Subgroups — C₂×#535×C₁₀₉ #28360700

Order 13952.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 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, complemented
2	C₂³×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#535×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
4	C₂×#127×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×#121×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₂×#120×C₁₀₉	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#41×C₁₀₉	—	2	8	no	—
11	C₂×#41×C₁₀₉	C₂²	1	8	yes	complemented
12	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₁₀₉	—	2	8	no	—
14	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂²×D₄×C₁₀₉	C₂²	1	8	yes	complemented
18	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄×C₁₀₉	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	complemented
22	#127×C₁₀₉	C₂²	1	16	yes	complemented
23	#124×C₁₀₉	C₂²	1	16	yes	complemented
24	#121×C₁₀₉	C₂²	1	32	yes	complemented
25	#120×C₁₀₉	C₂²	1	32	yes	complemented
26	#117×C₁₀₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
30	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	complemented
31	C₂⁴×C₁₀₉	—	2	8	no	—
32	C₂⁴×C₁₀₉	—	4	8	no	—
33	C₂⁴×C₁₀₉	C₂³	1	4	yes	—
34	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
35	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂⁴×C₁₀₉	C₂³	1	8	yes	—
37	#41×C₁₀₉	—	2	32	no	—
38	C₂×Q₈×C₁₀₉	—	2	4	no	—
39	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	complemented
40	C₂×D₄×C₁₀₉	—	2	8	no	—
41	C₂×D₄×C₁₀₉	—	2	8	no	—
42	C₂×D₄×C₁₀₉	—	2	8	no	—
43	C₂×D₄×C₁₀₉	—	2	4	no	—
44	C₂×D₄×C₁₀₉	—	2	32	no	—
45	C₂×D₄×C₁₀₉	—	4	16	no	—
46	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
47	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
48	C₂×D₄×C₁₀₉	C₂³	1	8	yes	—
49	C₂×D₄×C₁₀₉	C₂³	1	4	yes	complemented
50	C₂×D₄×C₁₀₉	C₂³	1	32	yes	complemented
51	C₂²×C₄×C₁₀₉	—	2	8	no	—
52	C₂²×C₄×C₁₀₉	D₄	1	8	yes	complemented
53	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
58	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	complemented
59	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	complemented
60	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
61	C₂²⋊C₄×C₁₀₉	C₂³	1	32	yes	complemented
62	C₄²×C₁₀₉	C₂³	1	8	yes	complemented
63	C₂³×C₁₀₉	—	2	4	no	—
64	C₂³×C₁₀₉	—	2	4	no	—
65	C₂³×C₁₀₉	—	2	16	no	—
66	C₂³×C₁₀₉	—	2	8	no	—
67	C₂³×C₁₀₉	—	2	8	no	—
68	C₂³×C₁₀₉	—	2	16	no	—
69	C₂³×C₁₀₉	—	4	16	no	—
70	C₂³×C₁₀₉	—	4	32	no	—
71	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
72	C₂³×C₁₀₉	C₂⁴	1	8	yes	—
73	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
74	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
75	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	2	8	no	—
77	D₄×C₁₀₉	—	2	16	no	—
78	D₄×C₁₀₉	—	2	16	no	—
79	D₄×C₁₀₉	—	2	16	no	—
80	D₄×C₁₀₉	—	2	8	no	—
81	D₄×C₁₀₉	—	4	32	no	—
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	2	8	no	—
84	C₂×C₄×C₁₀₉	—	2	16	no	—
85	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
88	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
89	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
92	C₂²×C₁₀₉	—	2	8	no	—
93	C₂²×C₁₀₉	—	2	4	no	—
94	C₂²×C₁₀₉	—	2	4	no	—
95	C₂²×C₁₀₉	—	2	8	no	—
96	C₂²×C₁₀₉	—	2	8	no	—
97	C₂²×C₁₀₉	—	2	16	no	—
98	C₂²×C₁₀₉	—	4	32	no	—
99	C₂²×C₁₀₉	—	4	8	no	—
100	C₂²×C₁₀₉	—	4	16	no	—
101	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
102	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
103	C₂²×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
104	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
105	C₄×C₁₀₉	—	2	4	no	—
