Subgroups — C₁₀₉×#9706 #42127700

Order 27904.

850 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2867	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#331×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	C₂×#127×C₁₀₉	—	2	1	no	—
6	C₂×#127×C₁₀₉	—	2	2	no	—
7	C₂×#127×C₁₀₉	—	2	1	no	—
8	C₂×#127×C₁₀₉	C₄	1	4	yes	—
9	C₂×#127×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#97×C₁₀₉	—	2	1	no	—
13	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
14	#336×C₁₀₉	—	2	1	no	—
15	#336×C₁₀₉	—	2	2	no	—
16	#331×C₁₀₉	—	2	8	no	—
17	C₂²×D₄×C₁₀₉	—	2	4	no	—
18	C₂²×D₄×C₁₀₉	—	2	4	no	—
19	C₂²×D₄×C₁₀₉	—	2	2	no	—
20	C₂²×D₄×C₁₀₉	—	2	2	no	—
21	C₂²×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	#127×C₁₀₉	—	2	4	no	—
25	#127×C₁₀₉	—	2	4	no	—
26	#127×C₁₀₉	—	4	4	no	—
27	#127×C₁₀₉	—	4	8	no	—
28	#127×C₁₀₉	—	4	8	no	—
29	#127×C₁₀₉	—	4	2	no	—
30	#127×C₁₀₉	—	4	2	no	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₀₉	—	2	4	no	—
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	complemented
39	#97×C₁₀₉	—	4	2	no	—
40	C₂⁴×C₁₀₉	—	2	4	no	—
41	C₂×D₄×C₁₀₉	—	2	8	no	—
42	C₂×D₄×C₁₀₉	—	4	8	no	—
43	C₂×D₄×C₁₀₉	—	4	8	no	—
44	C₂×D₄×C₁₀₉	—	4	8	no	—
45	C₂×D₄×C₁₀₉	—	4	8	no	—
46	C₂×D₄×C₁₀₉	—	4	8	no	—
47	C₂×D₄×C₁₀₉	—	4	8	no	—
48	C₂×D₄×C₁₀₉	—	4	8	no	—
49	C₂×D₄×C₁₀₉	—	4	4	no	—
50	C₂×D₄×C₁₀₉	—	4	2	no	—
51	C₂×D₄×C₁₀₉	—	4	2	no	—
52	C₂×D₄×C₁₀₉	—	4	2	no	—
53	C₂×D₄×C₁₀₉	—	4	2	no	—
54	C₂×D₄×C₁₀₉	—	4	4	no	—
55	C₂×D₄×C₁₀₉	—	4	8	no	—
56	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₀₉	—	2	4	no	—
62	C₂×C₈×C₁₀₉	—	4	2	no	—
63	C₂×C₈×C₁₀₉	—	4	2	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	4	no	—
66	C₄²×C₁₀₉	—	2	4	no	—
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	—	2	4	no	—
69	C₄²×C₁₀₉	D₈	1	2	yes	—
70	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
71	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
72	C₄²×C₁₀₉	#41	1	2	yes	—
73	C₂³×C₁₀₉	—	4	4	no	—
74	C₂³×C₁₀₉	—	4	8	no	—
75	C₂³×C₁₀₉	—	4	4	no	—
76	C₂³×C₁₀₉	—	4	8	no	—
77	C₂³×C₁₀₉	—	4	4	no	—
78	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	D₄×C₁₀₉	—	8	8	no	—
80	D₄×C₁₀₉	—	8	16	no	—
81	D₄×C₁₀₉	—	8	16	no	—
82	D₄×C₁₀₉	—	8	4	no	—
83	D₄×C₁₀₉	—	8	4	no	—
84	D₄×C₁₀₉	—	8	4	no	—
85	D₄×C₁₀₉	—	8	4	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	#100	1	4	yes	—
92	C₂×C₄×C₁₀₉	#102	1	4	yes	—
93	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₉	#121	1	2	yes	—
95	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
101	C₈×C₁₀₉	—	4	4	no	—
102	C₈×C₁₀₉	—	8	2	no	—
103	C₂²×C₁₀₉	—	8	8	no	—
104	C₂²×C₁₀₉	—	8	8	no	—
105	C₂²×C₁₀₉	—	8	4	no	—
106	C₂²×C₁₀₉	—	8	4	no	—
107	C₂²×C₁₀₉	—	8	4	no	—
108	C₂²×C₁₀₉	#390	1	1	yes	characteristic
109	C₂²×C₁₀₉	C₂×#100	1	2	yes	—
110	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
111	C₂²×C₁₀₉	#418	1	2	yes	—
112	C₂²×C₁₀₉	#418	1	1	yes	characteristic
113	C₄×C₁₀₉	—	2	4	no	—
114	C₄×C₁₀₉	—	2	4	no	—
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	#459	1	2	yes	—
117	C₄×C₁₀₉	#460	1	2	yes	—
118	C₄×C₁₀₉	#490	1	2	yes	—
119	C₄×C₁₀₉	#496	1	2	yes	—
120	#9706	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₀₉	—	16	4	no	—
122	C₂×C₁₀₉	#1545	1	2	yes	—
123	C₂×C₁₀₉	#1555	1	2	yes	—
124	C₂×C₁₀₉	#1961	1	1	yes	characteristic
125	C₂×C₁₀₉	#1965	1	1	yes	characteristic
126	C₂×C₁₀₉	#1982	1	1	yes	characteristic
