Subgroups — C₁₀₉×#20358 #42138352

Order 27904.

2046 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1742	C₂	1	6	yes	maximal, complemented
4	C₂×#343×C₁₀₉	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
6	#568×C₁₀₉	—	2	8	no	—
7	#568×C₁₀₉	C₄	1	16	yes	complemented
8	C₂×#131×C₁₀₉	—	2	8	no	—
9	C₂²×M₁₆×C₁₀₉	C₂²	1	12	yes	—
10	#520×C₁₀₉	—	2	8	no	—
11	C₂×C₄×Q₈×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄×D₄×C₁₀₉	C₄	1	6	yes	—
13	C₂×#117×C₁₀₉	C₂²	1	3	yes	—
14	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	#343×C₁₀₉	C₂²	1	12	yes	—
16	#343×C₁₀₉	C₂²	1	4	yes	—
17	C₂×#41×C₁₀₉	C₂×C₄	1	8	yes	—
18	C₂×#41×C₁₀₉	D₄	1	8	yes	—
19	C₂×#41×C₁₀₉	D₄	1	4	yes	—
20	C₂×#41×C₁₀₉	Q₈	1	4	yes	—
21	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₁₀₉	C₂×C₄	1	6	yes	—
24	C₂³×C₄×C₁₀₉	C₂³	1	3	yes	—
25	#131×C₁₀₉	—	2	64	no	—
26	C₂×M₁₆×C₁₀₉	—	2	12	no	—
27	C₂×M₁₆×C₁₀₉	—	2	24	no	—
28	C₂×M₁₆×C₁₀₉	C₂×C₄	1	48	yes	complemented
29	C₂×M₁₆×C₁₀₉	C₂³	1	8	yes	—
30	C₂²×C₈×C₁₀₉	—	2	12	no	—
31	C₄×Q₈×C₁₀₉	—	2	8	no	—
32	C₄×D₄×C₁₀₉	—	2	24	no	—
33	#117×C₁₀₉	—	2	12	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	12	no	—
35	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	6	yes	—
36	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
37	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
40	C₂⁴×C₁₀₉	C₂²×C₄	1	3	yes	—
41	#41×C₁₀₉	C₂²⋊C₄	1	32	yes	—
42	#41×C₁₀₉	C₄⋊C₄	1	32	yes	—
43	C₂×Q₈×C₁₀₉	C₄²	1	4	yes	—
44	C₂×Q₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
45	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
46	C₂×D₄×C₁₀₉	C₄²	1	12	yes	—
47	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	12	yes	—
48	C₂×D₄×C₁₀₉	C₄⋊C₄	1	12	yes	—
49	C₂²×C₄×C₁₀₉	—	2	12	no	—
50	C₂²×C₄×C₁₀₉	—	8	6	no	—
51	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₀₉	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
58	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₁₀₉	—	2	48	no	—
60	M₁₆×C₁₀₉	—	2	8	no	—
61	M₁₆×C₁₀₉	C₂²×C₄	1	16	yes	—
62	C₂×C₈×C₁₀₉	—	2	48	no	—
63	C₂×C₈×C₁₀₉	—	2	8	no	—
64	C₄⋊C₄×C₁₀₉	—	2	12	no	—
65	C₄⋊C₄×C₁₀₉	—	2	6	no	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₉	—	2	12	no	—
68	C₄²×C₁₀₉	—	2	12	no	—
69	C₂³×C₁₀₉	—	4	3	no	—
70	C₂³×C₁₀₉	—	4	3	no	—
71	C₂³×C₁₀₉	C₂×C₄²	1	6	yes	—
72	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	6	yes	—
73	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
74	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	#95	1	16	yes	—
76	D₄×C₁₀₉	#95	1	48	yes	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	6	no	—
79	C₂×C₄×C₁₀₉	—	4	6	no	—
80	C₂×C₄×C₁₀₉	—	4	2	no	—
81	C₂×C₄×C₁₀₉	—	8	12	no	—
82	C₂×C₄×C₁₀₉	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	24	yes	—
84	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	24	yes	—
86	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
87	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₁₀₉	—	2	16	no	—
92	C₂²×C₁₀₉	—	2	1	no	—
93	C₂²×C₁₀₉	—	2	6	no	—
94	C₂²×C₁₀₉	—	2	1	no	—
95	C₂²×C₁₀₉	—	2	6	no	—
96	C₂²×C₁₀₉	—	4	6	no	—
97	C₂²×C₁₀₉	—	4	6	no	—
98	C₂²×C₁₀₉	C₂×#95	1	12	yes	—
99	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
100	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₁₀₉	—	4	6	no	—
103	C₄×C₁₀₉	—	4	2	no	—
104	C₄×C₁₀₉	C₂×#95	1	12	yes	—
105	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	2	yes	—
106	C₄×C₁₀₉	C₂²×C₄⋊C₄	1	2	yes	—
107	#20358	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₀₉	—	2	1	no	—
109	C₂×C₁₀₉	—	2	1	no	—
110	C₂×C₁₀₉	—	2	12	no	—
111	C₂×C₁₀₉	C₂²×#95	1	1	yes	characteristic
112	C₂×C₁₀₉	#2888	1	1	yes	characteristic
113	C₂×C₁₀₉	#2889	1	1	yes	characteristic
114	C₂×#568	C₂×C₁₀₉	1	2	yes	—
115	C₂×#520	C₂×C₁₀₉	1	1	yes	characteristic
116	#1742	C₂×C₁₀₉	1	6	yes	complemented
117	C₂×#343	C₂×C₁₀₉	1	6	yes	complemented
118	C₁₀₉	#20358	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
120	#568	—	2	8	no	—
121	#568	C₄×C₁₀₉	1	16	yes	complemented
122	C₂×#131	—	2	8	no	—
123	C₂²×M₁₆	C₂²×C₁₀₉	1	12	yes	—
124	#520	—	2	8	no	—
