Subgroups — C₁₀₉×#20664 #42138658

Order 27904.

1438 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#2311	C₂	1	2	yes	maximal
4	C₁₀₉×#1809	C₂	1	4	yes	maximal, complemented
5	C₁₀₉×#1808	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1807	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1735	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#340×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#131×C₁₀₉	—	2	8	no	—
11	C₂×#131×C₁₀₉	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	#428×C₁₀₉	—	2	8	no	—
19	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
20	C₂×#106×C₁₀₉	C₂²	1	4	yes	—
21	#391×C₁₀₉	—	2	2	no	—
22	#390×C₁₀₉	—	2	2	no	—
23	#386×C₁₀₉	—	2	4	no	—
24	#380×C₁₀₉	C₂²	1	4	yes	—
25	#380×C₁₀₉	C₂²	1	4	yes	—
26	#340×C₁₀₉	C₂²	1	2	yes	—
27	#340×C₁₀₉	C₂²	1	2	yes	—
28	#340×C₁₀₉	C₂²	1	1	yes	characteristic
29	#340×C₁₀₉	C₂²	1	1	yes	characteristic
30	#340×C₁₀₉	C₂²	1	2	yes	—
31	C₂×#41×C₁₀₉	—	2	4	no	—
32	C₂×#41×C₁₀₉	—	2	4	no	—
33	C₂×#41×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₂×#41×C₁₀₉	Q₈	1	4	yes	—
35	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂²×D₄×C₁₀₉	D₄	1	2	yes	—
37	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₀₉	—	2	2	no	—
39	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	#131×C₁₀₉	—	2	32	no	—
42	C₂×M₁₆×C₁₀₉	—	2	8	no	—
43	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
45	C₂×M₁₆×C₁₀₉	D₄	1	8	yes	—
46	C₂²×C₈×C₁₀₉	—	2	4	no	—
47	C₂²×C₈×C₁₀₉	—	2	8	no	—
48	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
50	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
53	#117×C₁₀₉	—	2	4	no	—
54	#117×C₁₀₉	—	2	4	no	—
55	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
57	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
61	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
64	C₂×C₄²×C₁₀₉	—	2	2	no	—
65	#107×C₁₀₉	—	2	8	no	—
66	#106×C₁₀₉	—	2	8	no	—
67	#95×C₁₀₉	—	2	4	no	—
68	#95×C₁₀₉	—	2	4	no	—
69	C₂⁴×C₁₀₉	C₂×D₄	1	2	yes	—
70	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	#41×C₁₀₉	—	2	8	no	—
72	#41×C₁₀₉	—	2	8	no	—
73	#41×C₁₀₉	—	2	16	no	—
74	C₂×Q₈×C₁₀₉	—	2	4	no	—
75	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
76	C₂×D₄×C₁₀₉	—	2	8	no	—
77	C₂×D₄×C₁₀₉	—	2	4	no	—
78	C₂×D₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
79	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₁₀₉	—	2	4	no	—
81	C₂²×C₄×C₁₀₉	—	2	4	no	—
82	C₂²×C₄×C₁₀₉	—	2	2	no	—
83	C₂²×C₄×C₁₀₉	—	2	2	no	—
84	C₂²×C₄×C₁₀₉	—	2	4	no	—
85	C₂²×C₄×C₁₀₉	—	4	4	no	—
86	C₂²×C₄×C₁₀₉	—	8	2	no	—
87	C₂²×C₄×C₁₀₉	—	8	2	no	—
88	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
96	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₁₀₉	—	2	16	no	—
98	M₁₆×C₁₀₉	C₂²⋊C₄	1	16	yes	complemented
99	C₂×C₈×C₁₀₉	—	2	8	no	—
100	C₂×C₈×C₁₀₉	—	2	2	no	—
101	C₂×C₈×C₁₀₉	—	2	2	no	—
102	C₂×C₈×C₁₀₉	—	2	16	no	—
103	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₁₀₉	C₂×D₄	1	8	yes	—
107	C₄⋊C₄×C₁₀₉	—	2	2	no	—
108	C₄⋊C₄×C₁₀₉	—	2	4	no	—
109	C₄⋊C₄×C₁₀₉	—	2	2	no	—
110	C₄⋊C₄×C₁₀₉	—	2	4	no	—
111	C₄⋊C₄×C₁₀₉	—	2	2	no	—
