Subgroups — C₁₀₉×#23668 #42141662

Order 27904.

874 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	C₂²×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2317	C₂	1	2	yes	maximal
4	C₁₀₉×#2088	C₂	1	8	yes	maximal, complemented
5	C₁₀₉×#1736	C₂	1	2	yes	maximal
6	C₁₀₉×#1735	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#106×C₁₀₉	—	2	4	no	—
13	C₂×#106×C₁₀₉	C₂²	1	4	yes	—
14	#424×C₁₀₉	—	2	4	no	—
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
16	#389×C₁₀₉	C₂²	1	8	yes	—
17	#385×C₁₀₉	—	2	4	no	—
18	#384×C₁₀₉	—	2	2	no	—
19	#382×C₁₀₉	—	2	2	no	—
20	#340×C₁₀₉	C₂²	1	4	yes	—
21	#340×C₁₀₉	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₀₉	—	2	2	no	—
23	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₂×M₁₆×C₁₀₉	—	2	4	no	—
26	C₂²×C₈×C₁₀₉	—	2	4	no	—
27	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
28	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
30	#117×C₁₀₉	—	2	2	no	—
31	#117×C₁₀₉	—	2	4	no	—
32	#117×C₁₀₉	—	2	2	no	—
33	#117×C₁₀₉	Q₈	1	8	yes	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
38	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
43	C₂×C₄²×C₁₀₉	—	2	2	no	—
44	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
45	#106×C₁₀₉	—	2	8	no	—
46	C₄⋊C₈×C₁₀₉	—	2	8	no	—
47	#95×C₁₀₉	—	2	8	no	—
48	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	—	2	4	no	—
50	C₂²×C₄×C₁₀₉	—	2	4	no	—
51	C₂²×C₄×C₁₀₉	—	4	4	no	—
52	C₂²×C₄×C₁₀₉	—	8	4	no	—
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
59	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₀₉	—	4	4	no	—
61	C₂×C₈×C₁₀₉	—	2	8	no	—
62	C₂×C₈×C₁₀₉	—	2	4	no	—
63	C₂×C₈×C₁₀₉	—	2	2	no	—
64	C₂×C₈×C₁₀₉	—	2	4	no	—
65	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
66	C₄⋊C₄×C₁₀₉	—	2	4	no	—
67	C₄⋊C₄×C₁₀₉	—	2	4	no	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₄⋊C₄×C₁₀₉	—	2	4	no	—
70	C₄⋊C₄×C₁₀₉	—	4	4	no	—
71	C₄⋊C₄×C₁₀₉	—	8	8	no	—
72	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
73	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
75	C₄²×C₁₀₉	—	2	8	no	—
76	C₄²×C₁₀₉	—	2	2	no	—
77	C₄²×C₁₀₉	—	2	2	no	—
78	C₂³×C₁₀₉	—	2	4	no	—
79	C₂³×C₁₀₉	#121	1	2	yes	—
80	C₂³×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
81	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	—	2	4	no	—
83	C₂×C₄×C₁₀₉	—	2	8	no	—
84	C₂×C₄×C₁₀₉	—	2	8	no	—
85	C₂×C₄×C₁₀₉	—	4	4	no	—
86	C₂×C₄×C₁₀₉	—	4	4	no	—
87	C₂×C₄×C₁₀₉	—	4	4	no	—
88	C₂×C₄×C₁₀₉	—	4	8	no	—
89	C₂×C₄×C₁₀₉	—	8	8	no	—
90	C₂×C₄×C₁₀₉	#121	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
92	C₂×C₄×C₁₀₉	#123	1	4	yes	—
93	C₂×C₄×C₁₀₉	#125	1	8	yes	—
94	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	4	yes	—
95	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
99	C₈×C₁₀₉	—	2	4	no	—
100	C₈×C₁₀₉	—	4	2	no	—
101	C₂²×C₁₀₉	—	2	8	no	—
102	C₂²×C₁₀₉	—	2	4	no	—
103	C₂²×C₁₀₉	#398	1	4	yes	—
104	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
105	C₂²×C₁₀₉	C₂×C₂²⋊Q₈	1	2	yes	—
106	C₂²×C₁₀₉	#577	1	4	yes	—
107	C₄×C₁₀₉	—	4	8	no	—
108	C₄×C₁₀₉	—	4	4	no	—
109	C₄×C₁₀₉	—	8	4	no	—
110	C₄×C₁₀₉	#398	1	4	yes	—
111	C₄×C₁₀₉	C₂×#123	1	1	yes	characteristic
112	C₄×C₁₀₉	C₂×#125	1	2	yes	—
113	C₄×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	#23668	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₁₀₉	—	2	4	no	—
116	C₂×C₁₀₉	C₂×#398	1	1	yes	characteristic
117	C₂×C₁₀₉	#3057	1	2	yes	—
118	C₂×C₁₀₉	#3084	1	4	yes	—
