Subgroups — C₁₀₉×#32724 #42150718

Order 27904.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2855	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2068	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2056	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#2043	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#2036	C₂	1	2	yes	maximal
9	C₁₀₉×#2016	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#2001	C₂	1	2	yes	maximal
11	C₁₀₉×#1797	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
22	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	#391×C₁₀₉	C₂²	1	1	yes	characteristic
25	#391×C₁₀₉	C₂²	1	1	yes	characteristic
26	#391×C₁₀₉	C₂²	1	2	yes	—
27	#389×C₁₀₉	—	2	1	no	—
28	#389×C₁₀₉	—	2	1	no	—
29	#389×C₁₀₉	C₂²	1	1	yes	characteristic
30	#389×C₁₀₉	C₂²	1	1	yes	characteristic
31	#387×C₁₀₉	—	2	1	no	—
32	#387×C₁₀₉	—	2	1	no	—
33	#385×C₁₀₉	—	2	1	no	—
34	#384×C₁₀₉	C₂²	1	1	yes	characteristic
35	#381×C₁₀₉	—	2	1	no	—
36	#380×C₁₀₉	—	2	1	no	—
37	#380×C₁₀₉	—	2	1	no	—
38	#380×C₁₀₉	C₂²	1	1	yes	characteristic
39	#340×C₁₀₉	C₂²	1	2	yes	—
40	#340×C₁₀₉	C₂²	1	2	yes	—
41	#336×C₁₀₉	C₂²	1	1	yes	characteristic
42	#336×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₁₀₉	C₂³	1	2	yes	—
44	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
49	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
50	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
51	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
54	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
55	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
56	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
57	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂×C₄²×C₁₀₉	—	4	2	no	—
73	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
77	#107×C₁₀₉	—	4	2	no	—
78	C₄⋊C₈×C₁₀₉	—	4	2	no	—
79	#103×C₁₀₉	—	4	2	no	—
80	#103×C₁₀₉	—	4	2	no	—
81	#103×C₁₀₉	—	4	2	no	—
82	#103×C₁₀₉	—	4	2	no	—
83	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
84	#95×C₁₀₉	—	2	2	no	—
85	#95×C₁₀₉	—	2	2	no	—
86	#95×C₁₀₉	—	2	1	no	—
87	#95×C₁₀₉	—	2	1	no	—
88	#95×C₁₀₉	—	2	1	no	—
89	#95×C₁₀₉	—	2	1	no	—
90	#95×C₁₀₉	—	2	1	no	—
91	#95×C₁₀₉	—	2	1	no	—
92	#95×C₁₀₉	—	2	1	no	—
93	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂×Q₈×C₁₀₉	—	2	2	no	—
95	C₂×Q₈×C₁₀₉	—	2	2	no	—
96	C₂×Q₈×C₁₀₉	—	2	2	no	—
97	C₂×Q₈×C₁₀₉	—	2	2	no	—
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	—	2	2	no	—
100	C₂×Q₈×C₁₀₉	—	2	2	no	—
101	C₂×Q₈×C₁₀₉	—	2	1	no	—
102	C₂×Q₈×C₁₀₉	—	2	2	no	—
103	C₂×Q₈×C₁₀₉	—	2	1	no	—
104	C₂²×C₄×C₁₀₉	—	2	2	no	—
105	C₂²×C₄×C₁₀₉	—	2	1	no	—
106	C₂²×C₄×C₁₀₉	—	2	2	no	—
107	C₂²×C₄×C₁₀₉	—	2	2	no	—
108	C₂²×C₄×C₁₀₉	—	2	1	no	—
109	C₂²×C₄×C₁₀₉	—	2	1	no	—
110	C₂²×C₄×C₁₀₉	—	2	1	no	—
111	C₂²×C₄×C₁₀₉	—	2	1	no	—
112	C₂²×C₄×C₁₀₉	—	2	1	no	—
113	C₂²×C₄×C₁₀₉	—	2	2	no	—
114	C₂²×C₄×C₁₀₉	—	2	2	no	—
115	C₂²×C₄×C₁₀₉	—	4	1	no	—
116	C₂²×C₄×C₁₀₉	—	4	1	no	—
117	C₂²×C₄×C₁₀₉	—	4	1	no	—
118	C₂²×C₄×C₁₀₉	—	4	1	no	—
119	C₂²×C₄×C₁₀₉	Q₁₆	1	4	yes	complemented
120	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₁₀₉	—	4	8	no	—
