Subgroups — C₂×C₁₀₉×#2539 #42152813

Order 27904.

5726 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	C₂²×#121×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#120×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	C₂³×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2539	C₂	1	16	yes	maximal, direct product factor
6	C₂×#399×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₂×#392×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×#390×C₁₀₉	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁶×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂³×D₄×C₁₀₉	C₂²	1	4	yes	—
13	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#123×C₁₀₉	—	2	16	no	—
17	C₂×#123×C₁₀₉	C₂²	1	16	yes	complemented
18	C₂×#121×C₁₀₉	—	2	8	no	—
19	C₂×#121×C₁₀₉	C₂²	1	8	yes	complemented
20	C₂×#120×C₁₀₉	—	2	16	no	—
21	C₂×#120×C₁₀₉	C₂²	1	16	yes	complemented
22	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
23	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	C₂²×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
25	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
30	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
31	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
32	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	complemented
33	#399×C₁₀₉	C₂²	1	16	yes	complemented
34	#392×C₁₀₉	C₂²	1	16	yes	complemented
35	#390×C₁₀₉	C₂²	1	32	yes	complemented
36	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
37	C₂×#95×C₁₀₉	C₂²	1	2	yes	complemented
38	C₂⁵×C₁₀₉	—	2	4	no	—
39	C₂⁵×C₁₀₉	—	2	8	no	—
40	C₂⁵×C₁₀₉	—	2	8	no	—
41	C₂⁵×C₁₀₉	—	4	4	no	—
42	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
43	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
44	C₂⁵×C₁₀₉	C₂³	1	4	yes	—
45	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₀₉	—	2	8	no	—
47	C₂²×D₄×C₁₀₉	—	2	8	no	—
48	C₂²×D₄×C₁₀₉	—	2	8	no	—
49	C₂²×D₄×C₁₀₉	—	2	16	no	—
50	C₂²×D₄×C₁₀₉	—	2	16	no	—
51	C₂²×D₄×C₁₀₉	—	2	4	no	—
52	C₂²×D₄×C₁₀₉	—	4	8	no	—
53	C₂²×D₄×C₁₀₉	—	4	4	no	—
54	C₂²×D₄×C₁₀₉	C₂³	1	8	yes	—
55	C₂²×D₄×C₁₀₉	C₂³	1	16	yes	—
56	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	complemented
57	C₂³×C₄×C₁₀₉	—	2	4	no	—
58	C₂³×C₄×C₁₀₉	—	2	8	no	—
59	C₂³×C₄×C₁₀₉	—	4	4	no	—
60	C₂³×C₄×C₁₀₉	D₄	1	8	yes	complemented
61	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
62	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
63	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
64	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
66	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
68	#123×C₁₀₉	—	2	64	no	—
69	#121×C₁₀₉	—	2	32	no	—
70	#120×C₁₀₉	—	2	64	no	—
71	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
72	C₂×C₄⋊C₄×C₁₀₉	—	2	16	no	—
73	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
