Subgroups — C₂×C₁₀₉×#1388 #42130564

Order 27904.

554 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1388	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆×C₁₀₉	—	2	1	no	—
6	C₂²×M₁₆×C₁₀₉	—	2	1	no	—
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×M₃₂×C₁₀₉	—	2	1	no	—
9	C₂×M₃₂×C₁₀₉	—	2	1	no	—
10	C₂×M₃₂×C₁₀₉	—	2	1	no	—
11	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	—
12	C₂×M₃₂×C₁₀₉	C₄	1	2	yes	—
13	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₁₀₉	—	2	1	no	—
15	C₂×#108×C₁₀₉	—	2	1	no	—
16	C₂×#108×C₁₀₉	—	2	1	no	—
17	C₂×#108×C₁₀₉	—	2	1	no	—
18	C₂×#108×C₁₀₉	—	2	1	no	—
19	C₂×#108×C₁₀₉	—	2	1	no	—
20	C₂×#108×C₁₀₉	—	2	1	no	—
21	C₂×#108×C₁₀₉	C₄	1	2	yes	—
22	C₂×#108×C₁₀₉	C₄	1	2	yes	—
23	C₂×#108×C₁₀₉	C₄	1	2	yes	—
24	C₂×#108×C₁₀₉	C₄	1	2	yes	—
25	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×M₁₆×C₁₀₉	—	2	2	no	—
30	C₂×M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×M₁₆×C₁₀₉	—	2	2	no	—
32	C₂×M₁₆×C₁₀₉	—	2	2	no	—
33	C₂×M₁₆×C₁₀₉	—	2	2	no	—
34	C₂×M₁₆×C₁₀₉	—	4	1	no	—
35	C₂×M₁₆×C₁₀₉	—	4	1	no	—
36	C₂×M₁₆×C₁₀₉	—	4	1	no	—
37	C₂×M₁₆×C₁₀₉	—	4	1	no	—
38	C₂×M₁₆×C₁₀₉	—	4	1	no	—
39	C₂×M₁₆×C₁₀₉	—	4	1	no	—
40	C₂²×C₈×C₁₀₉	—	4	1	no	—
41	C₂²×C₈×C₁₀₉	—	4	1	no	—
42	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
54	C₂²×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
55	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	M₃₂×C₁₀₉	—	2	2	no	—
57	M₃₂×C₁₀₉	—	2	2	no	—
58	M₃₂×C₁₀₉	—	2	2	no	—
59	M₃₂×C₁₀₉	C₂×C₄	1	4	yes	—
60	C₂×C₁₆×C₁₀₉	—	2	2	no	—
61	C₂×C₁₆×C₁₀₉	—	2	2	no	—
62	C₂×C₁₆×C₁₀₉	—	2	2	no	—
63	#108×C₁₀₉	—	2	2	no	—
64	#108×C₁₀₉	—	2	2	no	—
65	#108×C₁₀₉	—	2	2	no	—
66	#108×C₁₀₉	—	2	2	no	—
67	#108×C₁₀₉	—	2	2	no	—
68	#108×C₁₀₉	—	2	2	no	—
69	#108×C₁₀₉	C₂×C₄	1	4	yes	—
70	#108×C₁₀₉	C₂×C₄	1	4	yes	—
71	C₂⁴×C₁₀₉	Q₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	M₁₆×C₁₀₉	—	2	4	no	—
87	M₁₆×C₁₀₉	—	2	4	no	—
88	M₁₆×C₁₀₉	—	4	2	no	—
89	M₁₆×C₁₀₉	—	4	2	no	—
90	M₁₆×C₁₀₉	—	4	2	no	—
91	M₁₆×C₁₀₉	—	4	2	no	—
92	M₁₆×C₁₀₉	—	4	2	no	—
93	M₁₆×C₁₀₉	—	4	2	no	—
94	C₂×C₈×C₁₀₉	—	4	2	no	—
95	C₂×C₈×C₁₀₉	—	4	2	no	—
96	C₂×C₈×C₁₀₉	—	4	2	no	—
97	C₂×C₈×C₁₀₉	—	4	2	no	—
98	C₂×C₈×C₁₀₉	—	4	2	no	—
99	C₂×C₈×C₁₀₉	—	4	2	no	—
100	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
102	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
103	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
128	C₁₆×C₁₀₉	—	2	4	no	—
129	C₂³×C₁₀₉	—	4	2	no	—
130	C₂³×C₁₀₉	#103	1	1	yes	characteristic
131	C₂³×C₁₀₉	#103	1	1	yes	characteristic
132	C₂³×C₁₀₉	#103	1	1	yes	characteristic
133	C₂³×C₁₀₉	#103	1	1	yes	characteristic
134	C₂³×C₁₀₉	#107	1	1	yes	characteristic
135	C₂³×C₁₀₉	#107	1	1	yes	characteristic
136	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
138	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
139	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
140	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
142	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
143	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
144	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
145	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
146	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
147	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
148	C₂×C₄×C₁₀₉	#102	1	1	yes	characteristic
149	C₂×C₄×C₁₀₉	#103	1	1	yes	characteristic
150	C₂×C₄×C₁₀₉	#103	1	1	yes	characteristic
151	C₂×C₄×C₁₀₉	#103	1	1	yes	characteristic
152	C₂×C₄×C₁₀₉	#103	1	1	yes	characteristic
153	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
154	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
156	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
157	C₂×C₄×C₁₀₉	#107	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	#107	1	1	yes	characteristic
159	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
165	C₈×C₁₀₉	—	4	4	no	—
166	C₈×C₁₀₉	—	4	4	no	—
167	C₈×C₁₀₉	#95	1	1	yes	characteristic
168	C₈×C₁₀₉	#95	1	1	yes	characteristic
169	C₈×C₁₀₉	#95	1	1	yes	characteristic
170	C₈×C₁₀₉	#95	1	1	yes	characteristic
171	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
172	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₈×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₀₉	—	2	2	no	—
176	C₂²×C₁₀₉	—	2	2	no	—
177	C₂²×C₁₀₉	—	2	2	no	—
178	C₂²×C₁₀₉	—	4	1	no	—
179	C₂²×C₁₀₉	—	4	1	no	—
180	C₂²×C₁₀₉	—	4	1	no	—
181	C₂²×C₁₀₉	—	4	1	no	—
182	C₂²×C₁₀₉	#340	1	1	yes	characteristic
183	C₂²×C₁₀₉	#340	1	1	yes	characteristic
184	C₂²×C₁₀₉	#340	1	1	yes	characteristic
185	C₂²×C₁₀₉	#340	1	1	yes	characteristic
186	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
