Subgroups — C₁₀₉×#7723 #42125717

Order 27904.

388 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1839	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1271	C₂	1	4	yes	maximal
5	C₂×#125×C₁₀₉	—	2	1	no	—
6	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
7	#389×C₁₀₉	—	2	1	no	—
8	#389×C₁₀₉	—	2	1	no	—
9	#389×C₁₀₉	C₄	1	2	yes	—
10	#389×C₁₀₉	C₂²	1	2	yes	—
11	#384×C₁₀₉	C₄	1	1	yes	characteristic
12	#384×C₁₀₉	C₄	1	1	yes	characteristic
13	#382×C₁₀₉	—	2	1	no	—
14	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
15	#336×C₁₀₉	—	2	1	no	—
16	#336×C₁₀₉	—	2	1	no	—
17	#336×C₁₀₉	—	2	1	no	—
18	#336×C₁₀₉	C₂²	1	2	yes	—
19	#336×C₁₀₉	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	2	no	—
22	#125×C₁₀₉	—	4	2	no	—
23	#125×C₁₀₉	—	4	2	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₉	—	4	2	no	—
40	#95×C₁₀₉	—	2	2	no	—
41	#95×C₁₀₉	—	2	2	no	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	—	2	2	no	—
44	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	—	4	2	no	—
52	C₂×C₈×C₁₀₉	—	4	2	no	—
53	C₂×C₈×C₁₀₉	—	4	2	no	—
54	C₂×C₈×C₁₀₉	—	4	2	no	—
55	C₂×C₈×C₁₀₉	—	4	2	no	—
56	C₂×C₈×C₁₀₉	—	4	2	no	—
57	C₄⋊C₄×C₁₀₉	—	2	4	no	—
58	C₄⋊C₄×C₁₀₉	—	4	2	no	—
59	C₄⋊C₄×C₁₀₉	—	4	2	no	—
60	C₄⋊C₄×C₁₀₉	—	4	2	no	—
61	C₄⋊C₄×C₁₀₉	—	4	2	no	—
62	C₄⋊C₄×C₁₀₉	—	4	2	no	—
63	C₄⋊C₄×C₁₀₉	—	4	4	no	—
64	C₄⋊C₄×C₁₀₉	—	4	2	no	—
65	C₄⋊C₄×C₁₀₉	—	4	2	no	—
66	C₄²×C₁₀₉	—	2	2	no	—
67	C₄²×C₁₀₉	—	2	2	no	—
68	C₄²×C₁₀₉	—	2	2	no	—
69	C₄²×C₁₀₉	—	4	1	no	—
70	C₄²×C₁₀₉	—	4	1	no	—
71	C₄²×C₁₀₉	—	4	1	no	—
72	C₄²×C₁₀₉	D₈	1	2	yes	—
73	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
74	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	—	2	1	no	—
76	C₂×C₄×C₁₀₉	—	2	1	no	—
77	C₂×C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄×C₁₀₉	—	2	1	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	—	4	2	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	#101	1	2	yes	—
97	C₂×C₄×C₁₀₉	#102	1	2	yes	—
98	C₂×C₄×C₁₀₉	#102	1	2	yes	—
99	C₂×C₄×C₁₀₉	#131	1	2	yes	—
100	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	#137	1	2	yes	—
103	C₈×C₁₀₉	—	8	2	no	—
104	C₈×C₁₀₉	—	8	2	no	—
105	C₂²×C₁₀₉	#408	1	1	yes	characteristic
106	C₂²×C₁₀₉	#410	1	1	yes	characteristic
107	C₂²×C₁₀₉	C₂×#101	1	1	yes	characteristic
108	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
109	C₂²×C₁₀₉	#419	1	1	yes	characteristic
110	C₂²×C₁₀₉	#421	1	1	yes	characteristic
111	C₂²×C₁₀₉	#421	1	1	yes	characteristic
112	C₄×C₁₀₉	—	2	2	no	—
113	C₄×C₁₀₉	—	2	2	no	—
114	C₄×C₁₀₉	—	2	2	no	—
115	C₄×C₁₀₉	—	2	2	no	—
116	C₄×C₁₀₉	—	2	1	no	—
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	—	4	1	no	—
120	C₄×C₁₀₉	—	8	2	no	—
121	C₄×C₁₀₉	—	8	2	no	—