74	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
75	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
77	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
78	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
80	C₂×C₂²⋊C₄×C₁₀₉	—	2	32	no	—
81	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	16	yes	complemented
82	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	—
85	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	complemented
87	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	16	yes	complemented
88	C₂×C₄²×C₁₀₉	—	4	4	no	—
89	C₂×C₄²×C₁₀₉	C₂³	1	8	yes	complemented
90	#95×C₁₀₉	C₂³	1	8	yes	complemented
91	C₂⁴×C₁₀₉	—	2	4	no	—
92	C₂⁴×C₁₀₉	—	2	4	no	—
93	C₂⁴×C₁₀₉	—	2	32	no	—
94	C₂⁴×C₁₀₉	—	2	8	no	—
95	C₂⁴×C₁₀₉	—	2	4	no	—
96	C₂⁴×C₁₀₉	—	2	8	no	—
97	C₂⁴×C₁₀₉	—	2	16	no	—
98	C₂⁴×C₁₀₉	—	2	4	no	—
99	C₂⁴×C₁₀₉	—	2	8	no	—
100	C₂⁴×C₁₀₉	—	2	4	no	—
101	C₂⁴×C₁₀₉	—	2	4	no	—
102	C₂⁴×C₁₀₉	—	2	16	no	—
103	C₂⁴×C₁₀₉	—	2	4	no	—
104	C₂⁴×C₁₀₉	—	2	8	no	—
105	C₂⁴×C₁₀₉	—	2	8	no	—
106	C₂⁴×C₁₀₉	—	4	8	no	—
107	C₂⁴×C₁₀₉	—	4	64	no	—
108	C₂⁴×C₁₀₉	—	4	16	no	—
109	C₂⁴×C₁₀₉	—	4	16	no	—
110	C₂⁴×C₁₀₉	C₂×D₄	1	8	yes	—
111	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
112	C₂⁴×C₁₀₉	#41	1	8	yes	—
113	C₂⁴×C₁₀₉	C₂⁴	1	4	yes	—
114	C₂⁴×C₁₀₉	C₂⁴	1	4	yes	—
115	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
116	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
117	C₂×D₄×C₁₀₉	—	2	16	no	—
118	C₂×D₄×C₁₀₉	—	2	16	no	—
119	C₂×D₄×C₁₀₉	—	2	16	no	—
120	C₂×D₄×C₁₀₉	—	2	32	no	—
121	C₂×D₄×C₁₀₉	—	2	32	no	—
122	C₂×D₄×C₁₀₉	—	2	8	no	—
123	C₂×D₄×C₁₀₉	—	4	16	no	—
124	C₂×D₄×C₁₀₉	—	4	8	no	—
125	C₂×D₄×C₁₀₉	—	4	64	no	—
126	C₂×D₄×C₁₀₉	—	4	16	no	—
127	C₂×D₄×C₁₀₉	—	4	32	no	—
128	C₂²×C₄×C₁₀₉	—	2	8	no	—
129	C₂²×C₄×C₁₀₉	—	2	4	no	—
130	C₂²×C₄×C₁₀₉	—	2	8	no	—
131	C₂²×C₄×C₁₀₉	—	2	4	no	—
132	C₂²×C₄×C₁₀₉	—	2	8	no	—
133	C₂²×C₄×C₁₀₉	—	2	4	no	—
134	C₂²×C₄×C₁₀₉	—	2	4	no	—
135	C₂²×C₄×C₁₀₉	—	2	8	no	—
136	C₂²×C₄×C₁₀₉	—	2	4	no	—
137	C₂²×C₄×C₁₀₉	—	2	4	no	—
138	C₂²×C₄×C₁₀₉	—	2	16	no	—
139	C₂²×C₄×C₁₀₉	—	4	16	no	—
140	C₂²×C₄×C₁₀₉	—	4	8	no	—
141	C₂²×C₄×C₁₀₉	—	4	16	no	—
142	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₁₀₉	C₂×D₄	1	16	yes	—
144	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₁₀₉	C₂×D₄	1	16	yes	complemented
146	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₁₀₉	C₂⁴	1	8	yes	—
150	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
151	C₄⋊C₄×C₁₀₉	—	2	8	no	—
152	C₄⋊C₄×C₁₀₉	—	2	32	no	—
153	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
154	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
155	C₂²⋊C₄×C₁₀₉	—	2	32	no	—
156	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
157	C₂²⋊C₄×C₁₀₉	—	2	32	no	—
158	C₂²⋊C₄×C₁₀₉	C₂×D₄	1	32	yes	complemented