112	C₄⋊C₄×C₁₀₉	—	2	2	no	—
113	C₄⋊C₄×C₁₀₉	—	4	4	no	—
114	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
115	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
116	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
117	C₄²×C₁₀₉	—	4	4	no	—
118	C₂³×C₁₀₉	—	2	1	no	—
119	C₂³×C₁₀₉	—	2	2	no	—
120	C₂³×C₁₀₉	—	2	1	no	—
121	C₂³×C₁₀₉	—	2	4	no	—
122	C₂³×C₁₀₉	—	2	2	no	—
123	C₂³×C₁₀₉	—	2	4	no	—
124	C₂³×C₁₀₉	—	2	2	no	—
125	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
126	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂³×C₁₀₉	#121	1	4	yes	—
128	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₁₀₉	—	2	8	no	—
130	D₄×C₁₀₉	—	2	16	no	—
131	D₄×C₁₀₉	—	2	8	no	—
132	C₂×C₄×C₁₀₉	—	2	4	no	—
133	C₂×C₄×C₁₀₉	—	2	4	no	—
134	C₂×C₄×C₁₀₉	—	2	4	no	—
135	C₂×C₄×C₁₀₉	—	2	8	no	—
136	C₂×C₄×C₁₀₉	—	4	2	no	—
137	C₂×C₄×C₁₀₉	—	4	2	no	—
138	C₂×C₄×C₁₀₉	—	4	4	no	—
139	C₂×C₄×C₁₀₉	—	4	2	no	—
140	C₂×C₄×C₁₀₉	—	4	4	no	—
141	C₂×C₄×C₁₀₉	—	4	2	no	—
142	C₂×C₄×C₁₀₉	—	4	4	no	—
143	C₂×C₄×C₁₀₉	—	4	2	no	—
144	C₂×C₄×C₁₀₉	—	4	2	no	—
145	C₂×C₄×C₁₀₉	—	8	8	no	—
146	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
147	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
148	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₁₀₉	#117	1	4	yes	—
150	C₂×C₄×C₁₀₉	#121	1	4	yes	—
151	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	8	yes	—
152	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
156	C₈×C₁₀₉	—	2	2	no	—
157	C₈×C₁₀₉	—	2	2	no	—
158	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	8	yes	—
159	C₂²×C₁₀₉	—	2	2	no	—
160	C₂²×C₁₀₉	—	2	2	no	—
161	C₂²×C₁₀₉	—	2	4	no	—
162	C₂²×C₁₀₉	—	2	4	no	—
163	C₂²×C₁₀₉	—	2	4	no	—
164	C₂²×C₁₀₉	—	2	4	no	—
165	C₂²×C₁₀₉	—	2	4	no	—
166	C₂²×C₁₀₉	—	4	4	no	—
167	C₂²×C₁₀₉	—	4	4	no	—
168	C₂²×C₁₀₉	#380	1	4	yes	—
169	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₁₀₉	C₂×#121	1	2	yes	—
171	C₂²×C₁₀₉	#576	1	1	yes	characteristic
172	C₂²×C₁₀₉	#577	1	2	yes	—
173	C₂²×C₁₀₉	#578	1	1	yes	characteristic
174	C₄×C₁₀₉	—	2	4	no	—
175	C₄×C₁₀₉	—	8	2	no	—
176	C₄×C₁₀₉	—	8	4	no	—
177	C₄×C₁₀₉	—	8	2	no	—
178	C₄×C₁₀₉	#380	1	4	yes	—
179	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₁₀₉	C₂×#117	1	1	yes	characteristic
181	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	2	yes	—
182	#20664	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂×C₁₀₉	—	2	4	no	—
184	C₂×C₁₀₉	—	4	4	no	—
185	C₂×C₁₀₉	C₂×#380	1	1	yes	characteristic
186	C₂×C₁₀₉	#2912	1	1	yes	characteristic
187	C₂×C₁₀₉	#2913	1	1	yes	characteristic
188	C₂×C₁₀₉	#3062	1	2	yes	—
189	C₂×C₁₀₉	#3063	1	2	yes	—
190	C₂²×#131	C₂×C₁₀₉	1	1	yes	characteristic
191	C₂×#428	C₂×C₁₀₉	1	2	yes	complemented
192	#2311	C₂×C₁₀₉	1	2	yes	—
193	#1809	C₂×C₁₀₉	1	4	yes	complemented
194	#1808	C₂×C₁₀₉	1	2	yes	complemented
195	#1807	C₂×C₁₀₉	1	2	yes	complemented
196	#1735	C₂×C₁₀₉	1	1	yes	characteristic, complemented
197	C₂×#340	C₂×C₁₀₉	1	1	yes	characteristic, complemented
198	C₁₀₉	#20664	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
200	C₂×#131	—	2	8	no	—
201	C₂×#131	C₄×C₁₀₉	1	8	yes	complemented
