Subgroups — C₁₀₉×#34189 #42152183

Order 27904.

2342 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2357	C₂	1	4	yes	maximal, complemented
3	C₁₀₉×#2356	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2353	C₂	1	8	yes	maximal, complemented
5	C₁₀₉×#2314	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#2311	C₂	1	2	yes	maximal
7	C₁₀₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#2283	C₂	1	4	yes	maximal, complemented
9	C₂×#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#2274	C₂	1	2	yes	maximal, complemented
11	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×#41×C₁₀₉	C₄	1	16	yes	complemented
13	C₂×C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
17	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	—
22	#391×C₁₀₉	C₂²	1	4	yes	—
23	#390×C₁₀₉	C₂²	1	4	yes	—
24	#388×C₁₀₉	C₂²	1	16	yes	—
25	#387×C₁₀₉	C₂²	1	16	yes	—
26	#386×C₁₀₉	C₂²	1	8	yes	—
27	#384×C₁₀₉	C₂²	1	2	yes	—
28	#383×C₁₀₉	C₂²	1	8	yes	—
29	#382×C₁₀₉	C₂²	1	2	yes	—
30	#382×C₁₀₉	C₂²	1	2	yes	—
31	#382×C₁₀₉	C₂²	1	4	yes	—
32	#382×C₁₀₉	C₂²	1	2	yes	—
33	#381×C₁₀₉	C₂²	1	8	yes	—
34	#380×C₁₀₉	C₂²	1	8	yes	—
35	#380×C₁₀₉	C₂²	1	4	yes	—
36	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	—
37	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	—
38	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
39	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
40	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
41	#376×C₁₀₉	C₂²	1	8	yes	—
42	C₂×#95×C₁₀₉	C₂²	1	4	yes	—
43	C₂×#41×C₁₀₉	—	2	4	no	—
44	C₂×#41×C₁₀₉	—	2	4	no	—
45	C₂×#41×C₁₀₉	C₂×C₄	1	8	yes	—
46	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
48	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
51	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
53	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₁₀₉	C₂×C₄	1	16	yes	complemented
55	C₄×D₄×C₁₀₉	C₂×C₄	1	32	yes	—
56	C₄×D₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
57	#117×C₁₀₉	—	2	8	no	—
58	#117×C₁₀₉	C₂×C₄	1	8	yes	—
59	#117×C₁₀₉	C₂×C₄	1	16	yes	complemented
60	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
61	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
68	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
69	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
71	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
72	C₂×C₄²×C₁₀₉	C₂×C₄	1	16	yes	complemented
73	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
74	C₂×C₄²×C₁₀₉	C₂³	1	16	yes	—
75	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
77	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
79	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
80	#95×C₁₀₉	C₂³	1	8	yes	—
81	#95×C₁₀₉	C₂³	1	8	yes	—
82	#95×C₁₀₉	C₂³	1	4	yes	—
83	#95×C₁₀₉	C₂³	1	16	yes	—
84	#95×C₁₀₉	C₂³	1	4	yes	—
85	#95×C₁₀₉	C₂³	1	16	yes	—
86	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
87	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	#41×C₁₀₉	—	2	32	no	—
89	C₂×Q₈×C₁₀₉	—	2	2	no	—
90	C₂×Q₈×C₁₀₉	—	2	2	no	—
91	C₂×Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₁₀₉	—	2	4	no	—
93	C₂×D₄×C₁₀₉	—	2	4	no	—
94	C₂×D₄×C₁₀₉	—	2	2	no	—
95	C₂×D₄×C₁₀₉	—	2	2	no	—
96	C₂×D₄×C₁₀₉	C₂²×C₄	1	8	yes	—
97	C₂×D₄×C₁₀₉	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₁₀₉	—	2	4	no	—
99	C₂²×C₄×C₁₀₉	—	2	4	no	—
100	C₂²×C₄×C₁₀₉	—	2	8	no	—
101	C₂²×C₄×C₁₀₉	—	4	8	no	—
102	C₂²×C₄×C₁₀₉	—	4	4	no	—
103	C₂²×C₄×C₁₀₉	—	4	8	no	—
104	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
105	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
106	C₂²×C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
107	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
109	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
110	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
112	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
117	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₁₀₉	—	2	8	no	—