159	C₄²×C₁₀₉	—	4	8	no	—
160	C₂³×C₁₀₉	—	2	8	no	—
161	C₂³×C₁₀₉	—	2	8	no	—
162	C₂³×C₁₀₉	—	2	32	no	—
163	C₂³×C₁₀₉	—	2	4	no	—
164	C₂³×C₁₀₉	—	2	4	no	—
165	C₂³×C₁₀₉	—	2	32	no	—
166	C₂³×C₁₀₉	—	2	16	no	—
167	C₂³×C₁₀₉	—	2	32	no	—
168	C₂³×C₁₀₉	—	2	4	no	—
169	C₂³×C₁₀₉	—	2	8	no	—
170	C₂³×C₁₀₉	—	2	4	no	—
171	C₂³×C₁₀₉	—	2	8	no	—
172	C₂³×C₁₀₉	—	2	4	no	—
173	C₂³×C₁₀₉	—	2	4	no	—
174	C₂³×C₁₀₉	—	2	4	no	—
175	C₂³×C₁₀₉	—	2	8	no	—
176	C₂³×C₁₀₉	—	2	8	no	—
177	C₂³×C₁₀₉	—	4	64	no	—
178	C₂³×C₁₀₉	—	4	16	no	—
179	C₂³×C₁₀₉	—	4	128	no	—
180	C₂³×C₁₀₉	—	4	16	no	—
181	C₂³×C₁₀₉	—	4	8	no	—
182	C₂³×C₁₀₉	—	4	8	no	—
183	C₂³×C₁₀₉	—	4	16	no	—
184	C₂³×C₁₀₉	—	4	8	no	—
185	C₂³×C₁₀₉	—	4	8	no	—
186	C₂³×C₁₀₉	—	4	8	no	—
187	C₂³×C₁₀₉	—	4	16	no	—
188	C₂³×C₁₀₉	—	4	16	no	—
189	C₂³×C₁₀₉	#127	1	16	yes	—
190	C₂³×C₁₀₉	C₂²×D₄	1	8	yes	—
191	C₂³×C₁₀₉	C₂²×D₄	1	4	yes	—
192	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
193	C₂³×C₁₀₉	C₂²×D₄	1	4	yes	—
194	C₂³×C₁₀₉	C₂×#41	1	8	yes	—
195	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
196	C₂³×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
197	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
198	D₄×C₁₀₉	—	4	64	no	—
199	D₄×C₁₀₉	—	4	16	no	—
200	D₄×C₁₀₉	—	4	32	no	—
201	C₂×C₄×C₁₀₉	—	2	8	no	—
202	C₂×C₄×C₁₀₉	—	2	8	no	—
203	C₂×C₄×C₁₀₉	—	2	32	no	—
204	C₂×C₄×C₁₀₉	—	2	8	no	—
205	C₂×C₄×C₁₀₉	—	2	4	no	—
206	C₂×C₄×C₁₀₉	—	2	8	no	—
207	C₂×C₄×C₁₀₉	—	2	4	no	—
208	C₂×C₄×C₁₀₉	—	2	4	no	—
209	C₂×C₄×C₁₀₉	—	2	4	no	—
210	C₂×C₄×C₁₀₉	—	2	8	no	—
211	C₂×C₄×C₁₀₉	—	2	4	no	—
212	C₂×C₄×C₁₀₉	—	2	4	no	—
213	C₂×C₄×C₁₀₉	—	4	16	no	—
214	C₂×C₄×C₁₀₉	—	4	16	no	—
215	C₂×C₄×C₁₀₉	—	4	8	no	—
216	C₂×C₄×C₁₀₉	—	4	8	no	—
217	C₂×C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₁₀₉	C₂²×D₄	1	16	yes	—
219	C₂×C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
220	C₂²×C₁₀₉	—	2	4	no	—
221	C₂²×C₁₀₉	—	2	4	no	—
222	C₂²×C₁₀₉	—	2	16	no	—
223	C₂²×C₁₀₉	—	2	32	no	—
224	C₂²×C₁₀₉	—	2	8	no	—
225	C₂²×C₁₀₉	—	2	16	no	—
226	C₂²×C₁₀₉	—	4	64	no	—
227	C₂²×C₁₀₉	—	4	4	no	—
228	C₂²×C₁₀₉	—	4	4	no	—
229	C₂²×C₁₀₉	—	4	8	no	—
230	C₂²×C₁₀₉	—	4	32	no	—
231	C₂²×C₁₀₉	—	4	4	no	—
232	C₂²×C₁₀₉	—	4	4	no	—
233	C₂²×C₁₀₉	—	4	4	no	—
234	C₂²×C₁₀₉	—	4	8	no	—
235	C₂²×C₁₀₉	—	4	8	no	—
236	C₂²×C₁₀₉	—	8	16	no	—
237	C₂²×C₁₀₉	#525	1	4	yes	—
238	C₂²×C₁₀₉	C₂×#127	1	8	yes	—
239	C₂²×C₁₀₉	C₂×#127	1	4	yes	—
240	C₂²×C₁₀₉	#534	1	4	yes	—
241	C₂²×C₁₀₉	#546	1	16	yes	—
242	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
243	C₂²×C₁₀₉	C₂³×D₄	1	2	yes	—
244	C₂²×C₁₀₉	C₂²×#41	1	2	yes	—
245	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
246	C₄×C₁₀₉	—	2	4	no	—
247	C₄×C₁₀₉	—	2	4	no	—
248	C₄×C₁₀₉	—	2	16	no	—
249	C₄×C₁₀₉	—	4	4	no	—
250	C₄×C₁₀₉	—	4	4	no	—
251	C₂×#2539	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₁₀₉	—	2	8	no	—
