Subgroups — C₁₀₉×#32740 #42150734

Order 27904.

702 conjugacy classes of subgroups.

The subgroup lattice has 472 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×#2088	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2077	C₂	1	2	yes	maximal
6	C₁₀₉×#2072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#2066	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#2058	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#2035	C₂	1	2	yes	maximal
10	C₁₀₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1797	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
17	C₂×#106×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×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
22	#391×C₁₀₉	—	2	1	no	—
23	#391×C₁₀₉	C₂²	1	1	yes	characteristic
24	#391×C₁₀₉	C₂²	1	1	yes	characteristic
25	#389×C₁₀₉	—	2	1	no	—
26	#389×C₁₀₉	C₂²	1	1	yes	characteristic
27	#389×C₁₀₉	C₂²	1	2	yes	—
28	#389×C₁₀₉	C₂²	1	1	yes	characteristic
29	#387×C₁₀₉	—	2	1	no	—
30	#387×C₁₀₉	—	2	1	no	—
31	#385×C₁₀₉	—	2	1	no	—
32	#384×C₁₀₉	C₂²	1	1	yes	characteristic
33	#381×C₁₀₉	—	2	1	no	—
34	#380×C₁₀₉	—	2	1	no	—
35	#380×C₁₀₉	C₂²	1	2	yes	—
36	#380×C₁₀₉	C₂²	1	1	yes	characteristic
37	#340×C₁₀₉	C₂²	1	2	yes	—
38	#340×C₁₀₉	C₂²	1	2	yes	—
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	—	2	1	no	—
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	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
51	C₂²⋊Q₈×C₁₀₉	—	4	2	no	—
52	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
53	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
68	C₂×C₄²×C₁₀₉	—	4	2	no	—
69	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	#106×C₁₀₉	—	4	2	no	—
74	#106×C₁₀₉	—	4	2	no	—
75	C₄⋊C₈×C₁₀₉	—	4	2	no	—
76	#103×C₁₀₉	—	4	2	no	—
77	#103×C₁₀₉	—	4	2	no	—
78	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
79	#95×C₁₀₉	—	2	2	no	—
80	#95×C₁₀₉	—	2	1	no	—
81	#95×C₁₀₉	—	2	2	no	—
82	#95×C₁₀₉	—	2	1	no	—
83	#95×C₁₀₉	—	2	1	no	—
84	#95×C₁₀₉	—	2	1	no	—
85	#95×C₁₀₉	—	2	1	no	—
86	#95×C₁₀₉	—	2	1	no	—
87	#95×C₁₀₉	—	2	1	no	—
88	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
89	C₂×Q₈×C₁₀₉	—	2	1	no	—
90	C₂×Q₈×C₁₀₉	—	2	2	no	—
91	C₂×Q₈×C₁₀₉	—	2	2	no	—
92	C₂×Q₈×C₁₀₉	—	2	1	no	—
93	C₂×Q₈×C₁₀₉	—	4	2	no	—
94	C₂×Q₈×C₁₀₉	—	4	2	no	—
95	C₂×Q₈×C₁₀₉	—	4	2	no	—
96	C₂²×C₄×C₁₀₉	—	2	2	no	—
97	C₂²×C₄×C₁₀₉	—	2	1	no	—
98	C₂²×C₄×C₁₀₉	—	2	2	no	—
99	C₂²×C₄×C₁₀₉	—	2	2	no	—
100	C₂²×C₄×C₁₀₉	—	2	1	no	—
101	C₂²×C₄×C₁₀₉	—	2	1	no	—
102	C₂²×C₄×C₁₀₉	—	2	1	no	—
103	C₂²×C₄×C₁₀₉	—	2	1	no	—
104	C₂²×C₄×C₁₀₉	—	2	1	no	—
105	C₂²×C₄×C₁₀₉	—	2	2	no	—
106	C₂²×C₄×C₁₀₉	—	2	2	no	—
107	C₂²×C₄×C₁₀₉	—	4	1	no	—
108	C₂²×C₄×C₁₀₉	—	4	1	no	—
109	C₂²×C₄×C₁₀₉	—	4	1	no	—
110	C₂²×C₄×C₁₀₉	—	4	1	no	—
111	C₂²×C₄×C₁₀₉	SD₁₆	1	4	yes	—
112	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₁₀₉	—	2	2	no	—
119	C₂×C₈×C₁₀₉	—	2	2	no	—
120	C₂×C₈×C₁₀₉	—	2	2	no	—
121	C₂×C₈×C₁₀₉	—	2	2	no	—
