Subgroups — C₁₀₉×#11163 #42129157

Order 27904.

958 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#418×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#100×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#343×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1736	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1729	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	#418×C₁₀₉	—	2	2	no	—
16	#418×C₁₀₉	—	2	2	no	—
17	C₂×#102×C₁₀₉	—	2	2	no	—
18	C₂×#102×C₁₀₉	—	2	2	no	—
19	C₂×#102×C₁₀₉	—	2	2	no	—
20	C₂×#102×C₁₀₉	—	2	2	no	—
21	C₂×#100×C₁₀₉	—	2	2	no	—
22	C₂×#100×C₁₀₉	—	2	2	no	—
23	C₂×#100×C₁₀₉	C₄	1	4	yes	complemented
24	#343×C₁₀₉	—	2	2	no	—
25	#341×C₁₀₉	—	2	2	no	—
26	#340×C₁₀₉	—	2	2	no	—
27	#336×C₁₀₉	—	2	2	no	—
28	C₂⁵×C₁₀₉	—	2	1	no	—
29	C₂²×D₄×C₁₀₉	—	2	1	no	—
30	C₂²×D₄×C₁₀₉	—	2	1	no	—
31	C₂²×D₄×C₁₀₉	—	2	1	no	—
32	C₂²×D₄×C₁₀₉	—	2	1	no	—
33	C₂²×D₄×C₁₀₉	—	2	1	no	—
34	C₂²×D₄×C₁₀₉	—	2	1	no	—
35	C₂²×D₄×C₁₀₉	—	2	1	no	—
36	C₂²×D₄×C₁₀₉	—	2	1	no	—
37	C₂²×D₄×C₁₀₉	—	2	1	no	—
38	C₂²×D₄×C₁₀₉	—	2	1	no	—
39	C₂²×D₄×C₁₀₉	—	2	1	no	—
40	C₂²×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₁₀₉	D₄	1	2	yes	—
42	C₂²×D₄×C₁₀₉	D₄	1	2	yes	—
43	C₂³×C₄×C₁₀₉	—	2	1	no	—
44	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₁₀₉	—	2	1	no	—
46	C₂×M₁₆×C₁₀₉	—	2	1	no	—
47	C₂×M₁₆×C₁₀₉	—	2	1	no	—
48	C₂×M₁₆×C₁₀₉	—	2	1	no	—
49	C₂×M₁₆×C₁₀₉	—	2	1	no	—
50	C₂×M₁₆×C₁₀₉	—	2	1	no	—
51	C₂×M₁₆×C₁₀₉	—	2	1	no	—
52	C₂×M₁₆×C₁₀₉	—	2	1	no	—
53	C₂×M₁₆×C₁₀₉	—	2	1	no	—
54	C₂×M₁₆×C₁₀₉	—	2	1	no	—
55	C₂×M₁₆×C₁₀₉	—	2	1	no	—
56	C₂×M₁₆×C₁₀₉	—	2	1	no	—
57	C₂×M₁₆×C₁₀₉	—	2	1	no	—
58	C₂×M₁₆×C₁₀₉	—	2	1	no	—
59	C₂×M₁₆×C₁₀₉	—	2	1	no	—
60	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
61	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
62	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
63	C₂²×C₈×C₁₀₉	—	2	1	no	—
64	C₂²×C₈×C₁₀₉	—	2	1	no	—
65	C₂²×C₈×C₁₀₉	—	2	1	no	—
66	C₂²×C₈×C₁₀₉	—	2	1	no	—
67	C₂²×C₈×C₁₀₉	—	2	1	no	—
68	C₂²×C₈×C₁₀₉	—	2	1	no	—
69	C₂²×C₈×C₁₀₉	—	2	1	no	—
70	C₂²×C₈×C₁₀₉	—	2	1	no	—
71	C₂²×C₈×C₁₀₉	—	2	1	no	—
72	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
73	#117×C₁₀₉	—	2	1	no	—
74	#117×C₁₀₉	—	2	1	no	—
75	#117×C₁₀₉	—	2	1	no	—
76	#117×C₁₀₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
84	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄²×C₁₀₉	—	2	1	no	—
86	#102×C₁₀₉	—	2	8	no	—
87	#102×C₁₀₉	—	2	8	no	—
88	#100×C₁₀₉	—	2	8	no	—
89	C₂⁴×C₁₀₉	—	2	2	no	—
90	C₂⁴×C₁₀₉	—	2	2	no	—
91	C₂⁴×C₁₀₉	—	2	2	no	—
92	C₂⁴×C₁₀₉	—	4	1	no	—
93	C₂⁴×C₁₀₉	—	4	1	no	—
94	C₂⁴×C₁₀₉	—	4	1	no	—
95	C₂⁴×C₁₀₉	—	4	1	no	—
96	C₂⁴×C₁₀₉	—	8	2	no	—
97	C₂⁴×C₁₀₉	—	8	2	no	—
98	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	—	2	2	no	—
100	C₂×D₄×C₁₀₉	—	2	2	no	—
101	C₂×D₄×C₁₀₉	—	2	2	no	—
102	C₂×D₄×C₁₀₉	—	2	2	no	—
103	C₂×D₄×C₁₀₉	—	2	2	no	—
104	C₂×D₄×C₁₀₉	—	2	2	no	—
105	C₂×D₄×C₁₀₉	—	2	2	no	—
106	C₂×D₄×C₁₀₉	—	2	2	no	—