127	C₂×C₈×C₁₀₉	—	2	2	no	—
128	C₂×C₈×C₁₀₉	—	2	2	no	—
129	C₂×C₈×C₁₀₉	—	2	2	no	—
130	C₂×C₈×C₁₀₉	—	2	2	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	—	2	2	no	—
133	C₄⋊C₄×C₁₀₉	—	2	1	no	—
134	C₄⋊C₄×C₁₀₉	—	2	2	no	—
135	C₄⋊C₄×C₁₀₉	—	2	2	no	—
136	C₄⋊C₄×C₁₀₉	—	2	2	no	—
137	C₄⋊C₄×C₁₀₉	—	2	2	no	—
138	C₄⋊C₄×C₁₀₉	—	2	1	no	—
139	C₄⋊C₄×C₁₀₉	—	4	2	no	—
140	C₄⋊C₄×C₁₀₉	—	4	2	no	—
141	C₄⋊C₄×C₁₀₉	—	4	2	no	—
142	C₄⋊C₄×C₁₀₉	—	4	2	no	—
143	C₄⋊C₄×C₁₀₉	—	4	4	no	—
144	C₄⋊C₄×C₁₀₉	—	4	2	no	—
145	C₄⋊C₄×C₁₀₉	—	4	2	no	—
146	C₄⋊C₄×C₁₀₉	—	4	2	no	—
147	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
148	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
149	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
150	C₄²×C₁₀₉	—	2	2	no	—
151	C₄²×C₁₀₉	—	2	2	no	—
152	C₄²×C₁₀₉	—	4	1	no	—
153	C₄²×C₁₀₉	—	4	1	no	—
154	C₂³×C₁₀₉	—	2	1	no	—
155	C₂³×C₁₀₉	—	2	1	no	—
156	C₂³×C₁₀₉	—	2	1	no	—
157	C₂³×C₁₀₉	—	2	1	no	—
158	C₂³×C₁₀₉	—	2	1	no	—
159	C₂³×C₁₀₉	—	2	1	no	—
160	C₂³×C₁₀₉	C₂×Q₁₆	1	2	yes	—
161	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₁₀₉	—	4	2	no	—
163	Q₈×C₁₀₉	—	4	2	no	—
164	Q₈×C₁₀₉	—	4	2	no	—
165	Q₈×C₁₀₉	—	4	2	no	—
166	Q₈×C₁₀₉	—	4	2	no	—
167	Q₈×C₁₀₉	—	4	2	no	—
168	C₂×C₄×C₁₀₉	—	2	1	no	—
169	C₂×C₄×C₁₀₉	—	2	1	no	—
170	C₂×C₄×C₁₀₉	—	2	1	no	—
171	C₂×C₄×C₁₀₉	—	2	1	no	—
172	C₂×C₄×C₁₀₉	—	2	1	no	—
173	C₂×C₄×C₁₀₉	—	2	1	no	—
174	C₂×C₄×C₁₀₉	—	2	1	no	—
175	C₂×C₄×C₁₀₉	—	2	1	no	—
176	C₂×C₄×C₁₀₉	—	2	1	no	—
177	C₂×C₄×C₁₀₉	—	2	1	no	—
178	C₂×C₄×C₁₀₉	—	2	1	no	—
179	C₂×C₄×C₁₀₉	—	2	1	no	—
180	C₂×C₄×C₁₀₉	—	4	4	no	—
181	C₂×C₄×C₁₀₉	—	4	4	no	—
182	C₂×C₄×C₁₀₉	—	4	2	no	—
183	C₂×C₄×C₁₀₉	—	4	2	no	—
184	C₂×C₄×C₁₀₉	—	4	1	no	—
185	C₂×C₄×C₁₀₉	—	4	1	no	—
186	C₂×C₄×C₁₀₉	—	4	2	no	—
187	C₂×C₄×C₁₀₉	—	4	2	no	—
188	C₂×C₄×C₁₀₉	—	4	2	no	—
189	C₂×C₄×C₁₀₉	—	4	1	no	—
190	C₂×C₄×C₁₀₉	—	4	1	no	—
191	C₂×C₄×C₁₀₉	—	4	1	no	—
192	C₂×C₄×C₁₀₉	—	4	1	no	—
193	C₂×C₄×C₁₀₉	—	4	2	no	—
194	C₂×C₄×C₁₀₉	—	4	1	no	—
195	C₂×C₄×C₁₀₉	—	4	1	no	—
196	C₂×C₄×C₁₀₉	—	4	1	no	—
197	C₂×C₄×C₁₀₉	—	4	1	no	—
198	C₂×C₄×C₁₀₉	—	4	1	no	—
199	C₂×C₄×C₁₀₉	—	4	2	no	—
200	C₂×C₄×C₁₀₉	—	4	1	no	—
201	C₂×C₄×C₁₀₉	—	4	2	no	—
202	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
203	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
204	C₂×C₄×C₁₀₉	#137	1	2	yes	—
205	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
207	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
208	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₁₀₉	—	4	2	no	—
210	C₈×C₁₀₉	—	4	2	no	—
211	C₂²×C₁₀₉	—	2	2	no	—
212	C₂²×C₁₀₉	—	2	2	no	—
213	C₂²×C₁₀₉	—	2	2	no	—
214	C₂²×C₁₀₉	—	4	1	no	—
215	C₂²×C₁₀₉	—	4	1	no	—
216	C₂²×C₁₀₉	—	4	1	no	—
217	C₂²×C₁₀₉	—	4	1	no	—
218	C₂²×C₁₀₉	#534	1	1	yes	characteristic
219	C₂²×C₁₀₉	#535	1	1	yes	characteristic
220	C₂²×C₁₀₉	#537	1	1	yes	characteristic
221	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
222	C₂²×C₁₀₉	C₂×#137	1	1	yes	characteristic
223	C₂²×C₁₀₉	#575	1	1	yes	characteristic
224	C₂²×C₁₀₉	#575	1	1	yes	characteristic
225	C₄×C₁₀₉	—	2	2	no	—
226	C₄×C₁₀₉	—	2	2	no	—
227	C₄×C₁₀₉	—	2	2	no	—
228	C₄×C₁₀₉	—	4	1	no	—
229	C₄×C₁₀₉	—	4	1	no	—
230	C₄×C₁₀₉	—	8	1	no	—