119	C₄⋊C₄×C₁₀₉	—	2	8	no	—
120	C₄⋊C₄×C₁₀₉	—	2	8	no	—
121	C₄⋊C₄×C₁₀₉	—	4	8	no	—
122	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
123	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
124	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
125	C₂²⋊C₄×C₁₀₉	—	4	16	no	—
126	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
127	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
128	C₄²×C₁₀₉	—	2	8	no	—
129	C₄²×C₁₀₉	—	4	16	no	—
130	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
131	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
132	C₄²×C₁₀₉	C₂²×C₄	1	8	yes	—
133	C₄²×C₁₀₉	C₂²×C₄	1	8	yes	—
134	C₄²×C₁₀₉	C₂²×C₄	1	16	yes	—
135	C₂³×C₁₀₉	—	2	4	no	—
136	C₂³×C₁₀₉	—	2	2	no	—
137	C₂³×C₁₀₉	—	2	2	no	—
138	C₂³×C₁₀₉	—	2	8	no	—
139	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
140	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
141	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
142	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
143	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
144	Q₈×C₁₀₉	—	2	8	no	—
145	D₄×C₁₀₉	—	2	16	no	—
146	D₄×C₁₀₉	—	2	8	no	—
147	C₂×C₄×C₁₀₉	—	2	2	no	—
148	C₂×C₄×C₁₀₉	—	2	2	no	—
149	C₂×C₄×C₁₀₉	—	2	8	no	—
150	C₂×C₄×C₁₀₉	—	2	4	no	—
151	C₂×C₄×C₁₀₉	—	2	4	no	—
152	C₂×C₄×C₁₀₉	—	2	16	no	—
153	C₂×C₄×C₁₀₉	—	2	8	no	—
154	C₂×C₄×C₁₀₉	—	2	8	no	—
155	C₂×C₄×C₁₀₉	—	2	8	no	—
156	C₂×C₄×C₁₀₉	—	2	16	no	—
157	C₂×C₄×C₁₀₉	—	2	8	no	—
158	C₂×C₄×C₁₀₉	—	4	8	no	—
159	C₂×C₄×C₁₀₉	—	4	16	no	—
160	C₂×C₄×C₁₀₉	—	8	8	no	—
161	C₂×C₄×C₁₀₉	#117	1	16	yes	—
162	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
163	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
164	C₂×C₄×C₁₀₉	C₂³×C₄	1	8	yes	—
165	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
166	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
168	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
169	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
170	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
171	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
172	C₂²×C₁₀₉	—	2	4	no	—
173	C₂²×C₁₀₉	—	2	4	no	—
174	C₂²×C₁₀₉	—	2	4	no	—
175	C₂²×C₁₀₉	—	2	4	no	—
176	C₂²×C₁₀₉	—	2	8	no	—
177	C₂²×C₁₀₉	—	4	4	no	—
178	C₂²×C₁₀₉	—	4	4	no	—
179	C₂²×C₁₀₉	C₂×#117	1	4	yes	—
180	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
181	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
182	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
183	C₂²×C₁₀₉	#585	1	4	yes	—
184	C₄×C₁₀₉	—	2	4	no	—
185	C₄×C₁₀₉	—	2	4	no	—
186	C₄×C₁₀₉	—	2	8	no	—
187	C₄×C₁₀₉	—	2	4	no	—
188	C₄×C₁₀₉	—	4	8	no	—
189	C₄×C₁₀₉	—	4	8	no	—
190	C₄×C₁₀₉	C₂×#117	1	4	yes	—
191	C₄×C₁₀₉	C₂×#117	1	4	yes	—
192	#34189	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₁₀₉	—	2	4	no	—
194	C₂×C₁₀₉	—	4	4	no	—
195	C₂×C₁₀₉	C₂²×#117	1	1	yes	characteristic
196	C₂×C₁₀₉	#3428	1	2	yes	—
197	C₂×C₁₀₉	#3510	1	2	yes	—
198	C₂×C₁₀₉	#3512	1	2	yes	—
199	C₂×C₄×#41	C₂×C₁₀₉	1	1	yes	characteristic
200	#2357	C₂×C₁₀₉	1	4	yes	complemented
201	#2356	C₂×C₁₀₉	1	4	yes	complemented
202	#2353	C₂×C₁₀₉	1	8	yes	complemented
203	#2314	C₂×C₁₀₉	1	4	yes	complemented
204	#2311	C₂×C₁₀₉	1	2	yes	—
205	#2285	C₂×C₁₀₉	1	1	yes	characteristic, complemented
206	#2283	C₂×C₁₀₉	1	4	yes	complemented
207	C₂×#382	C₂×C₁₀₉	1	1	yes	characteristic, complemented
208	#2274	C₂×C₁₀₉	1	2	yes	complemented
209	C₁₀₉	#34189	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
211	C₄×#41	C₄×C₁₀₉	1	16	yes	complemented
212	C₂×C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
213	C₂×C₄×D₄	C₂²×C₁₀₉	1	4	yes	—
214	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
215	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
216	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
217	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
218	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
219	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
