Subgroups — C₂×#537×C₁₀₉ #28360702

Order 13952.

1640 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	C₂²×#41×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	#537×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	C₂×#121×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	C₂×#121×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₀₉	—	2	16	no	—
8	C₂×#41×C₁₀₉	C₂²	1	16	yes	complemented
9	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	—
11	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₀₉	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₁₀₉	C₂²	1	32	yes	complemented
17	#121×C₁₀₉	C₂²	1	32	yes	complemented
18	#121×C₁₀₉	C₂²	1	32	yes	complemented
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
22	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
23	C₂⁴×C₁₀₉	C₂³	1	4	yes	—
24	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
25	#41×C₁₀₉	—	2	64	no	—
26	C₂×Q₈×C₁₀₉	—	2	8	no	—
27	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
28	C₂×D₄×C₁₀₉	—	2	16	no	—
29	C₂×D₄×C₁₀₉	—	2	8	no	—
30	C₂×D₄×C₁₀₉	—	4	8	no	—
31	C₂×D₄×C₁₀₉	C₂³	1	16	yes	—
32	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
33	C₂×D₄×C₁₀₉	C₂³	1	16	yes	—
34	C₂²×C₄×C₁₀₉	—	2	16	no	—
35	C₂²×C₄×C₁₀₉	—	4	4	no	—
36	C₂²×C₄×C₁₀₉	D₄	1	8	yes	complemented
37	C₂²×C₄×C₁₀₉	C₂³	1	16	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
43	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
45	C₂²⋊C₄×C₁₀₉	C₂³	1	32	yes	—
46	C₂³×C₁₀₉	—	2	2	no	—
47	C₂³×C₁₀₉	—	2	2	no	—
48	C₂³×C₁₀₉	—	2	8	no	—
49	C₂³×C₁₀₉	—	2	8	no	—
50	C₂³×C₁₀₉	—	2	8	no	—
51	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
52	C₂³×C₁₀₉	C₂⁴	1	8	yes	—
53	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
54	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	Q₈×C₁₀₉	—	2	16	no	—
56	D₄×C₁₀₉	—	2	32	no	—
57	D₄×C₁₀₉	—	2	16	no	—
58	D₄×C₁₀₉	—	4	16	no	—
59	C₂×C₄×C₁₀₉	—	2	8	no	—
60	C₂×C₄×C₁₀₉	—	2	8	no	—
61	C₂×C₄×C₁₀₉	—	2	32	no	—
62	C₂×C₄×C₁₀₉	—	4	8	no	—
63	C₂×C₄×C₁₀₉	C₂×D₄	1	8	yes	—
64	C₂×C₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
65	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
66	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
67	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
68	C₂²×C₁₀₉	—	2	8	no	—
69	C₂²×C₁₀₉	—	2	2	no	—
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	—	2	8	no	—
72	C₂²×C₁₀₉	—	2	8	no	—
73	C₂²×C₁₀₉	—	2	8	no	—
74	C₂²×C₁₀₉	—	4	16	no	—
75	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
76	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
77	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
79	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
80	C₄×C₁₀₉	—	2	8	no	—
81	C₄×C₁₀₉	—	2	8	no	—
82	C₄×C₁₀₉	C₂²×D₄	1	8	yes	—
83	C₂×C₁₀₉	—	2	4	no	—
84	C₂×C₁₀₉	—	4	8	no	—
85	C₂×C₁₀₉	#537	1	4	yes	direct product factor
86	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
87	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
89	C₂×#537	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₁₀₉	C₂×#537	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×#41	C₂×C₁₀₉	1	2	yes	complemented
92	#537	C₂×C₁₀₉	1	16	yes	direct product factor
93	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	4	yes	complemented
94	C₂×#121	C₂×C₁₀₉	1	4	yes	complemented
95	C₂×#121	C₂×C₁₀₉	1	4	yes	complemented
96	C₂²×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
97	C₂×#41	—	2	16	no	—
98	C₂×#41	C₂²×C₁₀₉	1	16	yes	complemented
99	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
100	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	—
101	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
102	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	—
103	C₂³×C₄	C₂²×C₁₀₉	1	4	yes	—
104	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
105	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
106	C₂²⋊Q₈	C₂²×C₁₀₉	1	32	yes	complemented
107	#121	C₂²×C₁₀₉	1	32	yes	complemented
108	#121	C₂²×C₁₀₉	1	32	yes	complemented
109	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