122	C₂×C₈×C₁₀₉	—	2	2	no	—
123	C₂×C₈×C₁₀₉	—	2	2	no	—
124	C₄⋊C₄×C₁₀₉	—	2	2	no	—
125	C₄⋊C₄×C₁₀₉	—	2	2	no	—
126	C₄⋊C₄×C₁₀₉	—	2	1	no	—
127	C₄⋊C₄×C₁₀₉	—	2	2	no	—
128	C₄⋊C₄×C₁₀₉	—	2	2	no	—
129	C₄⋊C₄×C₁₀₉	—	2	2	no	—
130	C₄⋊C₄×C₁₀₉	—	2	2	no	—
131	C₄⋊C₄×C₁₀₉	—	2	1	no	—
132	C₄⋊C₄×C₁₀₉	—	4	2	no	—
133	C₄⋊C₄×C₁₀₉	—	4	2	no	—
134	C₄⋊C₄×C₁₀₉	—	4	2	no	—
135	C₄⋊C₄×C₁₀₉	—	4	2	no	—
136	C₄⋊C₄×C₁₀₉	—	4	2	no	—
137	C₄⋊C₄×C₁₀₉	—	4	4	no	—
138	C₄⋊C₄×C₁₀₉	—	4	2	no	—
139	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
140	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
141	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
142	C₄²×C₁₀₉	—	2	2	no	—
143	C₄²×C₁₀₉	—	2	2	no	—
144	C₄²×C₁₀₉	—	4	1	no	—
145	C₄²×C₁₀₉	—	4	1	no	—
146	C₂³×C₁₀₉	—	2	1	no	—
147	C₂³×C₁₀₉	—	2	1	no	—
148	C₂³×C₁₀₉	—	2	1	no	—
149	C₂³×C₁₀₉	—	2	1	no	—
150	C₂³×C₁₀₉	—	2	1	no	—
151	C₂³×C₁₀₉	—	2	1	no	—
152	C₂³×C₁₀₉	C₂×SD₁₆	1	2	yes	—
153	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₁₀₉	—	4	2	no	—
155	Q₈×C₁₀₉	—	4	2	no	—
156	Q₈×C₁₀₉	—	8	4	no	—
157	C₂×C₄×C₁₀₉	—	2	1	no	—
158	C₂×C₄×C₁₀₉	—	2	1	no	—
159	C₂×C₄×C₁₀₉	—	2	1	no	—
160	C₂×C₄×C₁₀₉	—	2	1	no	—
161	C₂×C₄×C₁₀₉	—	2	1	no	—
162	C₂×C₄×C₁₀₉	—	2	1	no	—
163	C₂×C₄×C₁₀₉	—	2	1	no	—
164	C₂×C₄×C₁₀₉	—	2	1	no	—
165	C₂×C₄×C₁₀₉	—	2	1	no	—
166	C₂×C₄×C₁₀₉	—	2	1	no	—
167	C₂×C₄×C₁₀₉	—	2	1	no	—
168	C₂×C₄×C₁₀₉	—	2	1	no	—
169	C₂×C₄×C₁₀₉	—	4	4	no	—
170	C₂×C₄×C₁₀₉	—	4	4	no	—
171	C₂×C₄×C₁₀₉	—	4	2	no	—
172	C₂×C₄×C₁₀₉	—	4	2	no	—
173	C₂×C₄×C₁₀₉	—	4	1	no	—
174	C₂×C₄×C₁₀₉	—	4	1	no	—
175	C₂×C₄×C₁₀₉	—	4	2	no	—
176	C₂×C₄×C₁₀₉	—	4	1	no	—
177	C₂×C₄×C₁₀₉	—	4	1	no	—
178	C₂×C₄×C₁₀₉	—	4	2	no	—
179	C₂×C₄×C₁₀₉	—	4	1	no	—
180	C₂×C₄×C₁₀₉	—	4	1	no	—
181	C₂×C₄×C₁₀₉	—	4	2	no	—
182	C₂×C₄×C₁₀₉	—	4	2	no	—
183	C₂×C₄×C₁₀₉	—	4	1	no	—
184	C₂×C₄×C₁₀₉	—	4	1	no	—
185	C₂×C₄×C₁₀₉	—	4	1	no	—
186	C₂×C₄×C₁₀₉	—	4	1	no	—
187	C₂×C₄×C₁₀₉	—	4	1	no	—
188	C₂×C₄×C₁₀₉	—	4	1	no	—
189	C₂×C₄×C₁₀₉	—	4	2	no	—
190	C₂×C₄×C₁₀₉	—	4	2	no	—
191	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
192	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
193	C₂×C₄×C₁₀₉	#137	1	2	yes	—
194	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
196	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
197	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₁₀₉	—	4	2	no	—
199	C₈×C₁₀₉	—	4	2	no	—
200	C₂²×C₁₀₉	—	2	2	no	—
201	C₂²×C₁₀₉	—	2	2	no	—
202	C₂²×C₁₀₉	—	2	2	no	—
203	C₂²×C₁₀₉	—	4	1	no	—
204	C₂²×C₁₀₉	—	4	1	no	—
205	C₂²×C₁₀₉	—	4	1	no	—
206	C₂²×C₁₀₉	—	4	1	no	—
207	C₂²×C₁₀₉	#534	1	1	yes	characteristic
208	C₂²×C₁₀₉	#536	1	1	yes	characteristic
209	C₂²×C₁₀₉	#537	1	1	yes	characteristic
210	C₂²×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
211	C₂²×C₁₀₉	C₂×#137	1	1	yes	characteristic
212	C₂²×C₁₀₉	#575	1	1	yes	characteristic
213	C₂²×C₁₀₉	#575	1	1	yes	characteristic
214	C₄×C₁₀₉	—	2	2	no	—
215	C₄×C₁₀₉	—	2	2	no	—