220	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	—
221	#391	C₂²×C₁₀₉	1	4	yes	—
222	#390	C₂²×C₁₀₉	1	4	yes	—
223	#388	C₂²×C₁₀₉	1	16	yes	—
224	#387	C₂²×C₁₀₉	1	16	yes	—
225	#386	C₂²×C₁₀₉	1	8	yes	—
226	#384	C₂²×C₁₀₉	1	2	yes	—
227	#383	C₂²×C₁₀₉	1	8	yes	—
228	#382	C₂²×C₁₀₉	1	2	yes	—
229	#382	C₂²×C₁₀₉	1	4	yes	—
230	#382	C₂²×C₁₀₉	1	2	yes	—
231	#382	C₂²×C₁₀₉	1	2	yes	—
232	#381	C₂²×C₁₀₉	1	8	yes	—
233	#380	C₂²×C₁₀₉	1	8	yes	—
234	#380	C₂²×C₁₀₉	1	4	yes	—
235	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	—
236	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	—
237	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
238	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
239	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
240	#376	C₂²×C₁₀₉	1	8	yes	—
241	C₂×#95	C₂²×C₁₀₉	1	4	yes	—
242	C₂×#41	—	2	4	no	—
243	C₂×#41	—	2	4	no	—
244	C₂×#41	C₂×C₄×C₁₀₉	1	8	yes	—
245	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
246	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
247	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
248	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
249	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
250	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
251	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
252	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
253	C₄×Q₈	C₂×C₄×C₁₀₉	1	16	yes	complemented
254	C₄×D₄	C₂×C₄×C₁₀₉	1	32	yes	—
255	C₄×D₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
256	#117	—	2	8	no	—
257	#117	C₂×C₄×C₁₀₉	1	8	yes	—
258	#117	C₂×C₄×C₁₀₉	1	16	yes	complemented
259	C₂×C₄⋊C₄	—	2	8	no	—
260	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
261	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
262	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
263	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	—
264	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
265	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
266	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
267	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
268	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
270	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
271	C₂×C₄²	C₂×C₄×C₁₀₉	1	16	yes	complemented
272	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
273	C₂×C₄²	C₂³×C₁₀₉	1	16	yes	—
274	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
275	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
277	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
278	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
279	#95	C₂³×C₁₀₉	1	8	yes	—
280	#95	C₂³×C₁₀₉	1	8	yes	—
281	#95	C₂³×C₁₀₉	1	4	yes	—
282	#95	C₂³×C₁₀₉	1	16	yes	—
283	#95	C₂³×C₁₀₉	1	4	yes	—
284	#95	C₂³×C₁₀₉	1	16	yes	—
285	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
286	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
287	#41	—	2	32	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	C₂²×C₄×C₁₀₉	1	4	yes	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	2	4	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	C₂²×C₄×C₁₀₉	1	8	yes	—
296	C₂×D₄	C₂²×C₄×C₁₀₉	1	4	yes	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	4	8	no	—
301	C₂²×C₄	—	4	4	no	—
302	C₂²×C₄	—	4	8	no	—
303	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
304	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
305	C₂²×C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
306	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
307	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
308	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
309	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
310	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
311	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
312	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
