Subgroups — C₁₀₉×#12607 #42130601

Order 27904.

494 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#349×C₁₀₉	C₂	1	2	yes	maximal
3	C₂×#347×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1376	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
9	C₄×M₁₆×C₁₀₉	—	2	4	no	—
10	C₂×#97×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#349×C₁₀₉	C₄	1	8	yes	complemented
15	#347×C₁₀₉	—	2	4	no	—
16	C₂³×C₄×C₁₀₉	—	2	1	no	—
17	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	—	2	2	no	—
20	C₂×M₁₆×C₁₀₉	—	2	2	no	—
21	C₂×M₁₆×C₁₀₉	—	2	2	no	—
22	C₂×M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	—	2	1	no	—
29	C₂×C₄²×C₁₀₉	—	2	2	no	—
30	C₂×C₄²×C₁₀₉	—	2	2	no	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₁₀₉	—	2	4	no	—
34	M₃₂×C₁₀₉	C₂×C₄	1	8	yes	complemented
35	C₂×C₁₆×C₁₀₉	—	2	4	no	—
36	#97×C₁₀₉	—	2	2	no	—
37	#97×C₁₀₉	D₄	1	4	yes	—
38	C₄×C₈×C₁₀₉	—	2	2	no	—
39	C₄×C₈×C₁₀₉	—	2	2	no	—
40	C₂⁴×C₁₀₉	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	—	2	1	no	—
42	C₂²×C₄×C₁₀₉	—	2	1	no	—
43	C₂²×C₄×C₁₀₉	—	2	1	no	—
44	C₂²×C₄×C₁₀₉	—	2	1	no	—
45	C₂²×C₄×C₁₀₉	—	2	1	no	—
46	C₂²×C₄×C₁₀₉	—	2	4	no	—
47	C₂²×C₄×C₁₀₉	—	2	1	no	—
48	C₂²×C₄×C₁₀₉	—	4	4	no	—
49	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
50	C₂²×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₁₀₉	—	2	8	no	—
54	M₁₆×C₁₀₉	—	2	8	no	—
55	C₂×C₈×C₁₀₉	—	2	1	no	—
56	C₂×C₈×C₁₀₉	—	2	1	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	C₄²	1	2	yes	—
62	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₀₉	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	#41	1	2	yes	—
67	C₄²×C₁₀₉	—	2	4	no	—
68	C₄²×C₁₀₉	—	2	1	no	—
69	C₄²×C₁₀₉	—	2	1	no	—
70	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₀₉	M₁₆	1	2	yes	—
72	C₁₆×C₁₀₉	—	2	8	no	—
73	C₂³×C₁₀₉	—	2	1	no	—
74	C₂³×C₁₀₉	—	2	1	no	—
75	C₂³×C₁₀₉	—	4	2	no	—
76	C₂³×C₁₀₉	#97	1	2	yes	—
77	C₂³×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	—	2	4	no	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	4	8	no	—
88	C₂×C₄×C₁₀₉	—	4	2	no	—
89	C₂×C₄×C₁₀₉	#97	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₁₀₉	—	2	2	no	—
97	C₈×C₁₀₉	—	2	2	no	—
98	C₈×C₁₀₉	—	2	2	no	—
99	C₈×C₁₀₉	C₄×D₄	1	4	yes	—
100	C₂²×C₁₀₉	—	2	2	no	—
101	C₂²×C₁₀₉	—	2	2	no	—
102	C₂²×C₁₀₉	—	4	2	no	—
103	C₂²×C₁₀₉	—	4	2	no	—
104	C₂²×C₁₀₉	—	4	2	no	—
105	C₂²×C₁₀₉	C₂×#97	1	1	yes	characteristic
106	C₂²×C₁₀₉	#407	1	1	yes	characteristic
107	C₂²×C₁₀₉	#432	1	1	yes	characteristic
108	C₄×C₁₀₉	—	2	4	no	—
109	C₄×C₁₀₉	—	2	2	no	—
110	C₄×C₁₀₉	—	4	2	no	—
111	C₄×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
113	C₄×C₁₀₉	#407	1	1	yes	characteristic
114	C₄×C₁₀₉	#432	1	1	yes	characteristic
115	#12607	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂×C₁₀₉	—	2	1	no	—
117	C₂×C₁₀₉	—	2	1	no	—
118	C₂×C₁₀₉	—	4	2	no	—
119	C₂×C₁₀₉	#1749	1	1	yes	characteristic
120	C₂×C₁₀₉	#2165	1	2	yes	—
121	C₂×C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
122	C₂×#349	C₂×C₁₀₉	1	2	yes	—
123	C₂×#347	C₂×C₁₀₉	1	2	yes	complemented
124	#1376	C₂×C₁₀₉	1	2	yes	complemented
125	C₁₀₉	#12607	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×M₁₆	C₄×C₁₀₉	1	1	yes	characteristic
127	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂²×C₄²	C₄×C₁₀₉	1	1	yes	characteristic
129	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
130	C₄×M₁₆	—	2	4	no	—
131	C₂×#97	C₄×C₁₀₉	1	1	yes	characteristic
132	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
135	#349	C₄×C₁₀₉	1	8	yes	complemented
136	#347	—	2	4	no	—
137	C₂³×C₄	—	2	1	no	—
138	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
145	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
147	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
154	M₃₂	—	2	4	no	—
155	M₃₂	C₂×C₄×C₁₀₉	1	8	yes	complemented
156	C₂×C₁₆	—	2	4	no	—
157	#97	—	2	2	no	—
158	#97	D₄×C₁₀₉	1	4	yes	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₂⁴	M₁₆×C₁₀₉	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
171	C₂²×C₄	M₁₆×C₁₀₉	1	1	yes	characteristic
172	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
174	M₁₆	—	2	8	no	—
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
183	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
184	C₂×C₈	M₁₆×C₁₀₉	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
186	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
187	C₂×C₈	#41×C₁₀₉	1	2	yes	—
188	C₄²	—	2	4	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
192	C₄²	M₁₆×C₁₀₉	1	2	yes	—
193	C₁₆	—	2	8	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#97×C₁₀₉	1	2	yes	—
198	C₂³	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	#97×C₁₀₉	1	2	yes	—
211	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
213	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	C₄×D₄×C₁₀₉	1	4	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂×#97×C₁₀₉	1	1	yes	characteristic
227	C₂²	#407×C₁₀₉	1	1	yes	characteristic
228	C₂²	#432×C₁₀₉	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₄×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
233	C₄	C₄×M₁₆×C₁₀₉	1	1	yes	characteristic
234	C₄	#407×C₁₀₉	1	1	yes	characteristic
235	C₄	#432×C₁₀₉	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	—	4	2	no	—
239	C₂	C₁₀₉×#1749	1	1	yes	characteristic
240	C₂	C₁₀₉×#2165	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#117
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#432
prime radical	C₂⁴×C₁₀₉	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#97×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#347×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1376	C₂	semidirect
14. #349×C₁₀₉	C₄	semidirect
34. M₃₂×C₁₀₉	C₂×C₄	semidirect
115. #12607	C₁₀₉	direct (N × H)
123. C₂×#347	C₂×C₁₀₉	semidirect
124. #1376	C₂×C₁₀₉	semidirect
125. C₁₀₉	#12607	direct (N × H)
135. #349	C₄×C₁₀₉	semidirect
155. M₃₂	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×C₄×M₁₆×C₁₀₉	2
C₂×#349×C₁₀₉	2
C₂×#347×C₁₀₉	2
C₁₀₉×#1376	2
#12607	109

Subgroups — C109×#12607 #42130601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12607 #42130601