Subgroups — #16828100

Order 3712.

440 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271929	C₂	1	2	yes	maximal
2	C₂×#11745263	C₂	1	1	yes	maximal, characteristic
3	#14271211	C₂	1	2	yes	maximal, complemented
4	#14271210	C₂	1	4	yes	maximal, complemented
5	C₂×#11745167	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₂₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#90805	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×#90777	C₂²	1	2	yes	—
12	C₂×#90777	C₂²	1	1	yes	characteristic
13	#11745269	—	2	4	no	—
14	M₁₆×D₂₉	C₂²	1	8	yes	—
15	#11745263	—	2	4	no	—
16	#11745263	C₄	1	4	yes	—
17	#11745263	C₄	1	4	yes	—
18	C₂×#90772	C₂²	1	2	yes	—
19	C₂×C₈×D₂₉	C₂²	1	2	yes	—
20	C₂₉×#108	C₂²	1	2	yes	—
21	C₂₉×#108	C₂²	1	2	yes	—
22	#11745170	C₂²	1	8	yes	—
23	#11745167	C₂²	1	2	yes	—
24	#11745167	C₂²	1	2	yes	—
25	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₂₉	Q₈	1	2	yes	—
27	#90805	C₂×C₄	1	2	yes	—
28	#90805	C₂×C₄	1	2	yes	—
29	#90805	D₄	1	2	yes	—
30	#90805	D₄	1	2	yes	—
31	C₂×D₁₁₆	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₂₉	C₂³	1	2	yes	—
33	C₂×Dic₅₈	Q₈	1	1	yes	characteristic
34	M₁₆×C₂₉	—	2	2	no	—
35	M₁₆×C₂₉	C₂³	1	4	yes	—
36	C₂×C₈×C₂₉	—	2	2	no	—
37	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
41	#90777	—	2	2	no	—
42	#90777	C₂×C₄	1	2	yes	—
43	#90777	C₂×C₄	1	2	yes	—
44	#90777	C₂³	1	4	yes	—
45	C₂×#6597	—	2	2	no	—
46	C₂×#6597	C₂³	1	2	yes	—
47	#90772	—	2	8	no	—
48	#90772	C₂×C₄	1	8	yes	—
49	C₈×D₂₉	—	2	8	no	—
50	C₈×D₂₉	C₂×C₄	1	8	yes	—
51	C₂³×C₂₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₂₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₂₉	C₄⋊C₄	1	4	yes	—
60	C₂²⋊D₂₉	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
62	D₁₁₆	C₄⋊C₄	1	2	yes	—
63	D₁₁₆	C₄⋊C₄	1	2	yes	—
64	C₄×D₂₉	C₂²×C₄	1	4	yes	—
65	C₄×D₂₉	C₂×D₄	1	4	yes	—
66	Dic₅₈	C₄⋊C₄	1	2	yes	—
67	Dic₅₈	C₄⋊C₄	1	2	yes	—
68	C₈×C₂₉	—	2	4	no	—
69	C₈×C₂₉	C₂²×C₄	1	2	yes	—
70	C₈×C₂₉	C₂²×C₄	1	2	yes	—
71	#6597	—	2	4	no	—
72	#6597	C₂²×C₄	1	4	yes	—
73	#2916	—	29	1	no	maximal, Sylow
74	C₂²×C₂₉	—	4	1	no	—
75	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
78	C₂×D₂₉	—	4	2	no	—
79	C₂×D₂₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
84	Dic₂₉	C₂×C₄⋊C₄	1	4	yes	—
85	#568	—	29	2	no	—
86	C₂×#131	—	29	1	no	—
87	#430	—	29	2	no	—
88	#430	—	29	4	no	—
89	C₂×#108	—	29	1	no	—
90	C₂×#108	—	29	4	no	—
91	C₂×#108	D₂₉	1	1	yes	characteristic, complemented
92	C₂×C₂₉	—	2	1	no	—
93	C₂×C₂₉	—	2	1	no	—
94	C₂×C₂₉	—	2	1	no	—
95	C₂×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	D₂₉	—	2	4	no	—
97	C₂×#41	—	29	1	no	—
98	#131	—	29	4	no	—
99	#131	—	29	4	no	—
100	#131	—	58	4	no	—
101	#131	—	58	4	no	—
102	C₂×M₁₆	—	29	1	no	—
103	C₂×M₁₆	—	29	2	no	—
104	C₂×M₁₆	—	29	2	no	—
105	C₂×M₁₆	—	29	8	no	—
106	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
107	C₂²×C₈	—	29	2	no	—
108	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
109	#108	—	29	8	no	—
110	#108	—	29	2	no	—
111	#108	—	29	2	no	—
112	#108	C₂×D₂₉	1	2	yes	—
113	#108	C₂×D₂₉	1	2	yes	—
114	C₂₉	#2916	1	1	yes	characteristic, complemented, Sylow
115	#41	—	29	2	no	—
116	#41	—	29	2	no	—
117	#41	—	29	2	no	—
118	#41	—	29	2	no	—
119	C₂×Q₈	—	29	1	no	—
120	C₂×D₄	—	29	2	no	—
121	C₂×D₄	—	29	1	no	—
122	C₂²×C₄	—	29	2	no	—
123	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	29	2	no	—
126	M₁₆	—	29	2	no	—
127	M₁₆	—	29	8	no	—
128	M₁₆	—	29	4	no	—
129	M₁₆	—	58	8	no	—
130	M₁₆	—	58	2	no	—
131	M₁₆	C₂²×D₂₉	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	29	2	no	—
134	C₂×C₈	—	29	8	no	—
135	C₂×C₈	—	58	8	no	—
136	C₂×C₈	—	58	2	no	—
137	C₂×C₈	C₄×D₂₉	1	2	yes	—
138	C₂×C₈	C₄×D₂₉	1	2	yes	—
139	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
141	C₂³	—	29	2	no	—
142	C₂³	Q₈×D₂₉	1	1	yes	characteristic
143	Q₈	—	29	2	no	—
144	Q₈	—	29	2	no	—
145	D₄	—	29	4	no	—
146	D₄	—	29	4	no	—
147	D₄	—	29	2	no	—
148	D₄	—	29	2	no	—
149	C₂×C₄	—	29	2	no	—
150	C₂×C₄	—	29	4	no	—
151	C₂×C₄	—	29	4	no	—
152	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
154	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
155	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	29	4	no	—
160	C₈	—	58	4	no	—
161	C₈	C₂×C₄×D₂₉	1	2	yes	—
162	C₈	C₂×C₄×D₂₉	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	29	4	no	—
165	C₂²	—	116	2	no	—
166	C₂²	C₄⋊C₄×D₂₉	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₂₉	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₂₉	1	1	yes	characteristic
169	C₄	—	29	4	no	—
170	C₄	C₄⋊C₄×D₂₉	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₂₉	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
173	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	58	4	no	—
178	C₂	C₂×C₄⋊C₄×D₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×#108	C₂
socle	C₂×C₂₉	C₂²×C₄⋊C₄
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#108	D₂₉
hypocenter	C₂₉	#2916
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #14271211	C₂	semidirect
4. #14271210	C₂	semidirect
5. C₂×#11745167	C₂	semidirect
6. #108×D₂₉	C₂	semidirect
7. C₂×C₂₉×#108	C₂	semidirect
91. C₂×#108	D₂₉	semidirect
114. C₂₉	#2916	semidirect

Maximal subgroups

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

subgroup	index
#14271929	2
C₂×#11745263	2
#14271211	2
#14271210	2
C₂×#11745167	2
#108×D₂₉	2
C₂×C₂₉×#108	2
#2916	29