106	C₄×C₁₀₉	—	2	8	no	—
107	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
108	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
109	C₂×C₁₀₉	—	2	4	no	—
110	C₂×C₁₀₉	—	4	4	no	—
111	C₂×C₁₀₉	—	4	8	no	—
112	C₂×C₁₀₉	#535	1	4	yes	direct product factor
113	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
114	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
115	C₂×#535	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₁₀₉	C₂×#535	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×#41	C₂×C₁₀₉	1	1	yes	characteristic, complemented
118	C₂³×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
119	#535	C₂×C₁₀₉	1	16	yes	direct product factor
120	C₂×#127	C₂×C₁₀₉	1	2	yes	complemented
121	C₂×#124	C₂×C₁₀₉	1	2	yes	complemented
122	C₂×#121	C₂×C₁₀₉	1	4	yes	complemented
123	C₂×#120	C₂×C₁₀₉	1	4	yes	complemented
124	C₂×#117	C₂×C₁₀₉	1	1	yes	characteristic, complemented
125	C₂⁵	C₂²×C₁₀₉	1	2	yes	—
126	C₂×#41	—	2	8	no	—
127	C₂×#41	C₂²×C₁₀₉	1	8	yes	complemented
128	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
129	C₂²×D₄	—	2	8	no	—
130	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
131	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
132	C₂²×D₄	C₂²×C₁₀₉	1	2	yes	—
133	C₂²×D₄	C₂²×C₁₀₉	1	8	yes	complemented
134	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
135	C₂²×D₄	C₂²×C₁₀₉	1	8	yes	complemented
136	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
137	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	complemented
138	#127	C₂²×C₁₀₉	1	16	yes	complemented
139	#124	C₂²×C₁₀₉	1	16	yes	complemented
140	#121	C₂²×C₁₀₉	1	32	yes	complemented
141	#120	C₂²×C₁₀₉	1	32	yes	complemented
142	#117	C₂²×C₁₀₉	1	8	yes	complemented
143	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
144	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
146	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	complemented
147	C₂⁴	—	2	8	no	—
148	C₂⁴	—	4	8	no	—
149	C₂⁴	C₂³×C₁₀₉	1	4	yes	—
150	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
151	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
152	C₂⁴	C₂³×C₁₀₉	1	8	yes	—
153	#41	—	2	32	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	complemented
156	C₂×D₄	—	2	8	no	—
157	C₂×D₄	—	2	8	no	—
158	C₂×D₄	—	2	8	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	32	no	—
161	C₂×D₄	—	4	16	no	—
162	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
163	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
164	C₂×D₄	C₂³×C₁₀₉	1	8	yes	—
165	C₂×D₄	C₂³×C₁₀₉	1	4	yes	complemented
166	C₂×D₄	C₂³×C₁₀₉	1	32	yes	complemented
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	D₄×C₁₀₉	1	8	yes	complemented
169	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
170	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
171	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
174	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	complemented
175	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	complemented
176	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
177	C₂²⋊C₄	C₂³×C₁₀₉	1	32	yes	complemented
178	C₄²	C₂³×C₁₀₉	1	8	yes	complemented
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	16	no	—
182	C₂³	—	2	8	no	—
183	C₂³	—	2	8	no	—
184	C₂³	—	2	16	no	—
185	C₂³	—	4	16	no	—
186	C₂³	—	4	32	no	—
187	C₂³	C₂×D₄×C₁₀₉	1	4	yes	—
188	C₂³	C₂⁴×C₁₀₉	1	8	yes	—
189	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
190	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
191	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
192	Q₈	—	2	8	no	—
193	D₄	—	2	16	no	—
194	D₄	—	2	16	no	—
195	D₄	—	2	16	no	—
196	D₄	—	2	8	no	—