107	C₂×D₄×C₁₀₉	—	2	2	no	—
108	C₂×D₄×C₁₀₉	—	2	2	no	—
109	C₂×D₄×C₁₀₉	—	2	2	no	—
110	C₂×D₄×C₁₀₉	—	2	2	no	—
111	C₂×D₄×C₁₀₉	—	2	2	no	—
112	C₂×D₄×C₁₀₉	—	2	2	no	—
113	C₂×D₄×C₁₀₉	—	2	2	no	—
114	C₂×D₄×C₁₀₉	—	4	1	no	—
115	C₂×D₄×C₁₀₉	—	4	1	no	—
116	C₂×D₄×C₁₀₉	—	4	1	no	—
117	C₂×D₄×C₁₀₉	—	4	1	no	—
118	C₂×D₄×C₁₀₉	—	4	1	no	—
119	C₂×D₄×C₁₀₉	—	4	1	no	—
120	C₂×D₄×C₁₀₉	—	4	1	no	—
121	C₂×D₄×C₁₀₉	—	4	1	no	—
122	C₂×D₄×C₁₀₉	—	4	1	no	—
123	C₂×D₄×C₁₀₉	—	4	1	no	—
124	C₂×D₄×C₁₀₉	—	4	1	no	—
125	C₂×D₄×C₁₀₉	—	4	1	no	—
126	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
127	C₂²×C₄×C₁₀₉	—	2	2	no	—
128	C₂²×C₄×C₁₀₉	—	2	2	no	—
129	C₂²×C₄×C₁₀₉	—	4	1	no	—
130	C₂²×C₄×C₁₀₉	—	4	1	no	—
131	C₂²×C₄×C₁₀₉	—	8	2	no	—
132	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
140	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
141	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
142	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
143	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
144	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
145	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
146	M₁₆×C₁₀₉	—	2	2	no	—
147	M₁₆×C₁₀₉	—	2	2	no	—
148	M₁₆×C₁₀₉	—	2	2	no	—
149	M₁₆×C₁₀₉	—	2	2	no	—
150	M₁₆×C₁₀₉	—	2	2	no	—
151	M₁₆×C₁₀₉	—	2	2	no	—
152	M₁₆×C₁₀₉	—	4	1	no	—
153	M₁₆×C₁₀₉	—	4	1	no	—
154	M₁₆×C₁₀₉	—	4	1	no	—
155	M₁₆×C₁₀₉	—	4	1	no	—
156	M₁₆×C₁₀₉	—	4	1	no	—
157	M₁₆×C₁₀₉	—	4	1	no	—
158	C₂×C₈×C₁₀₉	—	2	2	no	—
159	C₂×C₈×C₁₀₉	—	2	2	no	—
160	C₂×C₈×C₁₀₉	—	2	2	no	—
161	C₂×C₈×C₁₀₉	—	2	2	no	—
162	C₂×C₈×C₁₀₉	—	2	2	no	—
163	C₂×C₈×C₁₀₉	—	2	2	no	—
164	C₂×C₈×C₁₀₉	—	4	1	no	—
165	C₂×C₈×C₁₀₉	—	4	1	no	—
166	C₂×C₈×C₁₀₉	—	4	1	no	—
167	C₂×C₈×C₁₀₉	—	4	1	no	—
168	C₂×C₈×C₁₀₉	—	4	1	no	—
169	C₂×C₈×C₁₀₉	—	4	1	no	—
170	C₂×C₈×C₁₀₉	—	4	1	no	—
171	C₂×C₈×C₁₀₉	—	4	1	no	—
172	C₂×C₈×C₁₀₉	—	4	1	no	—
173	C₂×C₈×C₁₀₉	—	4	1	no	—
174	C₄⋊C₄×C₁₀₉	—	2	2	no	—
175	C₄⋊C₄×C₁₀₉	—	2	2	no	—
176	C₄⋊C₄×C₁₀₉	—	2	2	no	—
177	C₄⋊C₄×C₁₀₉	—	2	2	no	—
178	C₄⋊C₄×C₁₀₉	—	4	1	no	—
179	C₄⋊C₄×C₁₀₉	—	4	1	no	—
180	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
181	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
182	C₄²×C₁₀₉	—	4	1	no	—
183	C₄²×C₁₀₉	—	4	1	no	—
184	C₂³×C₁₀₉	—	2	2	no	—
185	C₂³×C₁₀₉	—	2	2	no	—
186	C₂³×C₁₀₉	—	2	4	no	—
187	C₂³×C₁₀₉	—	4	2	no	—
188	C₂³×C₁₀₉	—	4	2	no	—
189	C₂³×C₁₀₉	—	4	2	no	—
190	C₂³×C₁₀₉	—	4	2	no	—
191	C₂³×C₁₀₉	—	4	2	no	—
192	C₂³×C₁₀₉	—	4	2	no	—
193	C₂³×C₁₀₉	—	8	4	no	—
194	C₂³×C₁₀₉	—	8	4	no	—
195	C₂³×C₁₀₉	—	8	2	no	—
196	C₂³×C₁₀₉	—	8	2	no	—
197	C₂³×C₁₀₉	—	8	4	no	—
198	C₂³×C₁₀₉	—	8	4	no	—
199	C₂³×C₁₀₉	—	8	2	no	—
200	C₂³×C₁₀₉	—	8	2	no	—
201	C₂³×C₁₀₉	—	8	2	no	—
202	C₂³×C₁₀₉	—	8	2	no	—
203	C₂³×C₁₀₉	#117	1	1	yes	characteristic
204	C₂³×C₁₀₉	#121	1	1	yes	characteristic
205	C₂³×C₁₀₉	#123	1	1	yes	characteristic
206	C₂³×C₁₀₉	#123	1	1	yes	characteristic
207	C₂³×C₁₀₉	#123	1	1	yes	characteristic
208	C₂³×C₁₀₉	#124	1	1	yes	characteristic
