Subgroups — C₁₀₉×#9695 #42127689

Order 27904.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₁₀₉	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	1	no	—
7	C₂×#127×C₁₀₉	—	2	2	no	—
8	C₂×#127×C₁₀₉	C₄	1	4	yes	—
9	C₂×#127×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
14	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
15	#336×C₁₀₉	—	2	1	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	2	no	—
29	#127×C₁₀₉	—	4	4	no	—
30	#127×C₁₀₉	—	4	4	no	—
31	#127×C₁₀₉	—	4	2	no	—
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	—	2	4	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	4	4	no	—
40	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	complemented
41	C₄×C₈×C₁₀₉	—	4	2	no	—
42	C₂⁴×C₁₀₉	—	2	4	no	—
43	C₂×D₄×C₁₀₉	—	2	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	8	no	—
50	C₂×D₄×C₁₀₉	—	4	8	no	—
51	C₂×D₄×C₁₀₉	—	4	4	no	—
52	C₂×D₄×C₁₀₉	—	4	2	no	—
53	C₂×D₄×C₁₀₉	—	4	2	no	—
54	C₂×D₄×C₁₀₉	—	4	2	no	—
55	C₂×D₄×C₁₀₉	—	4	4	no	—
56	C₂×D₄×C₁₀₉	—	4	2	no	—
57	C₂×D₄×C₁₀₉	—	4	4	no	—
58	C₂×D₄×C₁₀₉	—	4	4	no	—
59	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	—	2	4	no	—
65	C₂×C₈×C₁₀₉	—	4	2	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	4	2	no	—
68	C₂×C₈×C₁₀₉	—	4	4	no	—
69	C₄²×C₁₀₉	—	2	4	no	—
70	C₄²×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	D₈	1	2	yes	—
72	C₄²×C₁₀₉	D₈	1	4	yes	—
73	C₄²×C₁₀₉	SD₁₆	1	4	yes	—
74	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
75	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
76	C₄²×C₁₀₉	#41	1	2	yes	—
77	C₂³×C₁₀₉	—	4	4	no	—
78	C₂³×C₁₀₉	—	4	8	no	—
79	C₂³×C₁₀₉	—	4	4	no	—
80	C₂³×C₁₀₉	—	4	8	no	—
81	C₂³×C₁₀₉	—	4	4	no	—
82	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	D₄×C₁₀₉	—	8	8	no	—
84	D₄×C₁₀₉	—	8	16	no	—
85	D₄×C₁₀₉	—	8	16	no	—
86	D₄×C₁₀₉	—	8	4	no	—
87	D₄×C₁₀₉	—	8	4	no	—
88	D₄×C₁₀₉	—	8	4	no	—
89	D₄×C₁₀₉	—	8	2	no	—
90	D₄×C₁₀₉	—	8	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	#100	1	4	yes	—
95	C₂×C₄×C₁₀₉	#102	1	4	yes	—
96	C₂×C₄×C₁₀₉	#102	1	4	yes	—
97	C₂×C₄×C₁₀₉	#102	1	4	yes	—
98	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	#121	1	2	yes	—
100	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
107	C₈×C₁₀₉	—	4	4	no	—
108	C₈×C₁₀₉	—	8	2	no	—
109	C₂²×C₁₀₉	—	8	8	no	—
110	C₂²×C₁₀₉	—	8	8	no	—
111	C₂²×C₁₀₉	—	8	4	no	—
112	C₂²×C₁₀₉	—	8	4	no	—
113	C₂²×C₁₀₉	—	8	4	no	—
114	C₂²×C₁₀₉	#390	1	1	yes	characteristic
115	C₂²×C₁₀₉	C₂×#100	1	2	yes	—
116	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
117	C₂²×C₁₀₉	C₂×#102	1	2	yes	—
118	C₂²×C₁₀₉	#418	1	1	yes	characteristic
119	C₄×C₁₀₉	—	2	4	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	#331	1	8	yes	complemented
122	C₄×C₁₀₉	#459	1	2	yes	—
123	C₄×C₁₀₉	#460	1	2	yes	—
124	C₄×C₁₀₉	#486	1	2	yes	—
125	C₄×C₁₀₉	#492	1	1	yes	characteristic
126	C₄×C₁₀₉	#493	1	1	yes	characteristic
127	#9695	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₁₀₉	—	16	4	no	—
129	C₂×C₁₀₉	C₂×#331	1	2	yes	—
130	C₂×C₁₀₉	#1545	1	2	yes	—
131	C₂×C₁₀₉	#1957	1	1	yes	characteristic
132	C₂×C₁₀₉	#1965	1	1	yes	characteristic
133	C₂×C₁₀₉	#1982	1	1	yes	characteristic
134	#2867	C₂×C₁₀₉	1	1	yes	characteristic
135	#1842	C₂×C₁₀₉	1	1	yes	characteristic, complemented
136	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
137	C₂×#331	C₂×C₁₀₉	1	4	yes	complemented
138	C₁₀₉	#9695	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#127	—	2	1	no	—
140	C₂×#127	—	2	1	no	—