202	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
203	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
204	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
205	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
206	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
207	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
208	#428	—	2	8	no	—
209	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
210	C₂×#106	C₂²×C₁₀₉	1	4	yes	—
211	#391	—	2	2	no	—
212	#390	—	2	2	no	—
213	#386	—	2	4	no	—
214	#380	C₂²×C₁₀₉	1	4	yes	—
215	#380	C₂²×C₁₀₉	1	4	yes	—
216	#340	C₂²×C₁₀₉	1	1	yes	characteristic
217	#340	C₂²×C₁₀₉	1	2	yes	—
218	#340	C₂²×C₁₀₉	1	1	yes	characteristic
219	#340	C₂²×C₁₀₉	1	2	yes	—
220	#340	C₂²×C₁₀₉	1	2	yes	—
221	C₂×#41	—	2	4	no	—
222	C₂×#41	—	2	4	no	—
223	C₂×#41	C₂×C₄×C₁₀₉	1	4	yes	—
224	C₂×#41	Q₈×C₁₀₉	1	4	yes	—
225	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
226	C₂²×D₄	D₄×C₁₀₉	1	2	yes	—
227	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
228	C₂³×C₄	—	2	2	no	—
229	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
230	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
231	#131	—	2	32	no	—
232	C₂×M₁₆	—	2	8	no	—
233	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
234	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	complemented
235	C₂×M₁₆	D₄×C₁₀₉	1	8	yes	—
236	C₂²×C₈	—	2	4	no	—
237	C₂²×C₈	—	2	8	no	—
238	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
239	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
240	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
243	#117	—	2	4	no	—
244	#117	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	—	2	4	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
249	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	—	2	4	no	—
253	C₂×C₂²⋊C₄	—	2	4	no	—
254	C₂×C₄²	—	2	2	no	—
255	#107	—	2	8	no	—
256	#106	—	2	8	no	—
257	#95	—	2	4	no	—
258	#95	—	2	4	no	—
259	C₂⁴	C₂×D₄×C₁₀₉	1	2	yes	—
260	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
261	#41	—	2	8	no	—
262	#41	—	2	8	no	—
263	#41	—	2	16	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
266	C₂×D₄	—	2	8	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	C₄⋊C₄×C₁₀₉	1	8	yes	—
269	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	4	4	no	—
276	C₂²×C₄	—	8	2	no	—
277	C₂²×C₄	—	8	2	no	—
278	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
279	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
282	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
283	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
284	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
285	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
286	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
287	M₁₆	—	2	16	no	—
288	M₁₆	C₂²⋊C₄×C₁₀₉	1	16	yes	complemented
289	C₂×C₈	—	2	8	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	16	no	—
293	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
294	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
295	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