127	#2867	C₂×C₁₀₉	1	1	yes	characteristic
128	#1831	C₂×C₁₀₉	1	1	yes	characteristic, complemented
129	#1767	C₂×C₁₀₉	1	1	yes	characteristic, complemented
130	C₂×#331	C₂×C₁₀₉	1	4	yes	complemented
131	C₁₀₉	#9706	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#127	—	2	1	no	—
133	C₂×#127	—	2	2	no	—
134	C₂×#127	—	2	1	no	—
135	C₂×#127	C₄×C₁₀₉	1	4	yes	—
136	C₂×#127	C₂²×C₁₀₉	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
138	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
139	C₂×#97	—	2	1	no	—
140	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
141	#336	—	2	2	no	—
142	#336	—	2	1	no	—
143	#331	—	2	8	no	—
144	C₂²×D₄	—	2	4	no	—
145	C₂²×D₄	—	2	4	no	—
146	C₂²×D₄	—	2	2	no	—
147	C₂²×D₄	—	2	2	no	—
148	C₂²×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	2	no	—
151	#127	—	2	4	no	—
152	#127	—	2	4	no	—
153	#127	—	4	8	no	—
154	#127	—	4	4	no	—
155	#127	—	4	8	no	—
156	#127	—	4	2	no	—
157	#127	—	4	2	no	—
158	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
159	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
161	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	complemented
166	#97	—	4	2	no	—
167	C₂⁴	—	2	4	no	—
168	C₂×D₄	—	2	8	no	—
169	C₂×D₄	—	4	8	no	—
170	C₂×D₄	—	4	8	no	—
171	C₂×D₄	—	4	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	8	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₈×C₁₀₉	1	2	yes	—
197	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
198	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
199	C₄²	#41×C₁₀₉	1	2	yes	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	8	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
206	D₄	—	8	8	no	—
207	D₄	—	8	16	no	—
208	D₄	—	8	16	no	—
209	D₄	—	8	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	#100×C₁₀₉	1	4	yes	—
219	C₂×C₄	#102×C₁₀₉	1	4	yes	—
220	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
221	C₂×C₄	#121×C₁₀₉	1	2	yes	—
222	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
226	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
227	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
228	C₈	—	4	4	no	—
229	C₈	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	8	no	—
234	C₂²	—	8	8	no	—
235	C₂²	#390×C₁₀₉	1	1	yes	characteristic
236	C₂²	C₂×#100×C₁₀₉	1	2	yes	—
237	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
238	C₂²	#418×C₁₀₉	1	2	yes	—
239	C₂²	#418×C₁₀₉	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	4	no	—
243	C₄	#459×C₁₀₉	1	2	yes	—
244	C₄	#460×C₁₀₉	1	2	yes	—
245	C₄	#490×C₁₀₉	1	2	yes	—
246	C₄	#496×C₁₀₉	1	2	yes	—
247	C₂	—	16	4	no	—
248	C₂	C₁₀₉×#1545	1	2	yes	—
249	C₂	C₁₀₉×#1555	1	2	yes	—
250	C₂	C₁₀₉×#1961	1	1	yes	characteristic
251	C₂	C₁₀₉×#1965	1	1	yes	characteristic
252	C₂	C₁₀₉×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₁₀₉×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#1831	C₂	semidirect
3. C₁₀₉×#1767	C₂	semidirect
4. C₂×#331×C₁₀₉	C₂	semidirect
38. C₄⋊C₈×C₁₀₉	D₄	semidirect
120. #9706	C₁₀₉	direct (N × H)
128. #1831	C₂×C₁₀₉	semidirect
129. #1767	C₂×C₁₀₉	semidirect
130. C₂×#331	C₂×C₁₀₉	semidirect
131. C₁₀₉	#9706	direct (N × H)
165. C₄⋊C₈	D₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2867	2
C₁₀₉×#1831	2
C₁₀₉×#1767	2
C₂×#331×C₁₀₉	2
#9706	109

Subgroups — C109×#9706 #42127700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9706 #42127700