Subgroups — C₁₀₉×#20574 #42138568

Order 27904.

806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2295	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1771	C₂	1	2	yes	maximal
5	C₁₀₉×#1770	C₂	1	2	yes	maximal
6	C₁₀₉×#1770	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₁₀₉×#1768	C₂	1	2	yes	maximal
9	C₁₀₉×#1767	C₂	1	1	yes	maximal, characteristic
10	C₁₀₉×#1746	C₂	1	2	yes	maximal
11	C₁₀₉×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#446×C₁₀₉	C₄	1	8	yes	complemented
14	C₈×Q₈×C₁₀₉	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	#391×C₁₀₉	C₄	1	2	yes	—
27	#389×C₁₀₉	C₄	1	2	yes	—
28	#387×C₁₀₉	C₄	1	4	yes	—
29	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
31	#376×C₁₀₉	C₂²	1	2	yes	—
32	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
33	#336×C₁₀₉	C₂²	1	1	yes	characteristic
34	#336×C₁₀₉	C₂²	1	1	yes	characteristic
35	#336×C₁₀₉	C₂²	1	1	yes	characteristic
36	#336×C₁₀₉	C₂²	1	1	yes	characteristic
37	#336×C₁₀₉	C₂²	1	2	yes	—
38	#336×C₁₀₉	C₂²	1	2	yes	—
39	#336×C₁₀₉	C₂²	1	2	yes	—
40	C₂²×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
47	C₄×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
58	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
60	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
61	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
62	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
64	#97×C₁₀₉	—	4	2	no	—
65	#97×C₁₀₉	—	4	2	no	—
66	#97×C₁₀₉	C₂×C₄	1	8	yes	—
67	C₄×C₈×C₁₀₉	—	4	2	no	—
68	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₁₀₉	C₂×C₄	1	8	yes	—
70	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
71	#95×C₁₀₉	C₂×C₄	1	4	yes	—
72	#95×C₁₀₉	C₂×C₄	1	4	yes	—
73	#95×C₁₀₉	C₂×C₄	1	2	yes	—
74	#95×C₁₀₉	C₂×C₄	1	4	yes	—
75	C₂×Q₈×C₁₀₉	—	2	4	no	—
76	C₂×Q₈×C₁₀₉	C₄²	1	4	yes	—
77	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₀₉	—	2	1	no	—
86	C₂×C₈×C₁₀₉	—	2	2	no	—
87	C₂×C₈×C₁₀₉	—	2	1	no	—
88	C₂×C₈×C₁₀₉	—	2	2	no	—
89	C₂×C₈×C₁₀₉	—	2	1	no	—
90	C₂×C₈×C₁₀₉	—	2	1	no	—
91	C₂×C₈×C₁₀₉	—	2	1	no	—
92	C₂×C₈×C₁₀₉	—	2	1	no	—
93	C₂×C₈×C₁₀₉	—	2	1	no	—
94	C₂×C₈×C₁₀₉	—	2	2	no	—
95	C₂×C₈×C₁₀₉	—	2	1	no	—
96	C₂×C₈×C₁₀₉	—	2	2	no	—
97	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
103	C₄⋊C₄×C₁₀₉	—	2	4	no	—
104	C₄⋊C₄×C₁₀₉	—	4	4	no	—
105	C₄⋊C₄×C₁₀₉	C₄²	1	8	yes	—
106	C₄⋊C₄×C₁₀₉	C₄²	1	4	yes	—
107	C₄²×C₁₀₉	—	2	4	no	—
108	C₄²×C₁₀₉	—	2	2	no	—
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	—	2	1	no	—
111	C₄²×C₁₀₉	—	2	1	no	—
112	C₄²×C₁₀₉	—	2	1	no	—
113	C₄²×C₁₀₉	—	2	2	no	—
114	C₄²×C₁₀₉	—	4	2	no	—
115	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
116	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
117	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
118	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
119	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₁₀₉	—	2	8	no	—
121	C₂×C₄×C₁₀₉	—	2	4	no	—
122	C₂×C₄×C₁₀₉	—	2	2	no	—
123	C₂×C₄×C₁₀₉	—	2	4	no	—
124	C₂×C₄×C₁₀₉	—	2	2	no	—
125	C₂×C₄×C₁₀₉	—	2	4	no	—
126	C₂×C₄×C₁₀₉	—	2	2	no	—
127	C₂×C₄×C₁₀₉	—	2	2	no	—
128	C₂×C₄×C₁₀₉	—	2	2	no	—
129	C₂×C₄×C₁₀₉	—	2	4	no	—
130	C₂×C₄×C₁₀₉	—	2	2	no	—
131	C₂×C₄×C₁₀₉	—	2	2	no	—
