Subgroups — C₁₀₉×#20663 #42138657

Order 27904.

1438 conjugacy classes of subgroups.

The subgroup lattice has 402 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×#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2311	C₂	1	2	yes	maximal
5	C₁₀₉×#1809	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1808	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#1736	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#131×C₁₀₉	—	2	4	no	—
10	C₂×#131×C₁₀₉	—	2	4	no	—
11	C₂×#131×C₁₀₉	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	#427×C₁₀₉	—	2	4	no	—
19	C₂×#107×C₁₀₉	—	2	4	no	—
20	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
22	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
23	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
24	#391×C₁₀₉	—	2	2	no	—
25	#390×C₁₀₉	—	2	2	no	—
26	#386×C₁₀₉	—	2	4	no	—
27	#380×C₁₀₉	C₂²	1	8	yes	—
28	#340×C₁₀₉	C₂²	1	8	yes	—
29	C₂×#41×C₁₀₉	—	2	2	no	—
30	C₂×#41×C₁₀₉	—	2	2	no	—
31	C₂×#41×C₁₀₉	—	2	2	no	—
32	C₂×#41×C₁₀₉	—	2	2	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	2	no	—
43	C₂×M₁₆×C₁₀₉	—	2	2	no	—
44	C₂×M₁₆×C₁₀₉	—	2	4	no	—
45	C₂×M₁₆×C₁₀₉	—	2	4	no	—
46	C₂×M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₁₀₉	—	2	4	no	—
49	C₂²×C₈×C₁₀₉	—	2	4	no	—
50	C₂²×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
52	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
53	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
54	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
57	#117×C₁₀₉	—	2	4	no	—
58	#117×C₁₀₉	—	2	4	no	—
59	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
60	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
61	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
64	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
66	C₂×C₄²×C₁₀₉	—	2	2	no	—
67	#107×C₁₀₉	—	2	4	no	—
68	#107×C₁₀₉	—	2	4	no	—
69	#107×C₁₀₉	—	2	4	no	—
70	#106×C₁₀₉	—	2	4	no	—
71	#95×C₁₀₉	—	2	8	no	—
72	C₂⁴×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
74	#41×C₁₀₉	—	2	8	no	—
75	#41×C₁₀₉	—	2	8	no	—
76	#41×C₁₀₉	—	2	8	no	—
77	#41×C₁₀₉	—	2	8	no	—
78	C₂×Q₈×C₁₀₉	—	2	2	no	—
79	C₂×Q₈×C₁₀₉	—	2	2	no	—
80	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
81	C₂×D₄×C₁₀₉	—	2	4	no	—
82	C₂×D₄×C₁₀₉	—	2	4	no	—
83	C₂×D₄×C₁₀₉	—	2	2	no	—
84	C₂×D₄×C₁₀₉	—	2	2	no	—
85	C₂×D₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
86	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
87	C₂²×C₄×C₁₀₉	—	2	2	no	—
88	C₂²×C₄×C₁₀₉	—	2	2	no	—
89	C₂²×C₄×C₁₀₉	—	2	2	no	—
90	C₂²×C₄×C₁₀₉	—	2	2	no	—
91	C₂²×C₄×C₁₀₉	—	2	4	no	—
92	C₂²×C₄×C₁₀₉	—	2	4	no	—
93	C₂²×C₄×C₁₀₉	—	4	4	no	—
94	C₂²×C₄×C₁₀₉	—	8	4	no	—
95	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
96	C₂²×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
102	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
104	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
105	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₁₀₉	—	2	8	no	—
107	M₁₆×C₁₀₉	—	2	16	no	—
108	C₂×C₈×C₁₀₉	—	2	2	no	—
109	C₂×C₈×C₁₀₉	—	2	2	no	—
110	C₂×C₈×C₁₀₉	—	2	16	no	—
111	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	16	yes	complemented
112	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
113	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
119	C₄⋊C₄×C₁₀₉	—	2	4	no	—
120	C₄⋊C₄×C₁₀₉	—	2	4	no	—
121	C₄⋊C₄×C₁₀₉	—	2	4	no	—
122	C₄⋊C₄×C₁₀₉	—	2	4	no	—
123	C₄⋊C₄×C₁₀₉	—	4	4	no	—
124	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
125	C₂²⋊C₄×C₁₀₉	—	8	8	no	—
126	C₄²×C₁₀₉	—	4	4	no	—
127	C₂³×C₁₀₉	—	2	2	no	—
128	C₂³×C₁₀₉	—	2	2	no	—
129	C₂³×C₁₀₉	—	2	2	no	—
130	C₂³×C₁₀₉	—	2	2	no	—
131	C₂³×C₁₀₉	—	2	4	no	—
132	C₂³×C₁₀₉	—	2	4	no	—
133	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
134	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂³×C₁₀₉	#121	1	2	yes	—
136	C₂³×C₁₀₉	#121	1	2	yes	—
137	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₁₀₉	—	2	8	no	—
139	D₄×C₁₀₉	—	2	16	no	—
140	D₄×C₁₀₉	—	2	8	no	—
141	C₂×C₄×C₁₀₉	—	2	2	no	—
142	C₂×C₄×C₁₀₉	—	2	2	no	—
143	C₂×C₄×C₁₀₉	—	2	4	no	—
144	C₂×C₄×C₁₀₉	—	2	4	no	—
145	C₂×C₄×C₁₀₉	—	2	4	no	—
146	C₂×C₄×C₁₀₉	—	2	4	no	—
147	C₂×C₄×C₁₀₉	—	4	4	no	—
148	C₂×C₄×C₁₀₉	—	4	4	no	—
149	C₂×C₄×C₁₀₉	—	4	4	no	—
150	C₂×C₄×C₁₀₉	—	4	8	no	—
151	C₂×C₄×C₁₀₉	—	4	4	no	—
152	C₂×C₄×C₁₀₉	—	8	8	no	—