209	C₂³×C₁₀₉	#126	1	1	yes	characteristic
210	D₄×C₁₀₉	—	2	4	no	—
211	D₄×C₁₀₉	—	2	4	no	—
212	D₄×C₁₀₉	—	4	2	no	—
213	D₄×C₁₀₉	—	4	2	no	—
214	D₄×C₁₀₉	—	4	2	no	—
215	D₄×C₁₀₉	—	4	2	no	—
216	D₄×C₁₀₉	—	4	2	no	—
217	D₄×C₁₀₉	—	4	2	no	—
218	C₂×C₄×C₁₀₉	—	4	2	no	—
219	C₂×C₄×C₁₀₉	—	4	2	no	—
220	C₂×C₄×C₁₀₉	—	4	2	no	—
221	C₂×C₄×C₁₀₉	—	4	2	no	—
222	C₂×C₄×C₁₀₉	—	8	2	no	—
223	C₂×C₄×C₁₀₉	—	8	2	no	—
224	C₂×C₄×C₁₀₉	—	8	2	no	—
225	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
227	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
228	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
229	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
232	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
233	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
234	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
235	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
236	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
237	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
238	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
239	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
240	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
241	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
242	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
243	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
244	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
245	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
246	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
247	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
248	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
249	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
250	C₂×C₄×C₁₀₉	#126	1	1	yes	characteristic
251	C₂×C₄×C₁₀₉	#126	1	1	yes	characteristic
252	C₂×C₄×C₁₀₉	#126	1	1	yes	characteristic
253	C₈×C₁₀₉	—	4	2	no	—
254	C₈×C₁₀₉	—	4	2	no	—
255	C₈×C₁₀₉	—	4	2	no	—
256	C₈×C₁₀₉	—	4	2	no	—
257	C₂²×C₁₀₉	—	2	2	no	—
258	C₂²×C₁₀₉	—	2	2	no	—
259	C₂²×C₁₀₉	—	2	2	no	—
260	C₂²×C₁₀₉	—	2	2	no	—
261	C₂²×C₁₀₉	—	2	1	no	—
262	C₂²×C₁₀₉	—	2	1	no	—
263	C₂²×C₁₀₉	—	2	1	no	—
264	C₂²×C₁₀₉	—	2	1	no	—
265	C₂²×C₁₀₉	—	4	4	no	—
266	C₂²×C₁₀₉	—	8	4	no	—
267	C₂²×C₁₀₉	—	8	4	no	—
268	C₂²×C₁₀₉	—	8	4	no	—
269	C₂²×C₁₀₉	—	8	4	no	—
270	C₂²×C₁₀₉	—	8	4	no	—
271	C₂²×C₁₀₉	—	8	4	no	—
272	C₂²×C₁₀₉	—	8	2	no	—
273	C₂²×C₁₀₉	—	8	2	no	—
274	C₂²×C₁₀₉	#381	1	1	yes	characteristic
275	C₂²×C₁₀₉	#388	1	1	yes	characteristic
276	C₂²×C₁₀₉	#388	1	1	yes	characteristic
277	C₂²×C₁₀₉	#394	1	1	yes	characteristic
278	C₂²×C₁₀₉	#394	1	1	yes	characteristic
279	C₂²×C₁₀₉	#397	1	1	yes	characteristic
280	C₂²×C₁₀₉	#397	1	1	yes	characteristic
281	C₂²×C₁₀₉	#471	1	2	yes	—
282	C₂²×C₁₀₉	#472	1	2	yes	—
283	C₄×C₁₀₉	—	8	2	no	—
284	C₄×C₁₀₉	—	8	2	no	—
285	C₄×C₁₀₉	#386	1	1	yes	characteristic
286	C₄×C₁₀₉	#386	1	1	yes	characteristic
287	C₄×C₁₀₉	#388	1	1	yes	characteristic
288	C₄×C₁₀₉	#388	1	1	yes	characteristic