132	C₂×C₄×C₁₀₉	—	2	4	no	—
133	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
135	C₂×C₄×C₁₀₉	C₂×C₄²	1	4	yes	—
136	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₁₀₉	#131	1	4	yes	—
138	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₁₀₉	—	2	2	no	—
145	C₈×C₁₀₉	—	2	2	no	—
146	C₈×C₁₀₉	—	2	2	no	—
147	C₈×C₁₀₉	—	2	4	no	—
148	C₈×C₁₀₉	—	2	4	no	—
149	C₈×C₁₀₉	—	2	2	no	—
150	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
151	C₂²×C₁₀₉	#519	1	1	yes	characteristic
152	C₂²×C₁₀₉	#520	1	1	yes	characteristic
153	C₂²×C₁₀₉	C₂×#131	1	2	yes	—
154	C₂²×C₁₀₉	#568	1	2	yes	—
155	C₄×C₁₀₉	—	2	4	no	—
156	C₄×C₁₀₉	—	2	2	no	—
157	C₄×C₁₀₉	—	2	4	no	—
158	C₄×C₁₀₉	—	4	2	no	—
159	C₄×C₁₀₉	—	4	2	no	—
160	C₄×C₁₀₉	—	4	4	no	—
161	C₄×C₁₀₉	#405	1	4	yes	—
162	C₄×C₁₀₉	#519	1	1	yes	characteristic
163	C₄×C₁₀₉	#519	1	1	yes	characteristic
164	C₄×C₁₀₉	#520	1	1	yes	characteristic
165	C₄×C₁₀₉	#520	1	1	yes	characteristic
166	#20574	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₁₀₉	#2276	1	1	yes	characteristic
168	C₂×C₁₀₉	C₂×#405	1	1	yes	characteristic
169	C₂×C₁₀₉	#2875	1	1	yes	characteristic
170	C₂×C₁₀₉	#2985	1	2	yes	—
171	C₂×C₁₀₉	#2993	1	2	yes	—
172	C₂×#446	C₂×C₁₀₉	1	1	yes	characteristic
173	C₂×C₈×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
174	#2295	C₂×C₁₀₉	1	1	yes	characteristic
175	#1771	C₂×C₁₀₉	1	2	yes	—
176	#1770	C₂×C₁₀₉	1	2	yes	—
177	#1770	C₂×C₁₀₉	1	1	yes	characteristic
178	C₈×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
179	#1768	C₂×C₁₀₉	1	2	yes	—
180	#1767	C₂×C₁₀₉	1	1	yes	characteristic
181	#1746	C₂×C₁₀₉	1	2	yes	—
182	#1744	C₂×C₁₀₉	1	1	yes	characteristic
183	C₁₀₉	#20574	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
185	#446	C₄×C₁₀₉	1	8	yes	complemented
186	C₈×Q₈	C₄×C₁₀₉	1	8	yes	complemented
187	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
188	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
189	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
191	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
192	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
193	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
194	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
197	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
198	#391	C₄×C₁₀₉	1	2	yes	—
199	#389	C₄×C₁₀₉	1	2	yes	—
200	#387	C₄×C₁₀₉	1	4	yes	—
201	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
202	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
203	#376	C₂²×C₁₀₉	1	2	yes	—
204	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
205	#336	C₂²×C₁₀₉	1	1	yes	characteristic
206	#336	C₂²×C₁₀₉	1	1	yes	characteristic
207	#336	C₂²×C₁₀₉	1	1	yes	characteristic
208	#336	C₂²×C₁₀₉	1	1	yes	characteristic
209	#336	C₂²×C₁₀₉	1	2	yes	—
210	#336	C₂²×C₁₀₉	1	2	yes	—
211	#336	C₂²×C₁₀₉	1	2	yes	—
212	C₂²×Q₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
219	C₄×Q₈	C₂×C₄×C₁₀₉	1	4	yes	—
220	C₄×Q₈	C₂×C₄×C₁₀₉	1	4	yes	—
221	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	4	yes	—
223	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
225	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
226	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
