Subgroups — C₂×#547×C₁₀₉ #28360718

Order 13952.

1632 conjugacy classes of subgroups.

The subgroup lattice has 240 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	#547×C₁₀₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#123×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂×#121×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	C₂×#41×C₁₀₉	—	2	16	no	—
11	C₂×#41×C₁₀₉	C₂²	1	16	yes	complemented
12	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	—
15	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₁₀₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
20	#123×C₁₀₉	C₂²	1	32	yes	complemented
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	16	yes	complemented
22	#121×C₁₀₉	C₂²	1	16	yes	complemented
23	C₄×D₄×C₁₀₉	C₂²	1	8	yes	complemented
24	C₄×D₄×C₁₀₉	C₂²	1	8	yes	complemented
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
28	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	16	yes	complemented
31	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
34	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
36	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
37	#41×C₁₀₉	—	2	64	no	—
38	C₂×Q₈×C₁₀₉	—	2	8	no	—
39	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
40	C₂×D₄×C₁₀₉	—	2	16	no	—
41	C₂×D₄×C₁₀₉	—	2	4	no	—
42	C₂×D₄×C₁₀₉	D₄	1	8	yes	complemented
43	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
44	C₂×D₄×C₁₀₉	C₂³	1	16	yes	—
45	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₁₀₉	—	2	16	no	—
47	C₂²×C₄×C₁₀₉	—	2	8	no	—
48	C₂²×C₄×C₁₀₉	—	2	8	no	—
49	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	16	yes	—
51	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
52	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
59	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
61	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
62	C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
63	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
64	C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
65	C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
66	C₄²×C₁₀₉	C₂³	1	4	yes	—
67	C₂³×C₁₀₉	—	2	4	no	—
68	C₂³×C₁₀₉	—	2	4	no	—
69	C₂³×C₁₀₉	—	2	4	no	—
70	C₂³×C₁₀₉	—	2	4	no	—
71	C₂³×C₁₀₉	—	2	4	no	—
72	C₂³×C₁₀₉	C₂×D₄	1	8	yes	—
73	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
74	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
75	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	2	16	no	—
77	D₄×C₁₀₉	—	2	32	no	—
78	D₄×C₁₀₉	—	2	8	no	—
79	D₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
80	C₂×C₄×C₁₀₉	—	2	4	no	—
81	C₂×C₄×C₁₀₉	—	2	8	no	—
82	C₂×C₄×C₁₀₉	—	2	32	no	—
83	C₂×C₄×C₁₀₉	—	2	4	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	16	no	—
86	C₂×C₄×C₁₀₉	—	2	16	no	—
87	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₀₉	#41	1	4	yes	complemented
89	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
91	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
92	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
93	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
95	C₂²×C₁₀₉	—	2	16	no	—
96	C₂²×C₁₀₉	—	2	4	no	—
97	C₂²×C₁₀₉	—	2	4	no	—
98	C₂²×C₁₀₉	—	2	4	no	—
99	C₂²×C₁₀₉	—	2	4	no	—
100	C₂²×C₁₀₉	—	2	4	no	—
101	C₂²×C₁₀₉	—	4	8	no	—
102	C₂²×C₁₀₉	C₂²×D₄	1	8	yes	—
103	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
104	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
105	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
106	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	4	no	—
108	C₄×C₁₀₉	—	2	8	no	—
109	C₄×C₁₀₉	—	2	4	no	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
112	C₄×C₁₀₉	C₂×#41	1	4	yes	complemented
113	C₂×C₁₀₉	—	2	8	no	—
114	C₂×C₁₀₉	—	4	4	no	—
115	C₂×C₁₀₉	#547	1	4	yes	direct product factor
116	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×#547	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₁₀₉	C₂×#547	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×#41	C₂×C₁₀₉	1	2	yes	complemented
122	#547	C₂×C₁₀₉	1	16	yes	direct product factor
123	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
124	C₂×#123	C₂×C₁₀₉	1	4	yes	complemented
125	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	2	yes	complemented
126	C₂×#121	C₂×C₁₀₉	1	2	yes	complemented
127	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	C₂×C₄×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	C₂²×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	complemented
130	C₂×#41	—	2	16	no	—
131	C₂×#41	C₂²×C₁₀₉	1	16	yes	complemented
132	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
133	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
134	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	—
135	C₂²×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
137	C₂³×C₄	C₂²×C₁₀₉	1	4	yes	—
138	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₁₀₉	1	8	yes	complemented
140	#123	C₂²×C₁₀₉	1	32	yes	complemented
141	C₂²⋊Q₈	C₂²×C₁₀₉	1	16	yes	complemented
142	#121	C₂²×C₁₀₉	1	16	yes	complemented
143	C₄×D₄	C₂²×C₁₀₉	1	8	yes	complemented