119	C₂²×#106	C₂×C₁₀₉	1	1	yes	characteristic
120	C₂×#424	C₂×C₁₀₉	1	1	yes	characteristic
121	#2317	C₂×C₁₀₉	1	2	yes	—
122	#2088	C₂×C₁₀₉	1	8	yes	complemented
123	#1736	C₂×C₁₀₉	1	2	yes	—
124	#1735	C₂×C₁₀₉	1	1	yes	characteristic
125	C₁₀₉	#23668	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
127	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
129	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
131	C₂×#106	—	2	4	no	—
132	C₂×#106	C₂²×C₁₀₉	1	4	yes	—
133	#424	—	2	4	no	—
134	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
135	#389	C₂²×C₁₀₉	1	8	yes	—
136	#385	—	2	4	no	—
137	#384	—	2	2	no	—
138	#382	—	2	2	no	—
139	#340	C₂²×C₁₀₉	1	4	yes	—
140	#340	C₂²×C₁₀₉	1	8	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
147	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
148	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
149	#117	—	2	2	no	—
150	#117	—	2	4	no	—
151	#117	—	2	2	no	—
152	#117	Q₈×C₁₀₉	1	8	yes	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	8	no	—
157	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
158	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
164	#106	—	2	8	no	—
165	C₄⋊C₈	—	2	8	no	—
166	#95	—	2	8	no	—
167	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	—	8	4	no	—
172	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
173	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
176	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
177	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
178	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
179	M₁₆	—	4	4	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	8	8	no	—
191	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	8	yes	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	8	no	—
194	C₄²	—	2	8	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	#121×C₁₀₉	1	2	yes	—
199	C₂³	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
200	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	8	8	no	—
209	C₂×C₄	#121×C₁₀₉	1	2	yes	—
210	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
211	C₂×C₄	#123×C₁₀₉	1	4	yes	—
212	C₂×C₄	#125×C₁₀₉	1	8	yes	—
213	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	4	yes	—
214	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
216	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
217	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	4	2	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	8	no	—
222	C₂²	#398×C₁₀₉	1	4	yes	—
223	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
224	C₂²	C₂×C₂²⋊Q₈×C₁₀₉	1	2	yes	—
225	C₂²	#577×C₁₀₉	1	4	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	4	8	no	—
228	C₄	—	8	4	no	—
229	C₄	#398×C₁₀₉	1	4	yes	—
230	C₄	C₂×#123×C₁₀₉	1	1	yes	characteristic
231	C₄	C₂×#125×C₁₀₉	1	2	yes	—
232	C₄	C₂×C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	C₂×#398×C₁₀₉	1	1	yes	characteristic
235	C₂	C₁₀₉×#3057	1	2	yes	—
236	C₂	C₁₀₉×#3084	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#2088	C₂	semidirect
114. #23668	C₁₀₉	direct (N × H)
122. #2088	C₂×C₁₀₉	semidirect
125. C₁₀₉	#23668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#106×C₁₀₉	2
C₂×#424×C₁₀₉	2
C₁₀₉×#2317	2
C₁₀₉×#2088	2
C₁₀₉×#1736	2
C₁₀₉×#1735	2
#23668	109

Subgroups — C109×#23668 #42141662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#23668 #42141662