216	C₄×C₁₀₉	—	2	2	no	—
217	C₄×C₁₀₉	—	4	1	no	—
218	C₄×C₁₀₉	—	4	1	no	—
219	C₄×C₁₀₉	—	8	2	no	—
220	C₄×C₁₀₉	—	8	1	no	—
221	C₄×C₁₀₉	—	8	1	no	—
222	C₄×C₁₀₉	—	8	2	no	—
223	C₄×C₁₀₉	—	8	1	no	—
224	C₄×C₁₀₉	—	8	1	no	—
225	C₄×C₁₀₉	#562	1	2	yes	—
226	C₄×C₁₀₉	#563	1	2	yes	—
227	#32740	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂×C₁₀₉	—	4	2	no	—
229	C₂×C₁₀₉	#2814	1	1	yes	characteristic
230	C₂×C₁₀₉	#3187	1	1	yes	characteristic
231	C₂×C₁₀₉	#3196	1	1	yes	characteristic
232	C₂×C₁₀₉	#3229	1	1	yes	characteristic
233	C₂×C₁₀₉	#3240	1	1	yes	characteristic
234	C₂×C₁₀₉	#3246	1	1	yes	characteristic
235	C₂×C₁₀₉	#3252	1	1	yes	characteristic
236	#2857	C₂×C₁₀₉	1	1	yes	characteristic
237	#2855	C₂×C₁₀₉	1	1	yes	characteristic
238	#2088	C₂×C₁₀₉	1	2	yes	complemented
239	#2084	C₂×C₁₀₉	1	1	yes	characteristic, complemented
240	#2077	C₂×C₁₀₉	1	2	yes	—
241	#2072	C₂×C₁₀₉	1	1	yes	characteristic, complemented
242	#2066	C₂×C₁₀₉	1	2	yes	complemented
243	#2058	C₂×C₁₀₉	1	1	yes	characteristic, complemented
244	#2035	C₂×C₁₀₉	1	2	yes	—
245	#2024	C₂×C₁₀₉	1	1	yes	characteristic, complemented
246	#1797	C₂×C₁₀₉	1	1	yes	characteristic
247	C₁₀₉	#32740	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
249	C₂×C₂²⋊Q₈	—	2	1	no	—
250	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
251	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
252	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
253	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
254	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
255	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
256	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
257	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
258	#391	—	2	1	no	—
259	#391	C₂²×C₁₀₉	1	1	yes	characteristic
260	#391	C₂²×C₁₀₉	1	1	yes	characteristic
261	#389	—	2	1	no	—
262	#389	C₂²×C₁₀₉	1	1	yes	characteristic
263	#389	C₂²×C₁₀₉	1	2	yes	—
264	#389	C₂²×C₁₀₉	1	1	yes	characteristic
265	#387	—	2	1	no	—
266	#387	—	2	1	no	—
267	#385	—	2	1	no	—
268	#384	C₂²×C₁₀₉	1	1	yes	characteristic
269	#381	—	2	1	no	—
270	#380	—	2	1	no	—
271	#380	C₂²×C₁₀₉	1	1	yes	characteristic
272	#380	C₂²×C₁₀₉	1	2	yes	—
273	#340	C₂²×C₁₀₉	1	2	yes	—
274	#340	C₂²×C₁₀₉	1	2	yes	—
275	#340	C₂²×C₁₀₉	1	2	yes	—
276	#340	C₂²×C₁₀₉	1	2	yes	—
277	#336	C₂²×C₁₀₉	1	1	yes	characteristic
278	#336	C₂²×C₁₀₉	1	1	yes	characteristic
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
281	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
282	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
283	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
285	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
286	C₄⋊Q₈	—	4	2	no	—
287	C₂²⋊Q₈	—	4	2	no	—
288	C₂²⋊Q₈	—	4	4	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₄²	—	4	2	no	—
305	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
309	#106	—	4	2	no	—
310	#106	—	4	2	no	—