144	C₄×D₄	C₂²×C₁₀₉	1	8	yes	complemented
145	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
147	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
148	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	16	yes	complemented
151	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
154	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
156	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
157	#41	—	2	64	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	C₂³×C₁₀₉	1	8	yes	—
160	C₂×D₄	—	2	16	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	D₄×C₁₀₉	1	8	yes	complemented
163	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
164	C₂×D₄	C₂³×C₁₀₉	1	16	yes	—
165	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
171	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
172	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
173	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
174	C₂²×C₄	C₂³×C₁₀₉	1	16	yes	—
175	C₂²×C₄	C₂³×C₁₀₉	1	8	yes	—
176	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
179	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
180	C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	—
181	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
182	C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
183	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
184	C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
185	C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
186	C₄²	C₂³×C₁₀₉	1	4	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×D₄×C₁₀₉	1	8	yes	—
193	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
194	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
195	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
196	Q₈	—	2	16	no	—
197	D₄	—	2	32	no	—
198	D₄	—	2	8	no	—
199	D₄	C₂×D₄×C₁₀₉	1	16	yes	complemented
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
208	C₂×C₄	#41×C₁₀₉	1	4	yes	complemented
209	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
210	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
211	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
212	C₂×C₄	C₂⁴×C₁₀₉	1	8	yes	—
213	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
214	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
215	C₂²	—	2	16	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	4	8	no	—
222	C₂²	C₂²×D₄×C₁₀₉	1	8	yes	—
223	C₂²	C₂²×D₄×C₁₀₉	1	2	yes	—
224	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
225	C₂²	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
226	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	8	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
232	C₄	C₂×#41×C₁₀₉	1	4	yes	complemented
233	C₂	—	2	8	no	—
234	C₂	—	4	4	no	—
235	C₂	#547×C₁₀₉	1	4	yes	direct product factor
236	C₂	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
237	C₂	C₂²×#41×C₁₀₉	1	1	yes	characteristic
238	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. #547×C₁₀₉	C₂	direct (N × H)
3. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
4. C₂×#123×C₁₀₉	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₁₀₉	C₂	semidirect
6. C₂×#121×C₁₀₉	C₂	semidirect
7. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
8. C₂×C₄×D₄×C₁₀₉	C₂	semidirect
9. C₂²×C₄⋊C₄×C₁₀₉	C₂	semidirect
11. C₂×#41×C₁₀₉	C₂²	semidirect
19. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
20. #123×C₁₀₉	C₂²	semidirect
21. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
22. #121×C₁₀₉	C₂²	semidirect
23. C₄×D₄×C₁₀₉	C₂²	semidirect
24. C₄×D₄×C₁₀₉	C₂²	semidirect
30. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
42. C₂×D₄×C₁₀₉	D₄	semidirect
79. D₄×C₁₀₉	C₂×D₄	semidirect
88. C₂×C₄×C₁₀₉	#41	semidirect
112. C₄×C₁₀₉	C₂×#41	semidirect
115. C₂×C₁₀₉	#547	direct (N × H)
119. C₂×#547	C₁₀₉	direct (N × H)
120. C₁₀₉	C₂×#547	direct (N × H)
121. C₂²×#41	C₂×C₁₀₉	semidirect
122. #547	C₂×C₁₀₉	direct (N × H)
123. C₂×C₄⋊Q₈	C₂×C₁₀₉	semidirect
124. C₂×#123	C₂×C₁₀₉	semidirect
125. C₂×C₂²⋊Q₈	C₂×C₁₀₉	semidirect
126. C₂×#121	C₂×C₁₀₉	semidirect
127. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
128. C₂×C₄×D₄	C₂×C₁₀₉	semidirect
129. C₂²×C₄⋊C₄	C₂×C₁₀₉	semidirect
131. C₂×#41	C₂²×C₁₀₉	semidirect
139. C₄⋊Q₈	C₂²×C₁₀₉	semidirect
140. #123	C₂²×C₁₀₉	semidirect
141. C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
142. #121	C₂²×C₁₀₉	semidirect
143. C₄×D₄	C₂²×C₁₀₉	semidirect
144. C₄×D₄	C₂²×C₁₀₉	semidirect
150. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
162. C₂×D₄	D₄×C₁₀₉	semidirect
199. D₄	C₂×D₄×C₁₀₉	semidirect
208. C₂×C₄	#41×C₁₀₉	semidirect
232. C₄	C₂×#41×C₁₀₉	semidirect
235. C₂	#547×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
#547×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#123×C₁₀₉	2
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂×#121×C₁₀₉	2
C₂×C₄×D₄×C₁₀₉	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂×#547	109

Subgroups — C2×#547×C109 #28360718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#547×C₁₀₉ #28360718