231	C₄×C₁₀₉	—	8	2	no	—
232	C₄×C₁₀₉	—	8	1	no	—
233	C₄×C₁₀₉	—	8	1	no	—
234	C₄×C₁₀₉	—	8	1	no	—
235	C₄×C₁₀₉	—	8	2	no	—
236	C₄×C₁₀₉	#560	1	2	yes	—
237	C₄×C₁₀₉	#562	1	2	yes	—
238	#32724	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₁₀₉	—	4	2	no	—
240	C₂×C₁₀₉	#2813	1	1	yes	characteristic
241	C₂×C₁₀₉	#3189	1	1	yes	characteristic
242	C₂×C₁₀₉	#3196	1	1	yes	characteristic
243	C₂×C₁₀₉	#3209	1	1	yes	characteristic
244	C₂×C₁₀₉	#3219	1	1	yes	characteristic
245	C₂×C₁₀₉	#3249	1	1	yes	characteristic
246	C₂×C₁₀₉	#3252	1	1	yes	characteristic
247	#2857	C₂×C₁₀₉	1	1	yes	characteristic
248	#2855	C₂×C₁₀₉	1	1	yes	characteristic
249	#2100	C₂×C₁₀₉	1	1	yes	characteristic, complemented
250	#2095	C₂×C₁₀₉	1	1	yes	characteristic, complemented
251	#2068	C₂×C₁₀₉	1	2	yes	complemented
252	#2056	C₂×C₁₀₉	1	2	yes	complemented
253	#2043	C₂×C₁₀₉	1	1	yes	characteristic, complemented
254	#2036	C₂×C₁₀₉	1	2	yes	—
255	#2016	C₂×C₁₀₉	1	1	yes	characteristic, complemented
256	#2001	C₂×C₁₀₉	1	2	yes	—
257	#1797	C₂×C₁₀₉	1	1	yes	characteristic
258	C₁₀₉	#32724	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
260	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
261	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
262	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
263	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
264	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
265	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
266	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
267	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
268	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
269	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
270	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
271	#391	C₂²×C₁₀₉	1	1	yes	characteristic
272	#391	C₂²×C₁₀₉	1	2	yes	—
273	#391	C₂²×C₁₀₉	1	1	yes	characteristic
274	#389	—	2	1	no	—
275	#389	—	2	1	no	—
276	#389	C₂²×C₁₀₉	1	1	yes	characteristic
277	#389	C₂²×C₁₀₉	1	1	yes	characteristic
278	#387	—	2	1	no	—
279	#387	—	2	1	no	—
280	#385	—	2	1	no	—
281	#384	C₂²×C₁₀₉	1	1	yes	characteristic
282	#381	—	2	1	no	—
283	#380	—	2	1	no	—
284	#380	—	2	1	no	—
285	#380	C₂²×C₁₀₉	1	1	yes	characteristic
286	#340	C₂²×C₁₀₉	1	2	yes	—
287	#340	C₂²×C₁₀₉	1	2	yes	—
288	#336	C₂²×C₁₀₉	1	1	yes	characteristic
289	#336	C₂²×C₁₀₉	1	1	yes	characteristic
290	C₂²×Q₈	C₂³×C₁₀₉	1	2	yes	—
291	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
292	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
293	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
295	C₂×Q₁₆	—	4	2	no	—
296	C₂×Q₁₆	—	4	2	no	—
297	C₂×Q₁₆	—	4	2	no	—
298	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
299	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
300	C₄⋊Q₈	—	4	2	no	—
301	C₂²⋊Q₈	—	4	2	no	—
302	C₂²⋊Q₈	—	4	2	no	—
303	C₂²⋊Q₈	—	4	2	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₄²	—	4	2	no	—
320	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
321	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
324	#107	—	4	2	no	—
325	C₄⋊C₈	—	4	2	no	—
326	#103	—	4	2	no	—
