Subgroups — C₁₀₉×#7874 #42125868

Order 27904.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1277	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₀₉	—	2	2	no	—
5	C₂³×C₈×C₁₀₉	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
7	#423×C₁₀₉	—	2	4	no	—
8	#423×C₁₀₉	C₄	1	8	yes	—
9	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₀₉	—	2	4	no	—
12	C₂×C₄×C₈×C₁₀₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
17	C₂×M₁₆×C₁₀₉	—	2	8	no	—
18	C₂×M₁₆×C₁₀₉	—	4	4	no	—
19	C₂×M₁₆×C₁₀₉	—	4	4	no	—
20	C₂²×C₈×C₁₀₉	—	2	4	no	—
21	C₂²×C₈×C₁₀₉	—	2	8	no	—
22	C₂²×C₈×C₁₀₉	—	2	4	no	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	D₄	1	4	yes	—
28	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₁₀₉	—	2	8	no	—
31	C₄⋊C₈×C₁₀₉	—	2	4	no	—
32	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₁₀₉	—	2	4	no	—
34	C₄×C₈×C₁₀₉	—	2	4	no	—
35	C₂⁴×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	—	2	4	no	—
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
46	C₂²×C₄×C₁₀₉	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	M₁₆×C₁₀₉	—	4	16	no	—
52	C₂×C₈×C₁₀₉	—	2	16	no	—
53	C₂×C₈×C₁₀₉	—	2	4	no	—
54	C₂×C₈×C₁₀₉	—	2	4	no	—
55	C₂×C₈×C₁₀₉	—	2	8	no	—
56	C₂×C₈×C₁₀₉	—	4	4	no	—
57	C₂×C₈×C₁₀₉	—	4	4	no	—
58	C₂×C₈×C₁₀₉	M₁₆	1	8	yes	complemented
59	C₄²×C₁₀₉	C₄²	1	2	yes	—
60	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₁₀₉	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₁₀₉	C₄⋊C₄	1	4	yes	—
63	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
64	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	—	2	2	no	—
67	C₂³×C₁₀₉	#95	1	2	yes	—
68	C₂³×C₁₀₉	#106	1	2	yes	—
69	C₂³×C₁₀₉	#107	1	2	yes	—
70	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	8	no	—
74	C₂×C₄×C₁₀₉	—	2	4	no	—
75	C₂×C₄×C₁₀₉	—	2	4	no	—
76	C₂×C₄×C₁₀₉	#95	1	2	yes	—
77	C₂×C₄×C₁₀₉	#95	1	2	yes	—
78	C₂×C₄×C₁₀₉	#95	1	2	yes	—
79	C₂×C₄×C₁₀₉	#102	1	8	yes	—
80	C₂×C₄×C₁₀₉	#103	1	8	yes	—
81	C₂×C₄×C₁₀₉	#106	1	2	yes	—
82	C₂×C₄×C₁₀₉	#107	1	2	yes	—
83	C₂×C₄×C₁₀₉	#108	1	4	yes	—
84	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×M₁₆	1	4	yes	—
89	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
92	C₈×C₁₀₉	—	2	8	no	—
93	C₈×C₁₀₉	—	4	8	no	—
94	C₂²×C₁₀₉	—	2	4	no	—
95	C₂²×C₁₀₉	—	2	4	no	—
96	C₂²×C₁₀₉	—	2	2	no	—
97	C₂²×C₁₀₉	—	2	2	no	—
98	C₂²×C₁₀₉	#340	1	4	yes	—
99	C₂²×C₁₀₉	C₂×#95	1	1	yes	characteristic
100	C₂²×C₁₀₉	#423	1	2	yes	—
101	C₂²×C₁₀₉	C₂×#106	1	1	yes	characteristic
102	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
103	C₂²×C₁₀₉	C₂×#108	1	2	yes	—
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	2	no	—
106	C₄×C₁₀₉	#340	1	4	yes	—
107	C₄×C₁₀₉	#341	1	4	yes	—
108	C₄×C₁₀₉	C₄×M₁₆	1	2	yes	—
109	C₄×C₁₀₉	#407	1	2	yes	—
110	C₄×C₁₀₉	C₂×#102	1	2	yes	—
111	C₄×C₁₀₉	C₂×#103	1	2	yes	—
112	#7874	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₉	—	2	4	no	—
114	C₂×C₁₀₉	#1566	1	2	yes	—
115	C₂×C₁₀₉	#1567	1	2	yes	—
116	C₂×C₁₀₉	#1728	1	1	yes	characteristic
117	C₂×C₁₀₉	C₂×#340	1	1	yes	characteristic
118	C₂×C₁₀₉	C₂×#341	1	1	yes	characteristic
119	C₂×#423	C₂×C₁₀₉	1	2	yes	—
120	C₂²×C₄×C₈	C₂×C₁₀₉	1	1	yes	characteristic
121	#1277	C₂×C₁₀₉	1	4	yes	complemented
122	C₁₀₉	#7874	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×M₁₆	—	2	2	no	—
124	C₂³×C₈	C₄×C₁₀₉	1	2	yes	—
125	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