253	C₂×C₁₀₉	—	4	8	no	—
254	C₂×C₁₀₉	—	8	4	no	—
255	C₂×C₁₀₉	#2539	1	8	yes	direct product factor
256	C₂×C₁₀₉	C₂×#525	1	1	yes	characteristic
257	C₂×C₁₀₉	C₂²×#127	1	1	yes	characteristic
258	C₂×C₁₀₉	C₂×#534	1	1	yes	characteristic
259	C₂×C₁₀₉	C₂×#546	1	4	yes	—
260	C₂²×#123	C₂×C₁₀₉	1	2	yes	complemented
261	C₂²×#121	C₂×C₁₀₉	1	1	yes	characteristic, complemented
262	C₂²×#120	C₂×C₁₀₉	1	2	yes	complemented
263	C₂³×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
264	#2539	C₂×C₁₀₉	1	16	yes	direct product factor
265	C₂×#399	C₂×C₁₀₉	1	2	yes	complemented
266	C₂×#392	C₂×C₁₀₉	1	2	yes	complemented
267	C₂×#390	C₂×C₁₀₉	1	4	yes	complemented
268	C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
269	C₁₀₉	C₂×#2539	1	1	yes	characteristic, direct product factor, Sylow
270	C₂⁶	C₂²×C₁₀₉	1	1	yes	characteristic
271	C₂³×D₄	C₂²×C₁₀₉	1	2	yes	—
272	C₂³×D₄	C₂²×C₁₀₉	1	4	yes	—
273	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
274	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
275	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
276	C₂×#123	—	2	16	no	—
277	C₂×#123	C₂²×C₁₀₉	1	16	yes	complemented
278	C₂×#121	—	2	8	no	—
279	C₂×#121	C₂²×C₁₀₉	1	8	yes	complemented
280	C₂×#120	—	2	16	no	—
281	C₂×#120	C₂²×C₁₀₉	1	16	yes	complemented
282	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
283	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
284	C₂²×C₂²⋊C₄	—	2	8	no	—
285	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
287	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
288	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
289	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
290	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
291	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
292	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	complemented
293	#399	C₂²×C₁₀₉	1	16	yes	complemented
294	#392	C₂²×C₁₀₉	1	16	yes	complemented
295	#390	C₂²×C₁₀₉	1	32	yes	complemented
296	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
297	C₂×#95	C₂²×C₁₀₉	1	2	yes	complemented
298	C₂⁵	—	2	4	no	—
299	C₂⁵	—	2	8	no	—
300	C₂⁵	—	2	8	no	—
301	C₂⁵	—	4	4	no	—
302	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
303	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
304	C₂⁵	C₂³×C₁₀₉	1	4	yes	—
305	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
306	C₂²×D₄	—	2	8	no	—
307	C₂²×D₄	—	2	8	no	—
308	C₂²×D₄	—	2	8	no	—
309	C₂²×D₄	—	2	16	no	—
310	C₂²×D₄	—	2	16	no	—
311	C₂²×D₄	—	2	4	no	—
312	C₂²×D₄	—	4	8	no	—
313	C₂²×D₄	—	4	4	no	—
314	C₂²×D₄	C₂³×C₁₀₉	1	8	yes	—
315	C₂²×D₄	C₂³×C₁₀₉	1	16	yes	—
316	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	complemented
317	C₂³×C₄	—	2	4	no	—
318	C₂³×C₄	—	2	8	no	—
319	C₂³×C₄	—	4	4	no	—
320	C₂³×C₄	D₄×C₁₀₉	1	8	yes	complemented