187	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
188	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
189	C₄×C₁₀₉	#340	1	1	yes	characteristic
190	C₄×C₁₀₉	#340	1	1	yes	characteristic
191	C₄×C₁₀₉	#340	1	1	yes	characteristic
192	C₄×C₁₀₉	#340	1	1	yes	characteristic
193	C₄×C₁₀₉	C₂×#95	1	1	yes	characteristic
194	C₄×C₁₀₉	C₂×#102	1	1	yes	characteristic
195	C₄×C₁₀₉	C₂×#102	1	1	yes	characteristic
196	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
197	C₂×#1388	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₁₀₉	—	2	1	no	—
199	C₂×C₁₀₉	—	2	1	no	—
200	C₂×C₁₀₉	—	2	1	no	—
201	C₂×C₁₀₉	—	2	1	no	—
202	C₂×C₁₀₉	—	2	1	no	—
203	C₂×C₁₀₉	—	2	1	no	—
204	C₂×C₁₀₉	#1388	1	2	yes	direct product factor
205	C₂×C₁₀₉	C₂×#340	1	1	yes	characteristic
206	C₂²×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic
207	C₂²×#108	C₂×C₁₀₉	1	1	yes	characteristic
208	C₂²×#108	C₂×C₁₀₉	1	1	yes	characteristic
209	#1388	C₂×C₁₀₉	1	4	yes	direct product factor
210	C₁₀₉	C₂×#1388	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×M₁₆	—	2	1	no	—
212	C₂²×M₁₆	—	2	1	no	—
213	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₂×M₃₂	—	2	1	no	—
215	C₂×M₃₂	—	2	1	no	—
216	C₂×M₃₂	—	2	1	no	—
217	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	—
218	C₂×M₃₂	C₄×C₁₀₉	1	2	yes	—
219	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
220	C₂²×C₁₆	—	2	1	no	—
221	C₂×#108	—	2	1	no	—
222	C₂×#108	—	2	1	no	—
223	C₂×#108	—	2	1	no	—
224	C₂×#108	—	2	1	no	—
225	C₂×#108	—	2	1	no	—
226	C₂×#108	—	2	1	no	—
227	C₂×#108	C₄×C₁₀₉	1	2	yes	—
228	C₂×#108	C₄×C₁₀₉	1	2	yes	—
229	C₂×#108	C₄×C₁₀₉	1	2	yes	—
230	C₂×#108	C₄×C₁₀₉	1	2	yes	—
231	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
232	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
233	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	4	1	no	—
241	C₂×M₁₆	—	4	1	no	—
242	C₂×M₁₆	—	4	1	no	—
243	C₂×M₁₆	—	4	1	no	—
244	C₂×M₁₆	—	4	1	no	—
245	C₂×M₁₆	—	4	1	no	—
246	C₂²×C₈	—	4	1	no	—
247	C₂²×C₈	—	4	1	no	—
248	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
249	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
250	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
251	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
252	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
253	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
254	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
255	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
257	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
258	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
259	C₂²×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
260	C₂²×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
262	M₃₂	—	2	2	no	—
263	M₃₂	—	2	2	no	—
264	M₃₂	—	2	2	no	—
265	M₃₂	C₂×C₄×C₁₀₉	1	4	yes	—
266	C₂×C₁₆	—	2	2	no	—
267	C₂×C₁₆	—	2	2	no	—
268	C₂×C₁₆	—	2	2	no	—
269	#108	—	2	2	no	—
270	#108	—	2	2	no	—
271	#108	—	2	2	no	—
272	#108	—	2	2	no	—
273	#108	—	2	2	no	—
274	#108	—	2	2	no	—
275	#108	C₂×C₄×C₁₀₉	1	4	yes	—
276	#108	C₂×C₄×C₁₀₉	1	4	yes	—
277	C₂⁴	Q₁₆×C₁₀₉	1	1	yes	characteristic
278	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
279	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
280	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
281	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
283	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
285	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
286	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
287	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
288	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
289	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
290	C₂²×C₄	Q₁₆×C₁₀₉	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
292	M₁₆	—	2	4	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	4	2	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	M₁₆	—	4	2	no	—
299	M₁₆	—	4	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	4	2	no	—
304	C₂×C₈	—	4	2	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
307	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
308	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
309	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
310	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
311	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
312	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
313	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
314	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
315	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
316	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