122	#7723	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂×C₁₀₉	#1470	1	1	yes	characteristic
124	C₂×C₁₀₉	#1480	1	1	yes	characteristic
125	C₂×C₁₀₉	#1497	1	1	yes	characteristic
126	C₂×C₁₀₉	#1515	1	1	yes	characteristic
127	C₂×C₁₀₉	#1517	1	1	yes	characteristic
128	C₂×C₁₀₉	#1534	1	1	yes	characteristic
129	C₂×C₁₀₉	#1546	1	1	yes	characteristic
130	#2859	C₂×C₁₀₉	1	1	yes	characteristic
131	#1839	C₂×C₁₀₉	1	1	yes	characteristic
132	#1831	C₂×C₁₀₉	1	1	yes	characteristic
133	#1271	C₂×C₁₀₉	1	4	yes	—
134	C₁₀₉	#7723	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#125	—	2	1	no	—
136	C₂×C₄⋊C₈	—	2	1	no	—
137	#389	—	2	1	no	—
138	#389	—	2	1	no	—
139	#389	C₄×C₁₀₉	1	2	yes	—
140	#389	C₂²×C₁₀₉	1	2	yes	—
141	#384	C₄×C₁₀₉	1	1	yes	characteristic
142	#384	C₄×C₁₀₉	1	1	yes	characteristic
143	#382	—	2	1	no	—
144	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
145	#336	—	2	1	no	—
146	#336	—	2	1	no	—
147	#336	—	2	1	no	—
148	#336	C₂²×C₁₀₉	1	2	yes	—
149	#336	C₂²×C₁₀₉	1	2	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	#125	—	4	2	no	—
153	#125	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
169	C₄⋊C₈	—	4	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	4	1	no	—
200	C₄²	—	4	1	no	—
201	C₄²	—	4	1	no	—
202	C₄²	D₈×C₁₀₉	1	2	yes	—
203	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
204	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	#101×C₁₀₉	1	2	yes	—
227	C₂×C₄	#102×C₁₀₉	1	2	yes	—
228	C₂×C₄	#102×C₁₀₉	1	2	yes	—
229	C₂×C₄	#131×C₁₀₉	1	2	yes	—
230	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
231	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄	#137×C₁₀₉	1	2	yes	—
233	C₈	—	8	2	no	—
234	C₈	—	8	2	no	—
235	C₂²	#408×C₁₀₉	1	1	yes	characteristic
236	C₂²	#410×C₁₀₉	1	1	yes	characteristic
237	C₂²	C₂×#101×C₁₀₉	1	1	yes	characteristic
238	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
239	C₂²	#419×C₁₀₉	1	1	yes	characteristic
240	C₂²	#421×C₁₀₉	1	1	yes	characteristic
241	C₂²	#421×C₁₀₉	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₂	C₁₀₉×#1470	1	1	yes	characteristic
253	C₂	C₁₀₉×#1480	1	1	yes	characteristic
254	C₂	C₁₀₉×#1497	1	1	yes	characteristic
255	C₂	C₁₀₉×#1515	1	1	yes	characteristic
256	C₂	C₁₀₉×#1517	1	1	yes	characteristic
257	C₂	C₁₀₉×#1534	1	1	yes	characteristic
258	C₂	C₁₀₉×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
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
122. #7723	C₁₀₉	direct (N × H)
134. C₁₀₉	#7723	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×#2859	2
C₁₀₉×#1839	2
C₁₀₉×#1831	2
C₁₀₉×#1271	2
#7723	109

Subgroups — C109×#7723 #42125717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7723 #42125717