311	C₄⋊C₈	—	4	2	no	—
312	#103	—	4	2	no	—
313	#103	—	4	2	no	—
314	C₂²⋊C₈	—	4	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	1	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	1	no	—
320	#95	—	2	1	no	—
321	#95	—	2	1	no	—
322	#95	—	2	1	no	—
323	#95	—	2	1	no	—
324	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	1	no	—
329	C₂×Q₈	—	4	2	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	—	4	1	no	—
346	C₂²×C₄	—	4	1	no	—
347	C₂²×C₄	SD₁₆×C₁₀₉	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	4	1	no	—
381	C₄²	—	4	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	C₂×SD₁₆×C₁₀₉	1	2	yes	—
389	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
390	Q₈	—	4	2	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	8	4	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	4	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
428	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
429	C₂×C₄	#137×C₁₀₉	1	2	yes	—
430	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
431	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
432	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
433	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
434	C₈	—	4	2	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	#534×C₁₀₉	1	1	yes	characteristic
444	C₂²	#536×C₁₀₉	1	1	yes	characteristic
445	C₂²	#537×C₁₀₉	1	1	yes	characteristic
446	C₂²	C₂²×SD₁₆×C₁₀₉	1	1	yes	characteristic
447	C₂²	C₂×#137×C₁₀₉	1	1	yes	characteristic
448	C₂²	#575×C₁₀₉	1	1	yes	characteristic
449	C₂²	#575×C₁₀₉	1	1	yes	characteristic
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	4	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	8	1	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	1	no	—
460	C₄	—	8	2	no	—
461	C₄	#562×C₁₀₉	1	2	yes	—
462	C₄	#563×C₁₀₉	1	2	yes	—
463	C₂	—	4	2	no	—
464	C₂	C₁₀₉×#2814	1	1	yes	characteristic
465	C₂	C₁₀₉×#3187	1	1	yes	characteristic
466	C₂	C₁₀₉×#3196	1	1	yes	characteristic
467	C₂	C₁₀₉×#3229	1	1	yes	characteristic
468	C₂	C₁₀₉×#3240	1	1	yes	characteristic
469	C₂	C₁₀₉×#3246	1	1	yes	characteristic
470	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×#2088	C₂	semidirect
4. C₁₀₉×#2084	C₂	semidirect
6. C₁₀₉×#2072	C₂	semidirect
7. C₁₀₉×#2066	C₂	semidirect
8. C₁₀₉×#2058	C₂	semidirect
10. C₁₀₉×#2024	C₂	semidirect
227. #32740	C₁₀₉	direct (N × H)
238. #2088	C₂×C₁₀₉	semidirect
239. #2084	C₂×C₁₀₉	semidirect
241. #2072	C₂×C₁₀₉	semidirect
242. #2066	C₂×C₁₀₉	semidirect
243. #2058	C₂×C₁₀₉	semidirect
245. #2024	C₂×C₁₀₉	semidirect
247. C₁₀₉	#32740	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×#2857	2
C₁₀₉×#2855	2
C₁₀₉×#2088	2
C₁₀₉×#2084	2
C₁₀₉×#2077	2
C₁₀₉×#2072	2
C₁₀₉×#2066	2
C₁₀₉×#2058	2
C₁₀₉×#2035	2
C₁₀₉×#2024	2
C₁₀₉×#1797	2
#32740	109

Subgroups — C109×#32740 #42150734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32740 #42150734