296	C₂×C₈	C₂×D₄×C₁₀₉	1	8	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	4	no	—
304	C₂²⋊C₄	—	4	4	no	—
305	C₂²⋊C₄	—	8	4	no	—
306	C₂²⋊C₄	—	8	4	no	—
307	C₄²	—	4	4	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	2	2	no	—
310	C₂³	—	2	1	no	—
311	C₂³	—	2	4	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	4	no	—
314	C₂³	—	2	2	no	—
315	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
316	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
317	C₂³	#121×C₁₀₉	1	4	yes	—
318	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
319	Q₈	—	2	8	no	—
320	D₄	—	2	16	no	—
321	D₄	—	2	8	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	8	no	—
336	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
337	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
338	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
339	C₂×C₄	#117×C₁₀₉	1	4	yes	—
340	C₂×C₄	#121×C₁₀₉	1	4	yes	—
341	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	8	yes	—
342	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
344	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
345	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	8	yes	—
349	C₂²	—	2	4	no	—
350	C₂²	—	2	4	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	4	no	—
357	C₂²	—	4	4	no	—
358	C₂²	#380×C₁₀₉	1	4	yes	—
359	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
360	C₂²	C₂×#121×C₁₀₉	1	2	yes	—
361	C₂²	#576×C₁₀₉	1	1	yes	characteristic
362	C₂²	#577×C₁₀₉	1	2	yes	—
363	C₂²	#578×C₁₀₉	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	8	4	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	2	no	—
368	C₄	#380×C₁₀₉	1	4	yes	—
369	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
370	C₄	C₂×#117×C₁₀₉	1	1	yes	characteristic
371	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	2	yes	—
372	C₂	—	2	4	no	—
373	C₂	—	4	4	no	—
374	C₂	C₂×#380×C₁₀₉	1	1	yes	characteristic
375	C₂	C₁₀₉×#2912	1	1	yes	characteristic
376	C₂	C₁₀₉×#2913	1	1	yes	characteristic
377	C₂	C₁₀₉×#3062	1	2	yes	—
378	C₂	C₁₀₉×#3063	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×#41×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1809	C₂	semidirect
5. C₁₀₉×#1808	C₂	semidirect
6. C₁₀₉×#1807	C₂	semidirect
7. C₁₀₉×#1735	C₂	semidirect
8. C₂×#340×C₁₀₉	C₂	semidirect
11. C₂×#131×C₁₀₉	C₄	semidirect
44. C₂×M₁₆×C₁₀₉	C₂×C₄	semidirect
49. C₂²×C₈×C₁₀₉	C₂×C₄	semidirect
98. M₁₆×C₁₀₉	C₂²⋊C₄	semidirect
182. #20664	C₁₀₉	direct (N × H)
191. C₂×#428	C₂×C₁₀₉	semidirect
193. #1809	C₂×C₁₀₉	semidirect
194. #1808	C₂×C₁₀₉	semidirect
195. #1807	C₂×C₁₀₉	semidirect
196. #1735	C₂×C₁₀₉	semidirect
197. C₂×#340	C₂×C₁₀₉	semidirect
198. C₁₀₉	#20664	direct (N × H)
201. C₂×#131	C₄×C₁₀₉	semidirect
234. C₂×M₁₆	C₂×C₄×C₁₀₉	semidirect
239. C₂²×C₈	C₂×C₄×C₁₀₉	semidirect
288. M₁₆	C₂²⋊C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₀₉	2
C₂×#428×C₁₀₉	2
C₁₀₉×#2311	2
C₁₀₉×#1809	2
C₁₀₉×#1808	2
C₁₀₉×#1807	2
C₁₀₉×#1735	2
C₂×#340×C₁₀₉	2
#20664	109

Subgroups — C109×#20664 #42138658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20664 #42138658