289	C₄×C₁₀₉	#394	1	1	yes	characteristic
290	C₄×C₁₀₉	#394	1	1	yes	characteristic
291	C₄×C₁₀₉	#397	1	1	yes	characteristic
292	C₄×C₁₀₉	#397	1	1	yes	characteristic
293	#11163	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
294	C₂×C₁₀₉	—	2	1	no	—
295	C₂×C₁₀₉	—	2	1	no	—
296	C₂×C₁₀₉	—	2	1	no	—
297	C₂×C₁₀₉	—	2	1	no	—
298	C₂×C₁₀₉	—	8	4	no	—
299	C₂×C₁₀₉	#1440	1	1	yes	characteristic
300	C₂×C₁₀₉	#1931	1	2	yes	—
301	C₂×C₁₀₉	#2045	1	1	yes	characteristic
302	C₂×C₁₀₉	#2046	1	1	yes	characteristic
303	C₂×C₁₀₉	#2051	1	2	yes	—
304	C₂×#418	C₂×C₁₀₉	1	1	yes	characteristic
305	C₂²×#102	C₂×C₁₀₉	1	1	yes	characteristic
306	C₂²×#100	C₂×C₁₀₉	1	1	yes	characteristic
307	C₂×#343	C₂×C₁₀₉	1	1	yes	characteristic, complemented
308	C₂×#341	C₂×C₁₀₉	1	1	yes	characteristic, complemented
309	#1736	C₂×C₁₀₉	1	1	yes	characteristic, complemented
310	#1729	C₂×C₁₀₉	1	1	yes	characteristic, complemented
311	C₁₀₉	#11163	1	1	yes	characteristic, direct product factor, Sylow
312	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
313	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
314	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
315	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
316	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
317	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
319	#418	—	2	2	no	—
320	#418	—	2	2	no	—
321	C₂×#102	—	2	2	no	—
322	C₂×#102	—	2	2	no	—
323	C₂×#102	—	2	2	no	—
324	C₂×#102	—	2	2	no	—
325	C₂×#100	—	2	2	no	—
326	C₂×#100	—	2	2	no	—
327	C₂×#100	C₄×C₁₀₉	1	4	yes	complemented
328	#343	—	2	2	no	—
329	#341	—	2	2	no	—
330	#340	—	2	2	no	—
331	#336	—	2	2	no	—
332	C₂⁵	—	2	1	no	—
333	C₂²×D₄	—	2	1	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	—	2	1	no	—
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	—	2	1	no	—
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	—	2	1	no	—
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
345	C₂²×D₄	D₄×C₁₀₉	1	2	yes	—
346	C₂²×D₄	D₄×C₁₀₉	1	2	yes	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
365	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
366	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
367	C₂²×C₈	—	2	1	no	—
368	C₂²×C₈	—	2	1	no	—
369	C₂²×C₈	—	2	1	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₂²×C₈	—	2	1	no	—
374	C₂²×C₈	—	2	1	no	—
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
377	#117	—	2	1	no	—
378	#117	—	2	1	no	—
379	#117	—	2	1	no	—
380	#117	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
387	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₄²	—	2	1	no	—
390	#102	—	2	8	no	—
391	#102	—	2	8	no	—
392	#100	—	2	8	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	—	2	2	no	—
395	C₂⁴	—	2	2	no	—
396	C₂⁴	—	4	1	no	—
397	C₂⁴	—	4	1	no	—
398	C₂⁴	—	4	1	no	—
399	C₂⁴	—	4	1	no	—
400	C₂⁴	—	8	2	no	—
401	C₂⁴	—	8	2	no	—
402	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	4	1	no	—