314	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
315	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
316	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	2	8	no	—
320	C₄⋊C₄	—	4	8	no	—
321	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
322	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
323	C₂²⋊C₄	—	2	8	no	—
324	C₂²⋊C₄	—	4	16	no	—
325	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
326	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
327	C₄²	—	2	8	no	—
328	C₄²	—	4	16	no	—
329	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
330	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
331	C₄²	C₂²×C₄×C₁₀₉	1	8	yes	—
332	C₄²	C₂²×C₄×C₁₀₉	1	8	yes	—
333	C₄²	C₂²×C₄×C₁₀₉	1	16	yes	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	8	no	—
338	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
339	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
340	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
341	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
342	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
343	Q₈	—	2	8	no	—
344	D₄	—	2	16	no	—
345	D₄	—	2	8	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	16	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	8	no	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	2	16	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	4	8	no	—
358	C₂×C₄	—	4	16	no	—
359	C₂×C₄	—	8	8	no	—
360	C₂×C₄	#117×C₁₀₉	1	16	yes	—
361	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
362	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
363	C₂×C₄	C₂³×C₄×C₁₀₉	1	8	yes	—
364	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
365	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
366	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
367	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
368	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
369	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
370	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	8	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	4	4	no	—
378	C₂²	C₂×#117×C₁₀₉	1	4	yes	—
379	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
380	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
381	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
382	C₂²	#585×C₁₀₉	1	4	yes	—
383	C₄	—	2	4	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	8	no	—
386	C₄	—	2	4	no	—
387	C₄	—	4	8	no	—
388	C₄	—	4	8	no	—
389	C₄	C₂×#117×C₁₀₉	1	4	yes	—
390	C₄	C₂×#117×C₁₀₉	1	4	yes	—
391	C₂	—	2	4	no	—
392	C₂	—	4	4	no	—
393	C₂	C₂²×#117×C₁₀₉	1	1	yes	characteristic
394	C₂	C₁₀₉×#3428	1	2	yes	—
395	C₂	C₁₀₉×#3510	1	2	yes	—
396	C₂	C₁₀₉×#3512	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2357	C₂	semidirect
3. C₁₀₉×#2356	C₂	semidirect
4. C₁₀₉×#2353	C₂	semidirect
5. C₁₀₉×#2314	C₂	semidirect
7. C₁₀₉×#2285	C₂	semidirect
8. C₁₀₉×#2283	C₂	semidirect
9. C₂×#382×C₁₀₉	C₂	semidirect
10. C₁₀₉×#2274	C₂	semidirect
12. C₄×#41×C₁₀₉	C₄	semidirect
54. C₄×Q₈×C₁₀₉	C₂×C₄	semidirect
56. C₄×D₄×C₁₀₉	C₂×C₄	semidirect
59. #117×C₁₀₉	C₂×C₄	semidirect
72. C₂×C₄²×C₁₀₉	C₂×C₄	semidirect
192. #34189	C₁₀₉	direct (N × H)
200. #2357	C₂×C₁₀₉	semidirect
201. #2356	C₂×C₁₀₉	semidirect
202. #2353	C₂×C₁₀₉	semidirect
203. #2314	C₂×C₁₀₉	semidirect
205. #2285	C₂×C₁₀₉	semidirect
206. #2283	C₂×C₁₀₉	semidirect
207. C₂×#382	C₂×C₁₀₉	semidirect
208. #2274	C₂×C₁₀₉	semidirect
209. C₁₀₉	#34189	direct (N × H)
211. C₄×#41	C₄×C₁₀₉	semidirect
253. C₄×Q₈	C₂×C₄×C₁₀₉	semidirect
255. C₄×D₄	C₂×C₄×C₁₀₉	semidirect
258. #117	C₂×C₄×C₁₀₉	semidirect
271. C₂×C₄²	C₂×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×C₄×#41×C₁₀₉	2
C₁₀₉×#2357	2
C₁₀₉×#2356	2
C₁₀₉×#2353	2
C₁₀₉×#2314	2
C₁₀₉×#2311	2
C₁₀₉×#2285	2
C₁₀₉×#2283	2
C₂×#382×C₁₀₉	2
C₁₀₉×#2274	2
#34189	109

Subgroups — C109×#34189 #42152183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#34189 #42152183