Subgroups — C₁₀₉×#9891 #42127885

Order 27904.

468 conjugacy classes of subgroups.

The subgroup lattice has 266 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×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1295	C₂	1	4	yes	maximal
5	C₂×#125×C₁₀₉	C₄	1	2	yes	—
6	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
7	C₂×#97×C₁₀₉	—	2	1	no	—
8	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
11	#389×C₁₀₉	—	2	1	no	—
12	#389×C₁₀₉	—	2	1	no	—
13	#389×C₁₀₉	—	2	1	no	—
14	#389×C₁₀₉	—	2	1	no	—
15	#384×C₁₀₉	C₄	1	1	yes	characteristic
16	#384×C₁₀₉	C₄	1	1	yes	characteristic
17	#382×C₁₀₉	C₂²	1	2	yes	—
18	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
19	#336×C₁₀₉	—	2	1	no	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	—	2	2	no	—
22	#125×C₁₀₉	—	4	4	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₀₉	D₄	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	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	—	2	4	no	—
39	#97×C₁₀₉	—	2	4	no	—
40	#97×C₁₀₉	—	2	2	no	—
41	#97×C₁₀₉	—	2	4	no	—
42	#97×C₁₀₉	Q₈	1	4	yes	—
43	C₄×C₈×C₁₀₉	—	2	4	no	—
44	#95×C₁₀₉	—	2	2	no	—
45	#95×C₁₀₉	—	2	2	no	—
46	C₂²×C₄×C₁₀₉	—	2	2	no	—
47	C₂²×C₄×C₁₀₉	—	2	2	no	—
48	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₀₉	—	2	4	no	—
56	C₂×C₈×C₁₀₉	—	2	4	no	—
57	C₂×C₈×C₁₀₉	—	2	4	no	—
58	C₂×C₈×C₁₀₉	—	2	4	no	—
59	C₂×C₈×C₁₀₉	—	4	2	no	—
60	C₂×C₈×C₁₀₉	—	4	2	no	—
61	C₄⋊C₄×C₁₀₉	—	4	2	no	—
62	C₄⋊C₄×C₁₀₉	—	4	2	no	—
63	C₄⋊C₄×C₁₀₉	—	4	2	no	—
64	C₄⋊C₄×C₁₀₉	—	4	4	no	—
65	C₄⋊C₄×C₁₀₉	—	4	4	no	—
66	C₄⋊C₄×C₁₀₉	—	4	2	no	—
67	C₄⋊C₄×C₁₀₉	—	4	2	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄²×C₁₀₉	—	2	2	no	—
70	C₄²×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	—	2	2	no	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	—	2	2	no	—
74	C₄²×C₁₀₉	—	4	1	no	—
75	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₀₉	#41	1	2	yes	—
77	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	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	—	4	2	no	—
97	C₂×C₄×C₁₀₉	#100	1	2	yes	—
98	C₂×C₄×C₁₀₉	#101	1	2	yes	—
99	C₂×C₄×C₁₀₉	#104	1	2	yes	—
100	C₂×C₄×C₁₀₉	#104	1	2	yes	—
101	C₂×C₄×C₁₀₉	#104	1	2	yes	—
102	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
106	C₈×C₁₀₉	—	2	8	no	—
107	C₈×C₁₀₉	—	4	4	no	—
108	C₂²×C₁₀₉	#391	1	1	yes	characteristic
109	C₂²×C₁₀₉	C₂×#100	1	1	yes	characteristic
110	C₂²×C₁₀₉	C₂×#101	1	1	yes	characteristic
111	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
112	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
113	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
114	C₂²×C₁₀₉	#421	1	1	yes	characteristic
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	2	1	no	—
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	—	2	2	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	4	1	no	—
122	C₄×C₁₀₉	—	8	2	no	—
123	C₄×C₁₀₉	—	8	2	no	—
124	C₄×C₁₀₉	#330	1	4	yes	—
125	#9891	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₁₀₉	C₂×#330	1	1	yes	characteristic
127	C₂×C₁₀₉	#1525	1	1	yes	characteristic
128	C₂×C₁₀₉	#1528	1	1	yes	characteristic
129	C₂×C₁₀₉	#1530	1	1	yes	characteristic
130	C₂×C₁₀₉	#1986	1	1	yes	characteristic
131	C₂×C₁₀₉	#1994	1	1	yes	characteristic
132	C₂×C₁₀₉	#1997	1	1	yes	characteristic
133	#2859	C₂×C₁₀₉	1	1	yes	characteristic
134	#1767	C₂×C₁₀₉	1	1	yes	characteristic
135	C₄×#97	C₂×C₁₀₉	1	1	yes	characteristic
136	#1295	C₂×C₁₀₉	1	4	yes	—
137	C₁₀₉	#9891	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#125	C₄×C₁₀₉	1	2	yes	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×#97	—	2	1	no	—
141	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
142	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
143	C₂×C₄×C₈	—	2	1	no	—
144	#389	—	2	1	no	—
145	#389	—	2	1	no	—
146	#389	—	2	1	no	—
147	#389	—	2	1	no	—
148	#384	C₄×C₁₀₉	1	1	yes	characteristic
149	#384	C₄×C₁₀₉	1	1	yes	characteristic
150	#382	C₂²×C₁₀₉	1	2	yes	—
151	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	#125	—	4	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×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	D₄×C₁₀₉	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	#97	—	2	4	no	—
173	#97	—	2	2	no	—
174	#97	—	2	4	no	—
175	#97	Q₈×C₁₀₉	1	4	yes	—
176	C₄×C₈	—	2	4	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
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	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	1	no	—
208	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
209	C₄²	#41×C₁₀₉	1	2	yes	—
210	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
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	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	#100×C₁₀₉	1	2	yes	—
231	C₂×C₄	#101×C₁₀₉	1	2	yes	—
232	C₂×C₄	#104×C₁₀₉	1	2	yes	—
233	C₂×C₄	#104×C₁₀₉	1	2	yes	—
234	C₂×C₄	#104×C₁₀₉	1	2	yes	—
235	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
236	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
237	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
239	C₈	—	2	8	no	—
240	C₈	—	4	4	no	—
241	C₂²	#391×C₁₀₉	1	1	yes	characteristic
242	C₂²	C₂×#100×C₁₀₉	1	1	yes	characteristic
243	C₂²	C₂×#101×C₁₀₉	1	1	yes	characteristic
244	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
245	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
246	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
247	C₂²	#421×C₁₀₉	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	8	2	no	—
256	C₄	—	8	2	no	—
257	C₄	#330×C₁₀₉	1	4	yes	—
258	C₂	C₂×#330×C₁₀₉	1	1	yes	characteristic
259	C₂	C₁₀₉×#1525	1	1	yes	characteristic
260	C₂	C₁₀₉×#1528	1	1	yes	characteristic
261	C₂	C₁₀₉×#1530	1	1	yes	characteristic
262	C₂	C₁₀₉×#1986	1	1	yes	characteristic
263	C₂	C₁₀₉×#1994	1	1	yes	characteristic
264	C₂	C₁₀₉×#1997	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
125. #9891	C₁₀₉	direct (N × H)
137. C₁₀₉	#9891	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×#1767	2
C₄×#97×C₁₀₉	2
C₁₀₉×#1295	2
#9891	109

Subgroups — C109×#9891 #42127885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9891 #42127885