Subgroups — C₂×#14271119 #16827905

Order 3712.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#90777	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6597	C₂	1	1	yes	maximal, characteristic
3	#14271119	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11745151	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6597	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#117	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
8	C₂₉×#117	C₄	1	4	yes	—
9	C₂₉×#117	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₂₉	C₄	1	2	yes	—
11	C₂×C₂²⋊C₄×C₂₉	C₄	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
13	C₂×#90777	C₄	1	8	yes	complemented
14	C₂×#90777	C₂²	1	4	yes	—
15	C₂²×#6597	C₄	1	8	yes	complemented
16	C₂²×#6597	C₂²	1	1	yes	characteristic
17	C₂²×#6597	C₂²	1	2	yes	—
18	C₂²×#6597	C₂²	1	1	yes	characteristic
19	C₂²×#6597	C₂²	1	4	yes	—
20	#11745151	C₂²	1	8	yes	complemented
21	C₄×#6597	C₂²	1	8	yes	complemented
22	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
31	C₂²⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
32	C₄²×C₂₉	C₂×C₄	1	4	yes	—
33	C₄²×C₂₉	C₂³	1	4	yes	—
34	#90777	C₂×C₄	1	16	yes	complemented
35	C₂×#6597	C₂×C₄	1	4	yes	—
36	C₂×#6597	C₂×C₄	1	4	yes	—
37	C₂×#6597	C₂×C₄	1	8	yes	—
38	C₂×#6597	C₂×C₄	1	16	yes	complemented
39	C₂×#6597	C₂³	1	2	yes	—
40	C₂×#6597	C₂³	1	2	yes	—
41	C₂×#6597	C₂³	1	4	yes	—
42	C₂³×C₂₉	—	2	4	no	—
43	C₂³×C₂₉	C₄²	1	4	yes	—
44	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
45	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	—	2	8	no	—
47	C₂×C₄×C₂₉	C₄²	1	8	yes	—
48	C₂×C₄×C₂₉	C₄²	1	4	yes	—
49	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₂₉	C₂²×C₄	1	8	yes	—
54	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
57	#6597	C₂²×C₄	1	4	yes	—
58	#6597	C₂²×C₄	1	4	yes	—
59	#6597	C₂²×C₄	1	8	yes	—
60	C₂×#405	—	29	1	no	maximal, Sylow
61	C₂²×C₂₉	—	2	4	no	—
62	C₂²×C₂₉	—	2	8	no	—
63	C₂²×C₂₉	C₂×C₄²	1	4	yes	—
64	C₂²×C₂₉	C₂×C₄²	1	2	yes	—
65	C₂²×C₂₉	#131	1	4	yes	—
66	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₂₉	—	2	8	no	—
68	C₄×C₂₉	C₂×C₄²	1	2	yes	—
69	C₄×C₂₉	C₂×C₄²	1	4	yes	—
70	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
71	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
72	C₂²×M₁₆	—	29	1	no	—
73	C₂³×C₈	—	29	1	no	—
74	C₂×#117	D₂₉	1	1	yes	characteristic
75	#405	—	29	8	no	—
76	C₂×#97	—	29	2	no	—
77	C₂×C₄×C₈	—	29	2	no	—
78	C₂×C₂₉	—	2	4	no	—
79	C₂×C₂₉	#405	1	4	yes	complemented
80	C₂×C₂₉	C₂²×C₄²	1	1	yes	characteristic
81	C₂×C₂₉	C₂×#131	1	2	yes	—
82	C₂³×C₄	C₂×D₂₉	1	1	yes	characteristic
83	C₂×M₁₆	—	29	8	no	—
84	C₂×M₁₆	—	29	4	no	—
85	C₂²×C₈	—	29	8	no	—
86	C₂²×C₈	—	29	4	no	—
87	C₂²×C₈	—	29	1	no	—
88	C₂²×C₈	—	29	1	no	—
89	C₂²×C₈	—	29	2	no	—
90	#117	Dic₂₉	1	4	yes	—
91	#117	C₂×D₂₉	1	4	yes	—
92	C₂×C₄⋊C₄	Dic₂₉	1	2	yes	—
93	C₂×C₂²⋊C₄	Dic₂₉	1	2	yes	—
94	C₂×C₄²	C₂×D₂₉	1	2	yes	—
95	#97	—	29	8	no	—
96	C₄×C₈	—	29	8	no	—
97	C₂₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
98	C₂⁴	C₂×Dic₂₉	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₂₉	1	4	yes	—
100	C₂²×C₄	C₄×D₂₉	1	4	yes	—
101	C₂²×C₄	C₂×Dic₂₉	1	4	yes	—
102	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
106	M₁₆	—	29	16	no	—
107	C₂×C₈	—	29	4	no	—
108	C₂×C₈	—	29	4	no	—
109	C₂×C₈	—	29	16	no	—
110	C₂×C₈	—	29	8	no	—
111	C₂×C₈	—	29	2	no	—
112	C₂×C₈	—	29	4	no	—
113	C₂×C₈	—	29	2	no	—
114	C₄⋊C₄	C₂×Dic₂₉	1	8	yes	—
115	C₂²⋊C₄	C₂×Dic₂₉	1	8	yes	—
116	C₄²	C₂×Dic₂₉	1	4	yes	—
117	C₄²	C₂²×D₂₉	1	4	yes	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₂₉	1	4	yes	—
120	C₂³	C₂²×Dic₂₉	1	2	yes	—
121	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	C₄×Dic₂₉	1	8	yes	—
124	C₂×C₄	C₄×Dic₂₉	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₂₉	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
127	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₂₉	1	4	yes	—
129	C₂×C₄	C₂²×Dic₂₉	1	8	yes	—
130	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×Dic₂₉	1	2	yes	—
132	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
133	C₈	—	29	4	no	—
134	C₈	—	29	4	no	—
135	C₈	—	29	8	no	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	4	no	—
138	C₂²	C₂×C₄×Dic₂₉	1	4	yes	—
139	C₂²	C₂×C₄×Dic₂₉	1	2	yes	—
140	C₂²	#11745310	1	4	yes	—
141	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
142	C₄	—	2	8	no	—
143	C₄	C₂×C₄×Dic₂₉	1	4	yes	—
144	C₄	C₂×C₄×Dic₂₉	1	2	yes	—
145	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
146	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#14271119	1	4	yes	direct product factor
149	C₂	C₂²×C₄×Dic₂₉	1	1	yes	characteristic
150	C₂	C₂×#11745310	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₉
commutator	C₂×C₂₉	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×#117	C₂
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁴×C₂₉	C₂×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#117	D₂₉
hypocenter	C₂₉	C₂×#405
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14271119	C₂	direct (N × H)
4. C₂×#11745151	C₂	semidirect
5. C₂×C₄×#6597	C₂	semidirect
13. C₂×#90777	C₄	semidirect
15. C₂²×#6597	C₄	semidirect
20. #11745151	C₂²	semidirect
21. C₄×#6597	C₂²	semidirect
34. #90777	C₂×C₄	semidirect
38. C₂×#6597	C₂×C₄	semidirect
79. C₂×C₂₉	#405	semidirect
97. C₂₉	C₂×#405	semidirect
148. C₂	#14271119	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#90777	2
C₂³×#6597	2
#14271119	2
C₂×#11745151	2
C₂×C₄×#6597	2
C₂×C₂₉×#117	2
C₂×#405	29

Subgroups — C2×#14271119 #16827905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14271119 #16827905