126	#423	—	2	4	no	—
127	#423	C₄×C₁₀₉	1	8	yes	—
128	C₂×C₄⋊C₈	—	2	2	no	—
129	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	C₄×C₁₀₉	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
133	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
134	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
135	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
136	C₂×M₁₆	—	2	8	no	—
137	C₂×M₁₆	—	4	4	no	—
138	C₂×M₁₆	—	4	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
143	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
145	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄²	D₄×C₁₀₉	1	4	yes	—
147	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
149	C₄⋊C₈	—	2	8	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	C₄²×C₁₀₉	1	2	yes	—
158	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
159	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
160	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
162	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
164	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
165	C₂²×C₄	SD₁₆×C₁₀₉	1	4	yes	—
166	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
170	M₁₆	—	4	16	no	—
171	C₂×C₈	—	2	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₂×C₈	M₁₆×C₁₀₉	1	8	yes	complemented
178	C₄²	C₄²×C₁₀₉	1	2	yes	—
179	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
180	C₄²	C₂²⋊C₄×C₁₀₉	1	4	yes	—
181	C₄²	C₄⋊C₄×C₁₀₉	1	4	yes	—
182	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
183	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#95×C₁₀₉	1	2	yes	—
187	C₂³	#106×C₁₀₉	1	2	yes	—
188	C₂³	#107×C₁₀₉	1	2	yes	—
189	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	#95×C₁₀₉	1	2	yes	—
196	C₂×C₄	#95×C₁₀₉	1	2	yes	—
197	C₂×C₄	#95×C₁₀₉	1	2	yes	—
198	C₂×C₄	#102×C₁₀₉	1	8	yes	—
199	C₂×C₄	#103×C₁₀₉	1	8	yes	—
200	C₂×C₄	#106×C₁₀₉	1	2	yes	—
201	C₂×C₄	#107×C₁₀₉	1	2	yes	—
202	C₂×C₄	#108×C₁₀₉	1	4	yes	—
203	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
207	C₂×C₄	C₂×M₁₆×C₁₀₉	1	4	yes	—
208	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
210	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	4	8	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#340×C₁₀₉	1	4	yes	—
218	C₂²	C₂×#95×C₁₀₉	1	1	yes	characteristic
219	C₂²	#423×C₁₀₉	1	2	yes	—
220	C₂²	C₂×#106×C₁₀₉	1	1	yes	characteristic
221	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
222	C₂²	C₂×#108×C₁₀₉	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#340×C₁₀₉	1	4	yes	—
226	C₄	#341×C₁₀₉	1	4	yes	—
227	C₄	C₄×M₁₆×C₁₀₉	1	2	yes	—
228	C₄	#407×C₁₀₉	1	2	yes	—
229	C₄	C₂×#102×C₁₀₉	1	2	yes	—
230	C₄	C₂×#103×C₁₀₉	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	C₁₀₉×#1566	1	2	yes	—
233	C₂	C₁₀₉×#1567	1	2	yes	—
234	C₂	C₁₀₉×#1728	1	1	yes	characteristic
235	C₂	C₂×#340×C₁₀₉	1	1	yes	characteristic
236	C₂	C₂×#341×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#1277	C₂	semidirect
58. C₂×C₈×C₁₀₉	M₁₆	semidirect
112. #7874	C₁₀₉	direct (N × H)
121. #1277	C₂×C₁₀₉	semidirect
122. C₁₀₉	#7874	direct (N × H)
177. C₂×C₈	M₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₀₉	2
C₂²×C₄×C₈×C₁₀₉	2
C₁₀₉×#1277	2
#7874	109

Subgroups — C109×#7874 #42125868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7874 #42125868