321	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
322	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
323	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
324	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
326	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
328	#123	—	2	64	no	—
329	#121	—	2	32	no	—
330	#120	—	2	64	no	—
331	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	—	2	16	no	—
333	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	complemented
334	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	16	yes	complemented
335	C₂×C₂²⋊C₄	—	2	8	no	—
336	C₂×C₂²⋊C₄	—	2	8	no	—
337	C₂×C₂²⋊C₄	—	2	16	no	—
338	C₂×C₂²⋊C₄	—	2	8	no	—
339	C₂×C₂²⋊C₄	—	2	16	no	—
340	C₂×C₂²⋊C₄	—	2	32	no	—
341	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	16	yes	complemented
342	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	complemented
347	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	16	yes	complemented
348	C₂×C₄²	—	4	4	no	—
349	C₂×C₄²	C₂³×C₁₀₉	1	8	yes	complemented
350	#95	C₂³×C₁₀₉	1	8	yes	complemented
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	32	no	—
354	C₂⁴	—	2	8	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	2	8	no	—
357	C₂⁴	—	2	16	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	—	2	8	no	—
360	C₂⁴	—	2	4	no	—
361	C₂⁴	—	2	4	no	—
362	C₂⁴	—	2	16	no	—
363	C₂⁴	—	2	4	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	2	8	no	—
366	C₂⁴	—	4	8	no	—
367	C₂⁴	—	4	64	no	—
368	C₂⁴	—	4	16	no	—
369	C₂⁴	—	4	16	no	—
370	C₂⁴	C₂×D₄×C₁₀₉	1	8	yes	—
371	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
372	C₂⁴	#41×C₁₀₉	1	8	yes	—
373	C₂⁴	C₂⁴×C₁₀₉	1	4	yes	—
374	C₂⁴	C₂⁴×C₁₀₉	1	4	yes	—
375	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
376	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
377	C₂×D₄	—	2	16	no	—
378	C₂×D₄	—	2	16	no	—
379	C₂×D₄	—	2	16	no	—
380	C₂×D₄	—	2	32	no	—
381	C₂×D₄	—	2	32	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	4	16	no	—
384	C₂×D₄	—	4	8	no	—
385	C₂×D₄	—	4	64	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	32	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	8	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	16	no	—
399	C₂²×C₄	—	4	16	no	—
400	C₂²×C₄	—	4	8	no	—
401	C₂²×C₄	—	4	16	no	—
402	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₁₀₉	1	16	yes	—
404	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
405	C₂²×C₄	C₂×D₄×C₁₀₉	1	16	yes	complemented
406	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₁₀₉	1	8	yes	—
410	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	32	no	—
413	C₂²⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	2	16	no	—
415	C₂²⋊C₄	—	2	32	no	—
416	C₂²⋊C₄	—	2	16	no	—
417	C₂²⋊C₄	—	2	32	no	—
418	C₂²⋊C₄	C₂×D₄×C₁₀₉	1	32	yes	complemented