419	C₂×D₄	—	4	1	no	—
420	C₂×D₄	—	4	1	no	—
421	C₂×D₄	—	4	1	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	—	4	1	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂×D₄	—	4	1	no	—
426	C₂×D₄	—	4	1	no	—
427	C₂×D₄	—	4	1	no	—
428	C₂×D₄	—	4	1	no	—
429	C₂×D₄	—	4	1	no	—
430	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	4	1	no	—
434	C₂²×C₄	—	4	1	no	—
435	C₂²×C₄	—	8	2	no	—
436	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
443	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
444	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
445	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
446	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
447	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
448	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
449	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	2	2	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	4	1	no	—
457	M₁₆	—	4	1	no	—
458	M₁₆	—	4	1	no	—
459	M₁₆	—	4	1	no	—
460	M₁₆	—	4	1	no	—
461	M₁₆	—	4	1	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	2	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	4	1	no	—
469	C₂×C₈	—	4	1	no	—
470	C₂×C₈	—	4	1	no	—
471	C₂×C₈	—	4	1	no	—
472	C₂×C₈	—	4	1	no	—
473	C₂×C₈	—	4	1	no	—
474	C₂×C₈	—	4	1	no	—
475	C₂×C₈	—	4	1	no	—
476	C₂×C₈	—	4	1	no	—
477	C₂×C₈	—	4	1	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₄²	—	4	1	no	—
487	C₄²	—	4	1	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	2	no	—
493	C₂³	—	4	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	8	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	—	8	4	no	—
501	C₂³	—	8	4	no	—
502	C₂³	—	8	2	no	—
503	C₂³	—	8	2	no	—
504	C₂³	—	8	4	no	—
505	C₂³	—	8	2	no	—
506	C₂³	—	8	2	no	—
507	C₂³	#117×C₁₀₉	1	1	yes	characteristic
508	C₂³	#121×C₁₀₉	1	1	yes	characteristic
509	C₂³	#123×C₁₀₉	1	1	yes	characteristic
510	C₂³	#123×C₁₀₉	1	1	yes	characteristic
511	C₂³	#123×C₁₀₉	1	1	yes	characteristic
512	C₂³	#124×C₁₀₉	1	1	yes	characteristic
513	C₂³	#126×C₁₀₉	1	1	yes	characteristic
514	D₄	—	2	4	no	—
515	D₄	—	2	4	no	—
516	D₄	—	4	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	8	2	no	—
527	C₂×C₄	—	8	2	no	—
528	C₂×C₄	—	8	2	no	—
529	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
530	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
531	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
532	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
533	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
534	C₂×C₄	C₄×D₄×C₁₀₉	1	1	yes	characteristic
535	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
536	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
537	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
538	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
539	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
540	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
