Subgroups — C₁₀₉×#8598 #42126592

Order 27904.

478 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×#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1797	C₂	1	2	yes	maximal
3	C₂×#334×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1277	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
6	#431×C₁₀₉	—	2	4	no	—
7	C₂×C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₈×C₁₀₉	—	2	2	no	—
10	C₂×C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#336×C₁₀₉	—	2	2	no	—
14	#334×C₁₀₉	—	2	4	no	—
15	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	—	2	4	no	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₀₉	—	2	2	no	—
22	C₂×C₄²×C₁₀₉	—	2	2	no	—
23	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₁₀₉	—	2	4	no	—
28	C₄⋊C₈×C₁₀₉	—	2	4	no	—
29	C₄⋊C₈×C₁₀₉	—	2	4	no	—
30	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
31	C₂²⋊C₈×C₁₀₉	—	4	4	no	—
32	C₄×C₈×C₁₀₉	—	2	2	no	—
33	C₄×C₈×C₁₀₉	D₄	1	4	yes	—
34	C₂⁴×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	—	2	4	no	—
41	C₂²×C₄×C₁₀₉	—	4	2	no	—
42	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	—	2	4	no	—
50	C₂×C₈×C₁₀₉	—	2	2	no	—
51	C₂×C₈×C₁₀₉	—	2	4	no	—
52	C₂×C₈×C₁₀₉	—	4	4	no	—
53	C₂×C₈×C₁₀₉	—	4	4	no	—
54	C₂×C₈×C₁₀₉	—	4	4	no	—
55	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₁₀₉	—	2	1	no	—
57	C₄²×C₁₀₉	—	2	1	no	—
58	C₄²×C₁₀₉	—	2	2	no	—
59	C₄²×C₁₀₉	—	2	2	no	—
60	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₄²×C₁₀₉	#41	1	2	yes	—
62	C₂³×C₁₀₉	—	2	1	no	—
63	C₂³×C₁₀₉	—	2	2	no	—
64	C₂³×C₁₀₉	—	2	1	no	—
65	C₂³×C₁₀₉	—	2	2	no	—
66	C₂³×C₁₀₉	#106	1	2	yes	—
67	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	—	2	1	no	—
69	C₂×C₄×C₁₀₉	—	2	1	no	—
70	C₂×C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₄×C₁₀₉	—	2	8	no	—
72	C₂×C₄×C₁₀₉	—	2	1	no	—
73	C₂×C₄×C₁₀₉	—	2	1	no	—
74	C₂×C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	—	4	4	no	—
77	C₂×C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄×C₁₀₉	—	4	2	no	—
79	C₂×C₄×C₁₀₉	#106	1	2	yes	—
80	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	#121	1	2	yes	—
83	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₀₉	#131	1	4	yes	—
85	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₁₀₉	#135	1	2	yes	—
87	C₈×C₁₀₉	—	2	4	no	—
88	C₈×C₁₀₉	—	4	2	no	—
89	C₈×C₁₀₉	—	8	4	no	—
90	C₂²×C₁₀₉	—	2	2	no	—
91	C₂²×C₁₀₉	—	2	4	no	—
92	C₂²×C₁₀₉	—	4	1	no	—
93	C₂²×C₁₀₉	—	4	1	no	—
94	C₂²×C₁₀₉	—	4	2	no	—
95	C₂²×C₁₀₉	#380	1	1	yes	characteristic
96	C₂²×C₁₀₉	#424	1	2	yes	—
97	C₂²×C₁₀₉	C₂×#106	1	1	yes	characteristic
98	C₂²×C₁₀₉	#427	1	1	yes	characteristic
99	C₂²×C₁₀₉	#430	1	2	yes	—
100	C₄×C₁₀₉	—	2	2	no	—
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	4	1	no	—
103	C₄×C₁₀₉	—	4	1	no	—
104	C₄×C₁₀₉	#408	1	2	yes	—
105	C₄×C₁₀₉	#433	1	2	yes	—
106	C₄×C₁₀₉	#465	1	4	yes	—
107	#8598	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₁₀₉	—	4	2	no	—
109	C₂×C₁₀₉	#1570	1	2	yes	—
110	C₂×C₁₀₉	#1572	1	2	yes	—
111	C₂×C₁₀₉	#1798	1	1	yes	characteristic
112	C₂×C₁₀₉	#1807	1	1	yes	characteristic
113	C₂×C₁₀₉	#1811	1	1	yes	characteristic
114	C₂×#431	C₂×C₁₀₉	1	1	yes	characteristic
115	#1797	C₂×C₁₀₉	1	2	yes	—
116	C₂×#334	C₂×C₁₀₉	1	2	yes	complemented
117	#1277	C₂×C₁₀₉	1	2	yes	complemented
118	C₁₀₉	#8598	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
120	#431	—	2	4	no	—
121	C₂×C₄⋊C₈	C₄×C₁₀₉	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
123	C₂×C₂²⋊C₈	—	2	2	no	—
124	C₂×C₂²⋊C₈	C₄×C₁₀₉	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
127	#336	—	2	2	no	—
128	#334	—	2	4	no	—
129	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
130	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
131	C₂³×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
134	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
138	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄²	Q₈×C₁₀₉	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	—	4	4	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	D₄×C₁₀₉	1	4	yes	—
148	C₂⁴	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
159	C₂²×C₄	SD₁₆×C₁₀₉	1	4	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
175	C₄²	#41×C₁₀₉	1	2	yes	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	2	no	—
180	C₂³	#106×C₁₀₉	1	2	yes	—
181	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#106×C₁₀₉	1	2	yes	—
194	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
195	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄	#121×C₁₀₉	1	2	yes	—
197	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
198	C₂×C₄	#131×C₁₀₉	1	4	yes	—
199	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
200	C₂×C₄	#135×C₁₀₉	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	4	2	no	—
203	C₈	—	8	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	#380×C₁₀₉	1	1	yes	characteristic
210	C₂²	#424×C₁₀₉	1	2	yes	—
211	C₂²	C₂×#106×C₁₀₉	1	1	yes	characteristic
212	C₂²	#427×C₁₀₉	1	1	yes	characteristic
213	C₂²	#430×C₁₀₉	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	#408×C₁₀₉	1	2	yes	—
219	C₄	#433×C₁₀₉	1	2	yes	—
220	C₄	#465×C₁₀₉	1	4	yes	—
221	C₂	—	4	2	no	—
222	C₂	C₁₀₉×#1570	1	2	yes	—
223	C₂	C₁₀₉×#1572	1	2	yes	—
224	C₂	C₁₀₉×#1798	1	1	yes	characteristic
225	C₂	C₁₀₉×#1807	1	1	yes	characteristic
226	C₂	C₁₀₉×#1811	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	C₄⋊C₄
LCS	G ▷ C₂²×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. C₂×#334×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1277	C₂	semidirect
107. #8598	C₁₀₉	direct (N × H)
116. C₂×#334	C₂×C₁₀₉	semidirect
117. #1277	C₂×C₁₀₉	semidirect
118. C₁₀₉	#8598	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₀₉	2
C₁₀₉×#1797	2
C₂×#334×C₁₀₉	2
C₁₀₉×#1277	2
#8598	109

Subgroups — C109×#8598 #42126592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8598 #42126592