419	C₄²	—	4	8	no	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	8	no	—
422	C₂³	—	2	32	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	32	no	—
426	C₂³	—	2	16	no	—
427	C₂³	—	2	32	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	8	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	8	no	—
432	C₂³	—	2	4	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	4	64	no	—
438	C₂³	—	4	128	no	—
439	C₂³	—	4	16	no	—
440	C₂³	—	4	16	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	8	no	—
443	C₂³	—	4	16	no	—
444	C₂³	—	4	8	no	—
445	C₂³	—	4	8	no	—
446	C₂³	—	4	8	no	—
447	C₂³	—	4	16	no	—
448	C₂³	—	4	16	no	—
449	C₂³	#127×C₁₀₉	1	16	yes	—
450	C₂³	C₂²×D₄×C₁₀₉	1	8	yes	—
451	C₂³	C₂²×D₄×C₁₀₉	1	4	yes	—
452	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
453	C₂³	C₂²×D₄×C₁₀₉	1	4	yes	—
454	C₂³	C₂×#41×C₁₀₉	1	8	yes	—
455	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
456	C₂³	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
457	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
458	D₄	—	4	64	no	—
459	D₄	—	4	16	no	—
460	D₄	—	4	32	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	32	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	16	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₁₀₉	1	16	yes	—
479	C₂×C₄	C₂²×D₄×C₁₀₉	1	4	yes	—
480	C₂²	—	2	16	no	—
481	C₂²	—	2	32	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	16	no	—
485	C₂²	—	2	8	no	—
486	C₂²	—	4	64	no	—
487	C₂²	—	4	8	no	—
488	C₂²	—	4	32	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	8	no	—
492	C₂²	—	4	8	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	8	16	no	—
497	C₂²	#525×C₁₀₉	1	4	yes	—
498	C₂²	C₂×#127×C₁₀₉	1	8	yes	—
499	C₂²	C₂×#127×C₁₀₉	1	4	yes	—
500	C₂²	#534×C₁₀₉	1	4	yes	—
501	C₂²	#546×C₁₀₉	1	16	yes	—
502	C₂²	C₂³×D₄×C₁₀₉	1	2	yes	—
503	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
504	C₂²	C₂²×#41×C₁₀₉	1	2	yes	—
505	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	2	yes	—
506	C₄	—	2	16	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	4	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₂	—	2	8	no	—
512	C₂	—	4	8	no	—
513	C₂	—	8	4	no	—
514	C₂	C₁₀₉×#2539	1	8	yes	direct product factor
515	C₂	C₂×#525×C₁₀₉	1	1	yes	characteristic
516	C₂	C₂²×#127×C₁₀₉	1	1	yes	characteristic
517	C₂	C₂×#534×C₁₀₉	1	1	yes	characteristic
518	C₂	C₂×#546×C₁₀₉	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#123×C₁₀₉	C₂	semidirect
2. C₂²×#121×C₁₀₉	C₂	semidirect
3. C₂²×#120×C₁₀₉	C₂	semidirect
4. C₂³×C₂²⋊C₄×C₁₀₉	C₂	semidirect
5. C₁₀₉×#2539	C₂	direct (N × H)
6. C₂×#399×C₁₀₉	C₂	semidirect
7. C₂×#392×C₁₀₉	C₂	semidirect
8. C₂×#390×C₁₀₉	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
13. C₂³×D₄×C₁₀₉	C₂²	semidirect
17. C₂×#123×C₁₀₉	C₂²	semidirect
