Subgroups — C₁₀₉×#12602 #42130596

Order 27904.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#349×C₁₀₉	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1377	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×#117×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
9	#405×C₁₀₉	—	2	2	no	—
10	#405×C₁₀₉	—	2	2	no	—
11	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
12	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#349×C₁₀₉	C₄	1	8	yes	complemented
16	#346×C₁₀₉	—	2	2	no	—
17	C₄×C₁₆×C₁₀₉	—	2	2	no	—
18	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	—	2	2	no	—
21	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₀₉	—	2	1	no	—
23	C₂²×C₈×C₁₀₉	—	2	1	no	—
24	C₂²×C₈×C₁₀₉	—	2	2	no	—
25	C₂²×C₈×C₁₀₉	C₈	1	4	yes	—
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
30	#117×C₁₀₉	—	2	2	no	—
31	#117×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
34	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₁₀₉	—	2	4	no	—
37	M₃₂×C₁₀₉	C₂×C₄	1	8	yes	—
38	C₂×C₁₆×C₁₀₉	—	2	4	no	—
39	#97×C₁₀₉	—	2	2	no	—
40	#97×C₁₀₉	—	2	2	no	—
41	C₄×C₈×C₁₀₉	—	2	1	no	—
42	C₄×C₈×C₁₀₉	—	2	1	no	—
43	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
44	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
45	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	—	2	1	no	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	2	1	no	—
49	C₂²×C₄×C₁₀₉	—	2	1	no	—
50	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₁₀₉	—	2	8	no	—
57	C₂×C₈×C₁₀₉	—	2	1	no	—
58	C₂×C₈×C₁₀₉	—	2	1	no	—
59	C₂×C₈×C₁₀₉	—	2	1	no	—
60	C₂×C₈×C₁₀₉	—	2	1	no	—
61	C₂×C₈×C₁₀₉	—	2	4	no	—
62	C₂×C₈×C₁₀₉	—	2	4	no	—
63	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₁₀₉	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₀₉	#41	1	2	yes	—
72	C₄⋊C₄×C₁₀₉	—	2	4	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₀₉	M₁₆	1	2	yes	—
78	C₁₆×C₁₀₉	—	2	8	no	—
79	C₂³×C₁₀₉	—	2	1	no	—
80	C₂³×C₁₀₉	—	2	1	no	—
81	C₂³×C₁₀₉	—	4	2	no	—
82	C₂³×C₁₀₉	C₄×C₈	1	2	yes	—
83	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₁₀₉	—	2	2	no	—
102	C₈×C₁₀₉	—	2	2	no	—
103	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
104	C₈×C₁₀₉	C₂²⋊C₈	1	2	yes	—
105	C₈×C₁₀₉	C₄×D₄	1	4	yes	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	—	4	2	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₁₀₉	#431	1	1	yes	characteristic
114	C₄×C₁₀₉	—	2	4	no	—
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	—	4	2	no	—
117	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
119	C₄×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₁₀₉	#431	1	1	yes	characteristic
121	#12602	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₁₀₉	—	2	1	no	—
123	C₂×C₁₀₉	—	2	1	no	—
124	C₂×C₁₀₉	—	4	2	no	—
125	C₂×C₁₀₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₁₀₉	#2115	1	2	yes	—
127	C₂×#405	C₂×C₁₀₉	1	1	yes	characteristic
128	C₂×#349	C₂×C₁₀₉	1	2	yes	—
129	C₄×M₃₂	C₂×C₁₀₉	1	2	yes	complemented
130	#1377	C₂×C₁₀₉	1	2	yes	complemented
131	C₁₀₉	#12602	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂³×C₈	C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×#117	C₄×C₁₀₉	1	1	yes	characteristic
135	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
136	#405	—	2	2	no	—
137	#405	—	2	2	no	—
138	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
139	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
142	#349	C₄×C₁₀₉	1	8	yes	complemented
143	#346	—	2	2	no	—
144	C₄×C₁₆	—	2	2	no	—
145	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	C₈×C₁₀₉	1	4	yes	—
153	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	C₂×C₄⋊C₄	C₈×C₁₀₉	1	2	yes	—
160	C₂×C₂²⋊C₄	C₈×C₁₀₉	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
163	M₃₂	—	2	4	no	—
164	M₃₂	C₂×C₄×C₁₀₉	1	8	yes	—
165	C₂×C₁₆	—	2	4	no	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
171	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
172	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
178	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
180	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
181	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
183	M₁₆	—	2	8	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
191	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
193	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
194	C₂×C₈	C₂×C₈×C₁₀₉	1	2	yes	—
195	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
197	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
198	C₂×C₈	#41×C₁₀₉	1	2	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
204	C₄²	M₁₆×C₁₀₉	1	2	yes	—
205	C₁₆	—	2	8	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₄×C₈×C₁₀₉	1	2	yes	—
210	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
221	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	4	yes	—
222	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
227	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
231	C₈	C₂²⋊C₈×C₁₀₉	1	2	yes	—
232	C₈	C₄×D₄×C₁₀₉	1	4	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
239	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
240	C₂²	#431×C₁₀₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
245	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
246	C₄	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
247	C₄	#431×C₁₀₉	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
252	C₂	C₁₀₉×#2115	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₀₉	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×M₃₂×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1377	C₂	semidirect
15. #349×C₁₀₉	C₄	semidirect
121. #12602	C₁₀₉	direct (N × H)
129. C₄×M₃₂	C₂×C₁₀₉	semidirect
130. #1377	C₂×C₁₀₉	semidirect
131. C₁₀₉	#12602	direct (N × H)
142. #349	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×#405×C₁₀₉	2
C₂×#349×C₁₀₉	2
C₄×M₃₂×C₁₀₉	2
C₁₀₉×#1377	2
#12602	109

Subgroups — C109×#12602 #42130596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12602 #42130596