Subgroups — C₂×#536×C₁₀₉ #28360701

Order 13952.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 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	1	yes	maximal, characteristic
2	C₂³×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#536×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×#123×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₀₉	—	2	8	no	—
10	C₂×#41×C₁₀₉	C₂²	1	8	yes	—
11	C₂²×Q₈×C₁₀₉	—	2	8	no	—
12	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₀₉	C₂²	1	8	yes	complemented
16	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₀₉	C₂²	1	16	yes	complemented
22	#124×C₁₀₉	C₂²	1	16	yes	complemented
23	#123×C₁₀₉	C₂²	1	32	yes	complemented
24	C₂²⋊Q₈×C₁₀₉	C₂²	1	32	yes	complemented
25	#117×C₁₀₉	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	—
28	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
30	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
31	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
32	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
33	#41×C₁₀₉	—	2	32	no	—
34	C₂×Q₈×C₁₀₉	—	2	8	no	—
35	C₂×Q₈×C₁₀₉	—	2	8	no	—
36	C₂×Q₈×C₁₀₉	—	2	32	no	—
37	C₂×Q₈×C₁₀₉	—	2	4	no	—
38	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
39	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
40	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
41	C₂×D₄×C₁₀₉	—	2	8	no	—
42	C₂×D₄×C₁₀₉	—	2	4	no	—
43	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
44	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₁₀₉	—	2	8	no	—
46	C₂²×C₄×C₁₀₉	—	2	8	no	—
47	C₂²×C₄×C₁₀₉	D₄	1	8	yes	—
48	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
49	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
57	C₄⋊C₄×C₁₀₉	C₂³	1	32	yes	—
58	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
59	C₂²⋊C₄×C₁₀₉	C₂³	1	32	yes	—
60	C₄²×C₁₀₉	C₂³	1	8	yes	—
61	C₂³×C₁₀₉	—	2	4	no	—
62	C₂³×C₁₀₉	—	2	4	no	—
63	C₂³×C₁₀₉	—	2	8	no	—
64	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
65	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
66	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
67	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₀₉	—	2	16	no	—
69	Q₈×C₁₀₉	—	2	16	no	—
70	Q₈×C₁₀₉	—	2	8	no	—
71	D₄×C₁₀₉	—	2	16	no	—
72	D₄×C₁₀₉	—	2	8	no	—
73	C₂×C₄×C₁₀₉	—	2	4	no	—
74	C₂×C₄×C₁₀₉	—	2	16	no	—
75	C₂×C₄×C₁₀₉	—	2	8	no	—
76	C₂×C₄×C₁₀₉	—	2	8	no	—
77	C₂×C₄×C₁₀₉	—	2	16	no	—
78	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₀₉	C₂×D₄	1	16	yes	—
81	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
82	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
83	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
84	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
86	C₂²×C₁₀₉	—	2	8	no	—
87	C₂²×C₁₀₉	—	2	4	no	—
88	C₂²×C₁₀₉	—	2	4	no	—
89	C₂²×C₁₀₉	—	2	8	no	—
90	C₂²×C₁₀₉	—	4	8	no	—
91	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
92	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
93	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
94	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
95	C₄×C₁₀₉	—	2	4	no	—
96	C₄×C₁₀₉	—	2	8	no	—
97	C₄×C₁₀₉	—	2	8	no	—
98	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
99	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
100	C₂×C₁₀₉	—	2	4	no	—
101	C₂×C₁₀₉	—	4	4	no	—
102	C₂×C₁₀₉	#536	1	4	yes	direct product factor
103	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	2	yes	—
105	C₂×#536	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₁₀₉	C₂×#536	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×#41	C₂×C₁₀₉	1	1	yes	characteristic
108	C₂³×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
109	#536	C₂×C₁₀₉	1	16	yes	direct product factor
110	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	2	yes	complemented
111	C₂×#124	C₂×C₁₀₉	1	2	yes	complemented
112	C₂×#123	C₂×C₁₀₉	1	4	yes	complemented
113	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	4	yes	complemented
114	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	C₂×#41	—	2	8	no	—
116	C₂×#41	C₂²×C₁₀₉	1	8	yes	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
119	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
120	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
121	C₂²×Q₈	C₂²×C₁₀₉	1	8	yes	complemented
122	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
123	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