327	#103	—	4	2	no	—
328	#103	—	4	2	no	—
329	#103	—	4	2	no	—
330	C₂²⋊C₈	—	4	2	no	—
331	#95	—	2	2	no	—
332	#95	—	2	2	no	—
333	#95	—	2	1	no	—
334	#95	—	2	1	no	—
335	#95	—	2	1	no	—
336	#95	—	2	1	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	1	no	—
340	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	4	1	no	—
363	C₂²×C₄	—	4	1	no	—
364	C₂²×C₄	—	4	1	no	—
365	C₂²×C₄	—	4	1	no	—
366	C₂²×C₄	Q₁₆×C₁₀₉	1	4	yes	complemented
367	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
373	Q₁₆	—	4	8	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	4	4	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	4	1	no	—
400	C₄²	—	4	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	2	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	1	no	—
407	C₂³	C₂×Q₁₆×C₁₀₉	1	2	yes	—
408	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
409	Q₈	—	4	2	no	—
410	Q₈	—	4	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
450	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
451	C₂×C₄	#137×C₁₀₉	1	2	yes	—
452	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
453	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
454	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
455	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
456	C₈	—	4	2	no	—
457	C₈	—	4	2	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	4	1	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	#534×C₁₀₉	1	1	yes	characteristic
466	C₂²	#535×C₁₀₉	1	1	yes	characteristic
467	C₂²	#537×C₁₀₉	1	1	yes	characteristic
468	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
469	C₂²	C₂×#137×C₁₀₉	1	1	yes	characteristic
470	C₂²	#575×C₁₀₉	1	1	yes	characteristic
471	C₂²	#575×C₁₀₉	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	1	no	—
477	C₄	—	8	1	no	—
478	C₄	—	8	2	no	—
479	C₄	—	8	1	no	—
480	C₄	—	8	1	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	2	no	—
483	C₄	#560×C₁₀₉	1	2	yes	—
484	C₄	#562×C₁₀₉	1	2	yes	—
485	C₂	—	4	2	no	—
486	C₂	C₁₀₉×#2813	1	1	yes	characteristic
487	C₂	C₁₀₉×#3189	1	1	yes	characteristic
488	C₂	C₁₀₉×#3196	1	1	yes	characteristic
489	C₂	C₁₀₉×#3209	1	1	yes	characteristic
490	C₂	C₁₀₉×#3219	1	1	yes	characteristic
491	C₂	C₁₀₉×#3249	1	1	yes	characteristic
492	C₂	C₁₀₉×#3252	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₉×#2100	C₂	semidirect
4. C₁₀₉×#2095	C₂	semidirect
5. C₁₀₉×#2068	C₂	semidirect
6. C₁₀₉×#2056	C₂	semidirect
7. C₁₀₉×#2043	C₂	semidirect
9. C₁₀₉×#2016	C₂	semidirect
119. C₂²×C₄×C₁₀₉	Q₁₆	semidirect
238. #32724	C₁₀₉	direct (N × H)
249. #2100	C₂×C₁₀₉	semidirect
250. #2095	C₂×C₁₀₉	semidirect
251. #2068	C₂×C₁₀₉	semidirect
252. #2056	C₂×C₁₀₉	semidirect
253. #2043	C₂×C₁₀₉	semidirect
255. #2016	C₂×C₁₀₉	semidirect
258. C₁₀₉	#32724	direct (N × H)
366. C₂²×C₄	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2857	2
C₁₀₉×#2855	2
C₁₀₉×#2100	2
C₁₀₉×#2095	2
C₁₀₉×#2068	2
C₁₀₉×#2056	2
C₁₀₉×#2043	2
C₁₀₉×#2036	2
C₁₀₉×#2016	2
C₁₀₉×#2001	2
C₁₀₉×#1797	2
#32724	109

Subgroups — C109×#32724 #42150718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32724 #42150718