	4	no	—
32	C₂²×Dic₂₉	—	4	4	no	—
33	C₂²×Dic₂₉	C₂³	1	2	yes	—
34	C₂²×Dic₂₉	C₂³	1	2	yes	—
35	C₂²×Dic₂₉	C₂³	1	4	yes	—
36	C₄⋊C₄×C₂₉	C₂×C₄	1	8	yes	complemented
37	C₂²⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
38	C₄²×C₂₉	C₂×C₄	1	8	yes	complemented
39	C₂²⋊Dic₂₉	—	2	4	no	—
40	C₄⋊Dic₂₉	—	2	4	no	—
41	C₄⋊Dic₂₉	—	2	4	no	—
42	C₄⋊Dic₂₉	—	2	2	no	—
43	C₄⋊Dic₂₉	—	2	2	no	—
44	C₄⋊Dic₂₉	—	4	8	no	—
45	C₄×Dic₂₉	—	2	4	no	—
46	C₂³×C₂₉	—	2	2	no	—
47	C₂³×C₂₉	—	2	2	no	—
48	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₉	C₂×D₄	1	2	yes	—
50	C₂³×C₂₉	C₂×Q₈	1	2	yes	—
51	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₉	—	2	4	no	—
53	C₂×C₄×C₂₉	—	2	4	no	—
54	C₂×C₄×C₂₉	C₄⋊C₄	1	16	yes	complemented
55	C₂×C₄×C₂₉	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
62	C₂×Dic₂₉	—	2	2	no	—
63	C₂×Dic₂₉	—	2	2	no	—
64	C₂×Dic₂₉	—	2	4	no	—
65	C₂×Dic₂₉	—	2	4	no	—
66	C₂×Dic₂₉	—	2	8	no	—
67	C₂×Dic₂₉	—	2	4	no	—
68	C₂×Dic₂₉	—	4	8	no	—
69	#2317	—	29	1	no	maximal, Sylow
70	C₂²×C₂₉	—	2	4	no	—
71	C₂²×C₂₉	—	2	2	no	—
72	C₂²×C₂₉	—	2	2	no	—
73	C₂²×C₂₉	—	2	4	no	—
74	C₂²×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
76	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₂₉	H(5,𝔽₂)	1	2	yes	—
79	C₂²×C₂₉	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₂₉	—	2	8	no	—
81	C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
82	C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
83	Dic₂₉	—	4	2	no	—
84	Dic₂₉	—	4	4	no	—
85	Dic₂₉	—	4	2	no	—
86	C₂×#117	—	29	1	no	—
87	C₂×#117	D₂₉	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	29	1	no	—
89	#389	—	29	4	no	—
90	#385	—	29	4	no	—
91	#384	—	29	2	no	—
92	#382	—	29	2	no	—
93	C₂×C₂₉	—	2	4	no	—
94	C₂×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₂₉	#520	1	2	yes	—
96	C₂×C₂₉	#535	1	1	yes	characteristic
97	C₂×C₂₉	#536	1	1	yes	characteristic
98	C₂×C₂₉	#544	1	2	yes	—
99	C₂³×C₄	—	29	2	no	—
100	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
101	#117	—	58	4	no	—
102	#117	Dic₂₉	1	8	yes	complemented
103	C₂×C₄⋊C₄	—	29	2	no	—
104	C₂×C₄⋊C₄	—	29	2	no	—
105	C₂×C₄⋊C₄	—	29	1	no	—
106	C₂×C₄⋊C₄	—	29	1	no	—
107	C₂×C₄⋊C₄	—	29	8	no	—
108	C₂×C₄⋊C₄	—	58	4	no	—
109	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
110	C₂×C₂²⋊C₄	—	29	2	no	—
111	C₂×C₂²⋊C₄	C₂×D₂₉	1	2	yes	—
112	C₂×C₄²	—	29	2	no	—
113	C₂×C₄²	C₂×D₂₉	1	2	yes	—
114	#95	—	29	8	no	—
115	C₂₉	#2317	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₂²×D₂₉	1	1	yes	characteristic
117	C₂²×C₄	—	29	2	no	—
118	C₂²×C₄	—	29	4	no	—
119	C₂²×C₄	—	29	2	no	—
120	C₂²×C₄	—	58	4	no	—
121	C₂²×C₄	—	116	4	no	—