141	C₂×#127	—	2	2	no	—
142	C₂×#127	C₄×C₁₀₉	1	4	yes	—
143	C₂×#127	C₂²×C₁₀₉	1	2	yes	—
144	C₂×C₄⋊C₈	—	2	2	no	—
145	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
147	C₂×C₄×C₈	—	2	1	no	—
148	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#331	—	2	8	no	—
151	C₂²×D₄	—	2	4	no	—
152	C₂²×D₄	—	2	4	no	—
153	C₂²×D₄	—	2	2	no	—
154	C₂²×D₄	—	2	2	no	—
155	C₂²×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	#127	—	2	4	no	—
159	#127	—	2	4	no	—
160	#127	—	4	8	no	—
161	#127	—	4	4	no	—
162	#127	—	4	2	no	—
163	#127	—	4	4	no	—
164	#127	—	4	4	no	—
165	#127	—	4	2	no	—
166	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
167	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
170	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	4	4	no	—
174	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	complemented
175	C₄×C₈	—	4	2	no	—
176	C₂⁴	—	2	4	no	—
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	8	no	—
180	C₂×D₄	—	4	8	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂×D₄	—	4	8	no	—
184	C₂×D₄	—	4	8	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	D₈×C₁₀₉	1	2	yes	—
206	C₄²	D₈×C₁₀₉	1	4	yes	—
207	C₄²	SD₁₆×C₁₀₉	1	4	yes	—
208	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
209	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
210	C₄²	#41×C₁₀₉	1	2	yes	—
211	C₂³	—	4	4	no	—
212	C₂³	—	4	8	no	—
213	C₂³	—	4	8	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	4	no	—
216	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
217	D₄	—	8	8	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	16	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	4	no	—
223	D₄	—	8	2	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#100×C₁₀₉	1	4	yes	—
229	C₂×C₄	#102×C₁₀₉	1	4	yes	—
230	C₂×C₄	#102×C₁₀₉	1	4	yes	—
231	C₂×C₄	#102×C₁₀₉	1	4	yes	—
232	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
233	C₂×C₄	#121×C₁₀₉	1	2	yes	—
234	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
238	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
240	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	8	2	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	8	no	—
247	C₂²	—	8	8	no	—
248	C₂²	#390×C₁₀₉	1	1	yes	characteristic
249	C₂²	C₂×#100×C₁₀₉	1	2	yes	—
250	C₂²	C₂×#102×C₁₀₉	1	2	yes	—
251	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
252	C₂²	#418×C₁₀₉	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	#331×C₁₀₉	1	8	yes	complemented
256	C₄	#459×C₁₀₉	1	2	yes	—
257	C₄	#460×C₁₀₉	1	2	yes	—
258	C₄	#486×C₁₀₉	1	2	yes	—
259	C₄	#492×C₁₀₉	1	1	yes	characteristic
260	C₄	#493×C₁₀₉	1	1	yes	characteristic
261	C₂	—	16	4	no	—
262	C₂	C₂×#331×C₁₀₉	1	2	yes	—
263	C₂	C₁₀₉×#1545	1	2	yes	—
264	C₂	C₁₀₉×#1957	1	1	yes	characteristic
265	C₂	C₁₀₉×#1965	1	1	yes	characteristic
266	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×#1842	C₂	semidirect
3. C₄×C₄⋊C₈×C₁₀₉	C₂	semidirect
4. C₂×#331×C₁₀₉	C₂	semidirect
40. C₄⋊C₈×C₁₀₉	D₄	semidirect
121. C₄×C₁₀₉	#331	semidirect
127. #9695	C₁₀₉	direct (N × H)
135. #1842	C₂×C₁₀₉	semidirect
136. C₄×C₄⋊C₈	C₂×C₁₀₉	semidirect
137. C₂×#331	C₂×C₁₀₉	semidirect
138. C₁₀₉	#9695	direct (N × H)
174. C₄⋊C₈	D₄×C₁₀₉	semidirect
255. C₄	#331×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×#1842	2
C₄×C₄⋊C₈×C₁₀₉	2
C₂×#331×C₁₀₉	2
#9695	109

Subgroups — C109×#9695 #42127689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9695 #42127689