197	D₄	—	4	32	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	16	no	—
201	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
202	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
203	C₂×C₄	C₂×D₄×C₁₀₉	1	16	yes	complemented
204	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
205	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
206	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	8	no	—
212	C₂²	—	2	8	no	—
213	C₂²	—	2	16	no	—
214	C₂²	—	4	32	no	—
215	C₂²	—	4	16	no	—
216	C₂²	—	4	8	no	—
217	C₂²	C₂²×D₄×C₁₀₉	1	4	yes	—
218	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
219	C₂²	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
220	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	8	no	—
223	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
224	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	4	8	no	—
227	C₂	—	4	4	no	—
228	C₂	#535×C₁₀₉	1	4	yes	direct product factor
229	C₂	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
230	C₂	C₂×H(5,𝔽₂)×C₁₀₉	1	2	yes	—

	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. C₂³×D₄×C₁₀₉	C₂	semidirect
3. #535×C₁₀₉	C₂	direct (N × H)
4. C₂×#127×C₁₀₉	C₂	semidirect
5. C₂×#124×C₁₀₉	C₂	semidirect
6. C₂×#121×C₁₀₉	C₂	semidirect
7. C₂×#120×C₁₀₉	C₂	semidirect
8. C₂×#117×C₁₀₉	C₂	semidirect
11. C₂×#41×C₁₀₉	C₂²	semidirect
12. C₂²×Q₈×C₁₀₉	C₂²	semidirect
17. C₂²×D₄×C₁₀₉	C₂²	semidirect
18. C₂²×D₄×C₁₀₉	C₂²	semidirect
19. C₂²×D₄×C₁₀₉	C₂²	semidirect
21. C₂³×C₄×C₁₀₉	C₂²	semidirect
22. #127×C₁₀₉	C₂²	semidirect
23. #124×C₁₀₉	C₂²	semidirect
24. #121×C₁₀₉	C₂²	semidirect
25. #120×C₁₀₉	C₂²	semidirect
26. #117×C₁₀₉	C₂²	semidirect
27. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
29. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
30. C₂×C₄²×C₁₀₉	C₂²	semidirect
39. C₂×Q₈×C₁₀₉	C₂³	semidirect
49. C₂×D₄×C₁₀₉	C₂³	semidirect
50. C₂×D₄×C₁₀₉	C₂³	semidirect
52. C₂²×C₄×C₁₀₉	D₄	semidirect
58. C₂²×C₄×C₁₀₉	C₂³	semidirect
59. C₄⋊C₄×C₁₀₉	C₂³	semidirect
61. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
62. C₄²×C₁₀₉	C₂³	semidirect
87. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
112. C₂×C₁₀₉	#535	direct (N × H)
115. C₂×#535	C₁₀₉	direct (N × H)
116. C₁₀₉	C₂×#535	direct (N × H)
117. C₂²×#41	C₂×C₁₀₉	semidirect
118. C₂³×D₄	C₂×C₁₀₉	semidirect
119. #535	C₂×C₁₀₉	direct (N × H)
120. C₂×#127	C₂×C₁₀₉	semidirect
121. C₂×#124	C₂×C₁₀₉	semidirect
122. C₂×#121	C₂×C₁₀₉	semidirect
123. C₂×#120	C₂×C₁₀₉	semidirect
124. C₂×#117	C₂×C₁₀₉	semidirect
127. C₂×#41	C₂²×C₁₀₉	semidirect
128. C₂²×Q₈	C₂²×C₁₀₉	semidirect
133. C₂²×D₄	C₂²×C₁₀₉	semidirect
134. C₂²×D₄	C₂²×C₁₀₉	semidirect
135. C₂²×D₄	C₂²×C₁₀₉	semidirect
137. C₂³×C₄	C₂²×C₁₀₉	semidirect
138. #127	C₂²×C₁₀₉	semidirect
139. #124	C₂²×C₁₀₉	semidirect
140. #121	C₂²×C₁₀₉	semidirect
141. #120	C₂²×C₁₀₉	semidirect
142. #117	C₂²×C₁₀₉	semidirect
143. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
145. C₂×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
146. C₂×C₄²	C₂²×C₁₀₉	semidirect
155. C₂×Q₈	C₂³×C₁₀₉	semidirect
165. C₂×D₄	C₂³×C₁₀₉	semidirect
166. C₂×D₄	C₂³×C₁₀₉	semidirect
168. C₂²×C₄	D₄×C₁₀₉	semidirect
174. C₂²×C₄	C₂³×C₁₀₉	semidirect
175. C₄⋊C₄	C₂³×C₁₀₉	semidirect
177. C₂²⋊C₄	C₂³×C₁₀₉	semidirect
178. C₄²	C₂³×C₁₀₉	semidirect
203. C₂×C₄	C₂×D₄×C₁₀₉	semidirect
228. C₂	#535×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×D₄×C₁₀₉	2
#535×C₁₀₉	2
C₂×#127×C₁₀₉	2
C₂×#124×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂×#120×C₁₀₉	2
C₂×#117×C₁₀₉	2
C₂×#535	109

Subgroups — C2×#535×C109 #28360700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#535×C₁₀₉ #28360700