122	C₂²×C₄	Dic₅₈	1	4	yes	—
123	C₂²×C₄	D₁₁₆	1	4	yes	—
124	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
125	C₂²×C₄	C₂²×D₂₉	1	4	yes	—
126	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
128	C₄⋊C₄	—	58	4	no	—
129	C₄⋊C₄	—	58	4	no	—
130	C₄⋊C₄	—	58	2	no	—
131	C₄⋊C₄	—	58	2	no	—
132	C₄⋊C₄	—	116	8	no	—
133	C₄⋊C₄	C₂×Dic₂₉	1	8	yes	complemented
134	C₂²⋊C₄	—	58	4	no	—
135	C₂²⋊C₄	C₂×Dic₂₉	1	8	yes	—
136	C₄²	—	58	4	no	—
137	C₄²	C₂×Dic₂₉	1	8	yes	complemented
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Dic₅₈	1	2	yes	—
141	C₂³	C₂×D₁₁₆	1	2	yes	—
142	C₂³	C₂²×Dic₂₉	1	2	yes	—
143	C₂³	C₂³×D₂₉	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	58	2	no	—
147	C₂×C₄	—	58	2	no	—
148	C₂×C₄	—	58	4	no	—
149	C₂×C₄	—	58	4	no	—
150	C₂×C₄	—	58	8	no	—
151	C₂×C₄	—	58	4	no	—
152	C₂×C₄	—	116	8	no	—
153	C₂×C₄	C₄⋊Dic₂₉	1	16	yes	complemented
154	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
155	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
156	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
157	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
158	C₂×C₄	C₂²×Dic₂₉	1	8	yes	—
159	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
160	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×C₄⋊Dic₂₉	1	4	yes	—
166	C₂²	C₂²×Dic₅₈	1	1	yes	characteristic
167	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
168	C₂²	#11745364	1	2	yes	—
169	C₂²	#11745365	1	2	yes	—
170	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
171	C₄	—	2	8	no	—
172	C₄	—	116	4	no	—
173	C₄	—	116	2	no	—
174	C₄	—	116	2	no	—
175	C₄	C₂×C₄⋊Dic₂₉	1	4	yes	—
176	C₄	C₂×C₄⋊Dic₂₉	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₄⋊Dic₂₉	1	1	yes	characteristic
179	C₂	#14271646	1	2	yes	—
180	C₂	#14271732	1	1	yes	characteristic
181	C₂	#14271739	1	1	yes	characteristic
182	C₂	#14271791	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Fitting	C₂×C₂₉×#117	C₂
socle, cosocle	C₂³×C₂₉	C₂⁴
prime radical	C₂⁴×C₂₉	C₂³
hypercenter	C₂×#117	D₂₉
hypocenter	C₂₉	#2317
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14271062	C₂	semidirect
5. #14271033	C₂	semidirect
6. #14271029	C₂	semidirect
10. C₂₉×#117	C₄	semidirect
36. C₄⋊C₄×C₂₉	C₂×C₄	semidirect
38. C₄²×C₂₉	C₂×C₄	semidirect
54. C₂×C₄×C₂₉	C₄⋊C₄	semidirect
102. #117	Dic₂₉	semidirect
115. C₂₉	#2317	semidirect
133. C₄⋊C₄	C₂×Dic₂₉	semidirect
137. C₄²	C₂×Dic₂₉	semidirect
153. C₂×C₄	C₄⋊Dic₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745288	2
C₂²×C₄⋊Dic₂₉	2
#14271062	2
#14271038	2
#14271033	2
#14271029	2
C₂×C₂₉×#117	2
#2317	29