227	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
230	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
232	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
233	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
234	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	—
235	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
236	#97	—	4	2	no	—
237	#97	—	4	2	no	—
238	#97	C₂×C₄×C₁₀₉	1	8	yes	—
239	C₄×C₈	—	4	2	no	—
240	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
241	C₄×C₈	C₂×C₄×C₁₀₉	1	8	yes	—
242	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
243	#95	C₂×C₄×C₁₀₉	1	4	yes	—
244	#95	C₂×C₄×C₁₀₉	1	4	yes	—
245	#95	C₂×C₄×C₁₀₉	1	2	yes	—
246	#95	C₂×C₄×C₁₀₉	1	4	yes	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₄²×C₁₀₉	1	4	yes	—
249	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
251	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
270	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
271	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
272	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
273	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
274	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	C₄²×C₁₀₉	1	8	yes	—
278	C₄⋊C₄	C₄²×C₁₀₉	1	4	yes	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
288	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
289	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
290	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
291	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
306	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
307	C₂×C₄	C₂×C₄²×C₁₀₉	1	4	yes	—
308	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
309	C₂×C₄	#131×C₁₀₉	1	4	yes	—
310	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
312	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
313	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
314	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
315	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
323	C₂²	#519×C₁₀₉	1	1	yes	characteristic
324	C₂²	#520×C₁₀₉	1	1	yes	characteristic
325	C₂²	C₂×#131×C₁₀₉	1	2	yes	—
326	C₂²	#568×C₁₀₉	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	#405×C₁₀₉	1	4	yes	—
334	C₄	#519×C₁₀₉	1	1	yes	characteristic
335	C₄	#519×C₁₀₉	1	1	yes	characteristic
336	C₄	#520×C₁₀₉	1	1	yes	characteristic
337	C₄	#520×C₁₀₉	1	1	yes	characteristic
338	C₂	C₁₀₉×#2276	1	1	yes	characteristic
339	C₂	C₂×#405×C₁₀₉	1	1	yes	characteristic
340	C₂	C₁₀₉×#2875	1	1	yes	characteristic
341	C₂	C₁₀₉×#2985	1	2	yes	—
342	C₂	C₁₀₉×#2993	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. #446×C₁₀₉	C₄	semidirect
14. C₈×Q₈×C₁₀₉	C₄	semidirect
166. #20574	C₁₀₉	direct (N × H)
183. C₁₀₉	#20574	direct (N × H)
185. #446	C₄×C₁₀₉	semidirect
186. C₈×Q₈	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₀₉	2
C₂×C₈×Q₈×C₁₀₉	2
C₁₀₉×#2295	2
C₁₀₉×#1771	2
C₁₀₉×#1770	2
C₈×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1768	2
C₁₀₉×#1767	2
C₁₀₉×#1746	2
C₁₀₉×#1744	2
#20574	109

Subgroups — C109×#20574 #42138568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20574 #42138568