Subgroups — C₁₀₉×#8635 #42126629

Order 27904.

320 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1537	C₂	1	2	yes	maximal
5	C₁₀₉×#1307	C₂	1	2	yes	maximal
6	#533×C₁₀₉	C₂²	1	1	yes	characteristic
7	#446×C₁₀₉	—	2	1	no	—
8	#446×C₁₀₉	—	2	1	no	—
9	C₈×Q₈×C₁₀₉	—	2	1	no	—
10	C₈×Q₈×C₁₀₉	—	2	1	no	—
11	#432×C₁₀₉	C₂²	1	2	yes	—
12	#432×C₁₀₉	C₂²	1	2	yes	—
13	#424×C₁₀₉	C₄	1	2	yes	—
14	#423×C₁₀₉	C₂²	1	1	yes	characteristic
15	#405×C₁₀₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#335×C₁₀₉	—	2	1	no	—
18	#335×C₁₀₉	—	2	1	no	—
19	#334×C₁₀₉	—	2	1	no	—
20	#334×C₁₀₉	—	2	1	no	—
21	#330×C₁₀₉	—	2	4	no	—
22	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₄⋊Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
27	#125×C₁₀₉	D₄	1	2	yes	—
28	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
29	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
30	C₄×Q₈×C₁₀₉	—	2	1	no	—
31	C₄×Q₈×C₁₀₉	—	2	1	no	—
32	#117×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₀₉	—	2	2	no	—
35	C₄⋊C₈×C₁₀₉	—	2	2	no	—
36	C₄⋊C₈×C₁₀₉	—	2	1	no	—
37	C₄⋊C₈×C₁₀₉	—	2	1	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
41	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
42	#97×C₁₀₉	—	2	2	no	—
43	#97×C₁₀₉	—	2	2	no	—
44	#97×C₁₀₉	—	2	2	no	—
45	C₄×C₈×C₁₀₉	—	2	2	no	—
46	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₀₉	—	2	1	no	—
51	C₂×Q₈×C₁₀₉	—	2	1	no	—
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
55	M₁₆×C₁₀₉	—	4	2	no	—
56	C₂×C₈×C₁₀₉	—	2	2	no	—
57	C₂×C₈×C₁₀₉	—	2	2	no	—
58	C₂×C₈×C₁₀₉	—	4	1	no	—
59	C₂×C₈×C₁₀₉	—	4	2	no	—
60	C₂×C₈×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
70	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
71	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₁₀₉	#117	1	1	yes	characteristic
78	Q₈×C₁₀₉	—	4	2	no	—
79	Q₈×C₁₀₉	—	4	2	no	—
80	C₂×C₄×C₁₀₉	—	2	1	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	—	4	1	no	—
85	C₂×C₄×C₁₀₉	—	4	1	no	—
86	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
92	C₈×C₁₀₉	—	2	1	no	—
93	C₈×C₁₀₉	—	2	1	no	—
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	4	2	no	—
96	C₈×C₁₀₉	—	8	2	no	—
97	C₈×C₁₀₉	—	8	2	no	—
98	C₈×C₁₀₉	#101	1	2	yes	—
99	C₂²×C₁₀₉	—	2	1	no	—
100	C₂²×C₁₀₉	—	2	1	no	—
101	C₂²×C₁₀₉	—	2	1	no	—
102	C₂²×C₁₀₉	#380	1	1	yes	characteristic
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	2	1	no	—
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	—	2	1	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	4	2	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	C₂×#101	1	1	yes	characteristic
111	C₄×C₁₀₉	#413	1	1	yes	characteristic
112	#8635	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₀₉	—	4	1	no	—
114	C₂×C₁₀₉	#1801	1	1	yes	characteristic
115	C₂×C₁₀₉	#1814	1	1	yes	characteristic
116	C₂×C₁₀₉	#1814	1	1	yes	characteristic
117	#2960	C₂×C₁₀₉	1	1	yes	characteristic
118	#1568	C₂×C₁₀₉	1	1	yes	characteristic
119	#1568	C₂×C₁₀₉	1	1	yes	characteristic
120	#1537	C₂×C₁₀₉	1	2	yes	—
121	#1307	C₂×C₁₀₉	1	2	yes	—
122	C₁₀₉	#8635	1	1	yes	characteristic, direct product factor, Sylow
123	#533	C₂²×C₁₀₉	1	1	yes	characteristic
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	—	2	1	no	—
128	#432	C₂²×C₁₀₉	1	2	yes	—
129	#432	C₂²×C₁₀₉	1	2	yes	—
130	#424	C₄×C₁₀₉	1	2	yes	—
131	#423	C₂²×C₁₀₉	1	1	yes	characteristic
132	#405	C₄×C₁₀₉	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
134	#335	—	2	1	no	—
135	#335	—	2	1	no	—
136	#334	—	2	1	no	—
137	#334	—	2	1	no	—
138	#330	—	2	4	no	—
139	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
142	C₄⋊Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
143	C₄⋊Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
144	#125	D₄×C₁₀₉	1	2	yes	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	#117	C₂×C₄×C₁₀₉	1	2	yes	—
150	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	2	2	no	—
158	C₂²⋊C₈	—	2	2	no	—
159	#97	—	2	2	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
164	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
165	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
166	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
172	M₁₆	—	4	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
187	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
192	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
194	C₂³	#117×C₁₀₉	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
206	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
207	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
208	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	—	8	2	no	—
214	C₈	—	8	2	no	—
215	C₈	#101×C₁₀₉	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#380×C₁₀₉	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×#101×C₁₀₉	1	1	yes	characteristic
228	C₄	#413×C₁₀₉	1	1	yes	characteristic
229	C₂	—	4	1	no	—
230	C₂	C₁₀₉×#1801	1	1	yes	characteristic
231	C₂	C₁₀₉×#1814	1	1	yes	characteristic
232	C₂	C₁₀₉×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
112. #8635	C₁₀₉	direct (N × H)
122. C₁₀₉	#8635	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2960	2
C₁₀₉×#1568	2
C₁₀₉×#1537	2
C₁₀₉×#1307	2
#8635	109

Subgroups — C109×#8635 #42126629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8635 #42126629