110	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
111	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
112	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
113	C₂⁴	C₂³×C₁₀₉	1	4	yes	—
114	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
115	#41	—	2	64	no	—
116	C₂×Q₈	—	2	8	no	—
117	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
118	C₂×D₄	—	2	16	no	—
119	C₂×D₄	—	2	8	no	—
120	C₂×D₄	—	4	8	no	—
121	C₂×D₄	C₂³×C₁₀₉	1	16	yes	—
122	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
123	C₂×D₄	C₂³×C₁₀₉	1	16	yes	—
124	C₂²×C₄	—	2	16	no	—
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	D₄×C₁₀₉	1	8	yes	complemented
127	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
128	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
129	C₂²×C₄	C₂³×C₁₀₉	1	16	yes	—
130	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
131	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
132	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
133	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
134	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
135	C₂²⋊C₄	C₂³×C₁₀₉	1	32	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	8	no	—
139	C₂³	—	2	8	no	—
140	C₂³	—	2	8	no	—
141	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
142	C₂³	C₂⁴×C₁₀₉	1	8	yes	—
143	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
144	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
145	Q₈	—	2	16	no	—
146	D₄	—	2	32	no	—
147	D₄	—	2	16	no	—
148	D₄	—	4	16	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	32	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	C₂×D₄×C₁₀₉	1	8	yes	—
154	C₂×C₄	C₂×D₄×C₁₀₉	1	16	yes	complemented
155	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
156	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
157	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
158	C₂²	—	2	8	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	8	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	4	16	no	—
165	C₂²	C₂²×D₄×C₁₀₉	1	4	yes	—
166	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
167	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
168	C₂²	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
169	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
170	C₄	—	2	8	no	—
171	C₄	—	2	8	no	—
172	C₄	C₂²×D₄×C₁₀₉	1	8	yes	—
173	C₂	—	2	4	no	—
174	C₂	—	4	8	no	—
175	C₂	#537×C₁₀₉	1	4	yes	direct product factor
176	C₂	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
177	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
178	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₀₉	C₂	semidirect
2. #537×C₁₀₉	C₂	direct (N × H)
3. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
4. C₂×#121×C₁₀₉	C₂	semidirect
5. C₂×#121×C₁₀₉	C₂	semidirect
6. C₂²×C₄⋊C₄×C₁₀₉	C₂	semidirect
8. C₂×#41×C₁₀₉	C₂²	semidirect
16. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
17. #121×C₁₀₉	C₂²	semidirect
18. #121×C₁₀₉	C₂²	semidirect
21. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
36. C₂²×C₄×C₁₀₉	D₄	semidirect
64. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
85. C₂×C₁₀₉	#537	direct (N × H)
89. C₂×#537	C₁₀₉	direct (N × H)
90. C₁₀₉	C₂×#537	direct (N × H)
91. C₂²×#41	C₂×C₁₀₉	semidirect
92. #537	C₂×C₁₀₉	direct (N × H)
93. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
94. C₂×#121	C₂×C₁₀₉	semidirect
95. C₂×#121	C₂×C₁₀₉	semidirect
96. C₂²×C₄⋊C₄	C₂×C₁₀₉	semidirect
98. C₂×#41	C₂²×C₁₀₉	semidirect
106. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
107. #121	C₂²×C₁₀₉	semidirect
108. #121	C₂²×C₁₀₉	semidirect
111. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
126. C₂²×C₄	D₄×C₁₀₉	semidirect
154. C₂×C₄	C₂×D₄×C₁₀₉	semidirect
175. C₂	#537×C₁₀₉	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×#41×C₁₀₉	2
#537×C₁₀₉	2
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂×#537	109

Subgroups — C2×#537×C109 #28360702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#537×C₁₀₉ #28360702