317	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
318	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
319	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
320	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
321	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
322	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
323	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
324	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
325	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
326	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
327	C₂×C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
328	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
329	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
330	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
333	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
334	C₁₆	—	2	4	no	—
335	C₂³	—	4	2	no	—
336	C₂³	#103×C₁₀₉	1	1	yes	characteristic
337	C₂³	#103×C₁₀₉	1	1	yes	characteristic
338	C₂³	#103×C₁₀₉	1	1	yes	characteristic
339	C₂³	#103×C₁₀₉	1	1	yes	characteristic
340	C₂³	#107×C₁₀₉	1	1	yes	characteristic
341	C₂³	#107×C₁₀₉	1	1	yes	characteristic
342	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
343	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
344	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
345	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
346	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
347	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
348	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
349	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
350	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
351	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
352	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
353	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
354	C₂×C₄	#102×C₁₀₉	1	1	yes	characteristic
355	C₂×C₄	#103×C₁₀₉	1	1	yes	characteristic
356	C₂×C₄	#103×C₁₀₉	1	1	yes	characteristic
357	C₂×C₄	#103×C₁₀₉	1	1	yes	characteristic
358	C₂×C₄	#103×C₁₀₉	1	1	yes	characteristic
359	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
360	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
361	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
362	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
363	C₂×C₄	#107×C₁₀₉	1	1	yes	characteristic
364	C₂×C₄	#107×C₁₀₉	1	1	yes	characteristic
365	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
366	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
368	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
369	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
370	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
371	C₈	—	4	4	no	—
372	C₈	—	4	4	no	—
373	C₈	#95×C₁₀₉	1	1	yes	characteristic
374	C₈	#95×C₁₀₉	1	1	yes	characteristic
375	C₈	#95×C₁₀₉	1	1	yes	characteristic
376	C₈	#95×C₁₀₉	1	1	yes	characteristic
377	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
378	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
379	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
380	C₈	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	#340×C₁₀₉	1	1	yes	characteristic
389	C₂²	#340×C₁₀₉	1	1	yes	characteristic
390	C₂²	#340×C₁₀₉	1	1	yes	characteristic
391	C₂²	#340×C₁₀₉	1	1	yes	characteristic
392	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
393	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
394	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
395	C₄	#340×C₁₀₉	1	1	yes	characteristic
396	C₄	#340×C₁₀₉	1	1	yes	characteristic
397	C₄	#340×C₁₀₉	1	1	yes	characteristic
398	C₄	#340×C₁₀₉	1	1	yes	characteristic
399	C₄	C₂×#95×C₁₀₉	1	1	yes	characteristic
400	C₄	C₂×#102×C₁₀₉	1	1	yes	characteristic
401	C₄	C₂×#102×C₁₀₉	1	1	yes	characteristic
402	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
403	C₂	—	2	1	no	—
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	C₁₀₉×#1388	1	2	yes	direct product factor
410	C₂	C₂×#340×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	#102
commutator	C₈	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	#340
prime radical	C₂⁴×C₁₀₉	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1388	C₂	direct (N × H)
197. C₂×#1388	C₁₀₉	direct (N × H)
204. C₂×C₁₀₉	#1388	direct (N × H)
209. #1388	C₂×C₁₀₉	direct (N × H)
210. C₁₀₉	C₂×#1388	direct (N × H)
409. C₂	C₁₀₉×#1388	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×M₃₂×C₁₀₉	2
C₂²×#108×C₁₀₉	2
C₁₀₉×#1388	2
C₂×#1388	109

Subgroups — C2×C109×#1388 #42130564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#1388 #42130564