541	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
542	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
543	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
544	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
545	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
546	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
547	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
548	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
549	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
550	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
551	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
552	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
553	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
554	C₂×C₄	#126×C₁₀₉	1	1	yes	characteristic
555	C₂×C₄	#126×C₁₀₉	1	1	yes	characteristic
556	C₂×C₄	#126×C₁₀₉	1	1	yes	characteristic
557	C₈	—	4	2	no	—
558	C₈	—	4	2	no	—
559	C₈	—	4	2	no	—
560	C₈	—	4	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	8	4	no	—
571	C₂²	—	8	4	no	—
572	C₂²	—	8	4	no	—
573	C₂²	—	8	4	no	—
574	C₂²	—	8	4	no	—
575	C₂²	—	8	2	no	—
576	C₂²	—	8	2	no	—
577	C₂²	—	8	4	no	—
578	C₂²	#381×C₁₀₉	1	1	yes	characteristic
579	C₂²	#388×C₁₀₉	1	1	yes	characteristic
580	C₂²	#388×C₁₀₉	1	1	yes	characteristic
581	C₂²	#394×C₁₀₉	1	1	yes	characteristic
582	C₂²	#394×C₁₀₉	1	1	yes	characteristic
583	C₂²	#397×C₁₀₉	1	1	yes	characteristic
584	C₂²	#397×C₁₀₉	1	1	yes	characteristic
585	C₂²	#471×C₁₀₉	1	2	yes	—
586	C₂²	#472×C₁₀₉	1	2	yes	—
587	C₄	—	8	2	no	—
588	C₄	—	8	2	no	—
589	C₄	#386×C₁₀₉	1	1	yes	characteristic
590	C₄	#386×C₁₀₉	1	1	yes	characteristic
591	C₄	#388×C₁₀₉	1	1	yes	characteristic
592	C₄	#388×C₁₀₉	1	1	yes	characteristic
593	C₄	#394×C₁₀₉	1	1	yes	characteristic
594	C₄	#394×C₁₀₉	1	1	yes	characteristic
595	C₄	#397×C₁₀₉	1	1	yes	characteristic
596	C₄	#397×C₁₀₉	1	1	yes	characteristic
597	C₂	—	2	1	no	—
598	C₂	—	2	1	no	—
599	C₂	—	2	1	no	—
600	C₂	—	2	1	no	—
601	C₂	—	8	4	no	—
602	C₂	C₁₀₉×#1440	1	1	yes	characteristic
603	C₂	C₁₀₉×#1931	1	2	yes	—
604	C₂	C₁₀₉×#2045	1	1	yes	characteristic
605	C₂	C₁₀₉×#2046	1	1	yes	characteristic
606	C₂	C₁₀₉×#2051	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
prime radical	C₂³×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#343×C₁₀₉	C₂	semidirect
5. C₂×#341×C₁₀₉	C₂	semidirect
6. C₁₀₉×#1736	C₂	semidirect
7. C₁₀₉×#1729	C₂	semidirect
23. C₂×#100×C₁₀₉	C₄	semidirect
293. #11163	C₁₀₉	direct (N × H)
307. C₂×#343	C₂×C₁₀₉	semidirect
308. C₂×#341	C₂×C₁₀₉	semidirect
309. #1736	C₂×C₁₀₉	semidirect
310. #1729	C₂×C₁₀₉	semidirect
311. C₁₀₉	#11163	direct (N × H)
327. C₂×#100	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×#418×C₁₀₉	2
C₂²×#102×C₁₀₉	2
C₂²×#100×C₁₀₉	2
C₂×#343×C₁₀₉	2
C₂×#341×C₁₀₉	2
C₁₀₉×#1736	2
C₁₀₉×#1729	2
#11163	109

Subgroups — C109×#11163 #42129157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11163 #42129157