125	C₂×C₄×Q₈	C₄×C₁₀₉	1	2	yes	—
126	C₂×C₄×D₄	C₄×C₁₀₉	1	6	yes	—
127	C₂×#117	C₂²×C₁₀₉	1	3	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	3	yes	—
129	#343	C₂²×C₁₀₉	1	12	yes	—
130	#343	C₂²×C₁₀₉	1	4	yes	—
131	C₂×#41	C₂×C₄×C₁₀₉	1	8	yes	—
132	C₂×#41	D₄×C₁₀₉	1	8	yes	—
133	C₂×#41	D₄×C₁₀₉	1	4	yes	—
134	C₂×#41	Q₈×C₁₀₉	1	4	yes	—
135	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
136	C₂²×D₄	C₂×C₄×C₁₀₉	1	3	yes	—
137	C₂³×C₄	C₂×C₄×C₁₀₉	1	6	yes	—
138	C₂³×C₄	C₂³×C₁₀₉	1	3	yes	—
139	#131	—	2	64	no	—
140	C₂×M₁₆	—	2	12	no	—
141	C₂×M₁₆	—	2	24	no	—
142	C₂×M₁₆	C₂×C₄×C₁₀₉	1	48	yes	complemented
143	C₂×M₁₆	C₂³×C₁₀₉	1	8	yes	—
144	C₂²×C₈	—	2	12	no	—
145	C₄×Q₈	—	2	8	no	—
146	C₄×D₄	—	2	24	no	—
147	#117	—	2	12	no	—
148	C₂×C₄⋊C₄	—	2	12	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	6	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	6	yes	—
153	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
154	C₂⁴	C₂²×C₄×C₁₀₉	1	3	yes	—
155	#41	C₂²⋊C₄×C₁₀₉	1	32	yes	—
156	#41	C₄⋊C₄×C₁₀₉	1	32	yes	—
157	C₂×Q₈	C₄²×C₁₀₉	1	4	yes	—
158	C₂×Q₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
159	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
160	C₂×D₄	C₄²×C₁₀₉	1	12	yes	—
161	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	12	yes	—
162	C₂×D₄	C₄⋊C₄×C₁₀₉	1	12	yes	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	8	6	no	—
165	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
168	C₂²×C₄	C₂²×C₄×C₁₀₉	1	12	yes	—
169	C₂²×C₄	C₂×D₄×C₁₀₉	1	6	yes	—
170	C₂²×C₄	C₂×D₄×C₁₀₉	1	12	yes	—
171	C₂²×C₄	C₂×Q₈×C₁₀₉	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
173	M₁₆	—	2	48	no	—
174	M₁₆	—	2	8	no	—
175	M₁₆	C₂²×C₄×C₁₀₉	1	16	yes	—
176	C₂×C₈	—	2	48	no	—
177	C₂×C₈	—	2	8	no	—
178	C₄⋊C₄	—	2	12	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	12	no	—
182	C₄²	—	2	12	no	—
183	C₂³	—	4	3	no	—
184	C₂³	—	4	3	no	—
185	C₂³	C₂×C₄²×C₁₀₉	1	6	yes	—
186	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	6	yes	—
187	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
188	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
189	Q₈	#95×C₁₀₉	1	16	yes	—
190	D₄	#95×C₁₀₉	1	48	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	6	no	—
193	C₂×C₄	—	4	6	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	12	no	—
196	C₂×C₄	C₂×C₄²×C₁₀₉	1	6	yes	—
197	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	24	yes	—
198	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	6	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	24	yes	—
200	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	6	yes	—
201	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
202	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
204	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
205	C₈	—	2	16	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	6	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	6	no	—
210	C₂²	—	4	6	no	—
211	C₂²	—	4	6	no	—
212	C₂²	C₂×#95×C₁₀₉	1	12	yes	—
213	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
214	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
215	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
216	C₄	—	4	6	no	—
217	C₄	—	4	2	no	—
218	C₄	C₂×#95×C₁₀₉	1	12	yes	—
219	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	2	yes	—
220	C₄	C₂²×C₄⋊C₄×C₁₀₉	1	2	yes	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	—	2	12	no	—
224	C₂	C₂²×#95×C₁₀₉	1	1	yes	characteristic
225	C₂	C₁₀₉×#2888	1	1	yes	characteristic
226	C₂	C₁₀₉×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1742	C₂	semidirect
4. C₂×#343×C₁₀₉	C₂	semidirect
7. #568×C₁₀₉	C₄	semidirect
28. C₂×M₁₆×C₁₀₉	C₂×C₄	semidirect
107. #20358	C₁₀₉	direct (N × H)
116. #1742	C₂×C₁₀₉	semidirect
117. C₂×#343	C₂×C₁₀₉	semidirect
118. C₁₀₉	#20358	direct (N × H)
121. #568	C₄×C₁₀₉	semidirect
142. C₂×M₁₆	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₁₀₉	2
C₂×#520×C₁₀₉	2
C₁₀₉×#1742	2
C₂×#343×C₁₀₉	2
#20358	109

Subgroups — C109×#20358 #42138352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20358 #42138352