19. C₂×#121×C₁₀₉	C₂²	semidirect
21. C₂×#120×C₁₀₉	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₁₀₉	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₁₀₉	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
30. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
31. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
32. C₂²×C₄²×C₁₀₉	C₂²	semidirect
33. #399×C₁₀₉	C₂²	semidirect
34. #392×C₁₀₉	C₂²	semidirect
35. #390×C₁₀₉	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
37. C₂×#95×C₁₀₉	C₂²	semidirect
56. C₂²×D₄×C₁₀₉	C₂³	semidirect
60. C₂³×C₄×C₁₀₉	D₄	semidirect
73. C₂×C₄⋊C₄×C₁₀₉	C₂³	semidirect
74. C₂×C₄⋊C₄×C₁₀₉	C₂³	semidirect
81. C₂×C₂²⋊C₄×C₁₀₉	D₄	semidirect
86. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
87. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
89. C₂×C₄²×C₁₀₉	C₂³	semidirect
90. #95×C₁₀₉	C₂³	semidirect
145. C₂²×C₄×C₁₀₉	C₂×D₄	semidirect
158. C₂²⋊C₄×C₁₀₉	C₂×D₄	semidirect
251. C₂×#2539	C₁₀₉	direct (N × H)
255. C₂×C₁₀₉	#2539	direct (N × H)
260. C₂²×#123	C₂×C₁₀₉	semidirect
261. C₂²×#121	C₂×C₁₀₉	semidirect
262. C₂²×#120	C₂×C₁₀₉	semidirect
263. C₂³×C₂²⋊C₄	C₂×C₁₀₉	semidirect
264. #2539	C₂×C₁₀₉	direct (N × H)
265. C₂×#399	C₂×C₁₀₉	semidirect
266. C₂×#392	C₂×C₁₀₉	semidirect
267. C₂×#390	C₂×C₁₀₉	semidirect
268. C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
269. C₁₀₉	C₂×#2539	direct (N × H)
273. C₂³×D₄	C₂²×C₁₀₉	semidirect
277. C₂×#123	C₂²×C₁₀₉	semidirect
279. C₂×#121	C₂²×C₁₀₉	semidirect
281. C₂×#120	C₂²×C₁₀₉	semidirect
282. C₂²×C₄⋊C₄	C₂²×C₁₀₉	semidirect
283. C₂²×C₄⋊C₄	C₂²×C₁₀₉	semidirect
289. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
290. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
291. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
292. C₂²×C₄²	C₂²×C₁₀₉	semidirect
293. #399	C₂²×C₁₀₉	semidirect
294. #392	C₂²×C₁₀₉	semidirect
295. #390	C₂²×C₁₀₉	semidirect
296. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
297. C₂×#95	C₂²×C₁₀₉	semidirect
316. C₂²×D₄	C₂³×C₁₀₉	semidirect
320. C₂³×C₄	D₄×C₁₀₉	semidirect
333. C₂×C₄⋊C₄	C₂³×C₁₀₉	semidirect
334. C₂×C₄⋊C₄	C₂³×C₁₀₉	semidirect
341. C₂×C₂²⋊C₄	D₄×C₁₀₉	semidirect
346. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
347. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
349. C₂×C₄²	C₂³×C₁₀₉	semidirect
350. #95	C₂³×C₁₀₉	semidirect
405. C₂²×C₄	C₂×D₄×C₁₀₉	semidirect
418. C₂²⋊C₄	C₂×D₄×C₁₀₉	semidirect
514. C₂	C₁₀₉×#2539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#123×C₁₀₉	2
C₂²×#121×C₁₀₉	2
C₂²×#120×C₁₀₉	2
C₂³×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#2539	2
C₂×#399×C₁₀₉	2
C₂×#392×C₁₀₉	2
C₂×#390×C₁₀₉	2
C₂×C₄×C₂²⋊C₄×C₁₀₉	2
C₂×#2539	109

Subgroups — C2×C109×#2539 #42152813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#2539 #42152813