125	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
126	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₁₀₉	1	16	yes	complemented
128	#124	C₂²×C₁₀₉	1	16	yes	complemented
129	#123	C₂²×C₁₀₉	1	32	yes	complemented
130	C₂²⋊Q₈	C₂²×C₁₀₉	1	32	yes	complemented
131	#117	C₂²×C₁₀₉	1	8	yes	complemented
132	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
136	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
137	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
138	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
139	#41	—	2	32	no	—
140	C₂×Q₈	—	2	8	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	2	32	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
145	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
146	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
150	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	D₄×C₁₀₉	1	8	yes	—
154	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
155	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
156	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
157	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
158	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
159	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
162	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
163	C₄⋊C₄	C₂³×C₁₀₉	1	32	yes	—
164	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
165	C₂²⋊C₄	C₂³×C₁₀₉	1	32	yes	—
166	C₄²	C₂³×C₁₀₉	1	8	yes	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	4	no	—
169	C₂³	—	2	8	no	—
170	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
171	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
172	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
173	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
174	Q₈	—	2	16	no	—
175	Q₈	—	2	16	no	—
176	Q₈	—	2	8	no	—
177	D₄	—	2	16	no	—
178	D₄	—	2	8	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
185	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
186	C₂×C₄	C₂×D₄×C₁₀₉	1	16	yes	—
187	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
188	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
189	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
190	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
191	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	4	8	no	—
197	C₂²	C₂²×D₄×C₁₀₉	1	4	yes	—
198	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
199	C₂²	2₋¹⁺⁴×C₁₀₉	1	4	yes	—
200	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	2	8	no	—
203	C₄	—	2	8	no	—
204	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
205	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	4	4	no	—
208	C₂	#536×C₁₀₉	1	4	yes	direct product factor
209	C₂	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
210	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂
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
2. C₂³×Q₈×C₁₀₉	C₂	semidirect
3. #536×C₁₀₉	C₂	direct (N × H)
4. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
5. C₂×#124×C₁₀₉	C₂	semidirect
6. C₂×#123×C₁₀₉	C₂	semidirect
7. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
8. C₂×#117×C₁₀₉	C₂	semidirect
15. C₂²×Q₈×C₁₀₉	C₂²	semidirect
21. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
22. #124×C₁₀₉	C₂²	semidirect
23. #123×C₁₀₉	C₂²	semidirect
24. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
25. #117×C₁₀₉	C₂²	semidirect
102. C₂×C₁₀₉	#536	direct (N × H)
105. C₂×#536	C₁₀₉	direct (N × H)
106. C₁₀₉	C₂×#536	direct (N × H)
108. C₂³×Q₈	C₂×C₁₀₉	semidirect
109. #536	C₂×C₁₀₉	direct (N × H)
110. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
111. C₂×#124	C₂×C₁₀₉	semidirect
112. C₂×#123	C₂×C₁₀₉	semidirect
113. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
114. C₂×#117	C₂×C₁₀₉	semidirect
121. C₂²×Q₈	C₂²×C₁₀₉	semidirect
127. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
128. #124	C₂²×C₁₀₉	semidirect
129. #123	C₂²×C₁₀₉	semidirect
130. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
131. #117	C₂²×C₁₀₉	semidirect
208. C₂	#536×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
C₂³×Q₈×C₁₀₉	2
#536×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#124×C₁₀₉	2
C₂×#123×C₁₀₉	2
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂×#117×C₁₀₉	2
C₂×#536	109

Subgroups — C2×#536×C109 #28360701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#536×C₁₀₉ #28360701