153	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₁₀₉	#117	1	4	yes	—
157	C₂×C₄×C₁₀₉	#121	1	2	yes	—
158	C₂×C₄×C₁₀₉	#121	1	2	yes	—
159	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
160	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	—
161	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₀₉	—	2	4	no	—
168	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
169	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
170	C₂²×C₁₀₉	—	2	4	no	—
171	C₂²×C₁₀₉	—	2	2	no	—
172	C₂²×C₁₀₉	—	2	2	no	—
173	C₂²×C₁₀₉	—	2	4	no	—
174	C₂²×C₁₀₉	—	2	8	no	—
175	C₂²×C₁₀₉	—	2	4	no	—
176	C₂²×C₁₀₉	—	4	2	no	—
177	C₂²×C₁₀₉	—	4	2	no	—
178	C₂²×C₁₀₉	—	4	4	no	—
179	C₂²×C₁₀₉	#380	1	4	yes	—
180	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
182	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
183	C₂²×C₁₀₉	#576	1	2	yes	—
184	C₂²×C₁₀₉	#578	1	2	yes	—
185	C₄×C₁₀₉	—	2	4	no	—
186	C₄×C₁₀₉	—	8	4	no	—
187	C₄×C₁₀₉	—	8	4	no	—
188	C₄×C₁₀₉	#380	1	4	yes	—
189	C₄×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
190	C₄×C₁₀₉	C₂×#117	1	1	yes	characteristic
191	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	#20663	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₁₀₉	—	2	4	no	—
195	C₂×C₁₀₉	—	4	4	no	—
196	C₂×C₁₀₉	C₂×#380	1	1	yes	characteristic
197	C₂×C₁₀₉	#2913	1	2	yes	—
198	C₂×C₁₀₉	#3058	1	1	yes	characteristic
199	C₂×C₁₀₉	#3059	1	1	yes	characteristic
200	C₂×C₁₀₉	#3061	1	2	yes	—
201	C₂²×#131	C₂×C₁₀₉	1	1	yes	characteristic
202	C₂×#427	C₂×C₁₀₉	1	1	yes	characteristic, complemented
203	C₂²×#107	C₂×C₁₀₉	1	1	yes	characteristic, complemented
204	#2311	C₂×C₁₀₉	1	2	yes	—
205	#1809	C₂×C₁₀₉	1	4	yes	complemented
206	#1808	C₂×C₁₀₉	1	4	yes	complemented
207	#1736	C₂×C₁₀₉	1	2	yes	complemented
208	C₁₀₉	#20663	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
210	C₂×#131	—	2	4	no	—
211	C₂×#131	—	2	4	no	—
212	C₂×#131	C₄×C₁₀₉	1	8	yes	complemented
213	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
215	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
216	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
217	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
218	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
219	#427	—	2	4	no	—
220	C₂×#107	—	2	4	no	—
221	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
222	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
223	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
224	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
225	#391	—	2	2	no	—
226	#390	—	2	2	no	—
227	#386	—	2	4	no	—
228	#380	C₂²×C₁₀₉	1	8	yes	—
229	#340	C₂²×C₁₀₉	1	8	yes	—
230	C₂×#41	—	2	2	no	—
231	C₂×#41	—	2	2	no	—
232	C₂×#41	—	2	2	no	—
233	C₂×#41	—	2	2	no	—
234	C₂×#41	C₂×C₄×C₁₀₉	1	4	yes	—
235	C₂×#41	Q₈×C₁₀₉	1	4	yes	—
236	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
237	C₂²×D₄	D₄×C₁₀₉	1	2	yes	—
238	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
239	C₂³×C₄	—	2	2	no	—
240	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
241	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
242	#131	—	2	32	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	4	no	—
246	C₂×M₁₆	—	2	4	no	—
247	C₂×M₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
248	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	complemented
249	C₂²×C₈	—	2	4	no	—
250	C₂²×C₈	—	2	4	no	—
251	C₂²×C₈	C₂×C₄×C₁₀₉	1	4	yes	—
252	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
253	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
254	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
255	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
256	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
258	#117	—	2	4	no	—
259	#117	—	2	4	no	—
260	C₂×C₄⋊C₄	—	2	4	no	—
261	C₂×C₄⋊C₄	—	2	4	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
264	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	8	no	—
267	C₂×C₄²	—	2	2	no	—
268	#107	—	2	4	no	—
269	#107	—	2	4	no	—
270	#107	—	2	4	no	—
271	#106	—	2	4	no	—
272	#95	—	2	8	no	—
273	C₂⁴	C₂×D₄×C₁₀₉	1	2	yes	—
274	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
275	#41	—	2	8	no	—
276	#41	—	2	8	no	—
277	#41	—	2	8	no	—
278	#41	—	2	8	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	4	yes	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	4	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	C₄⋊C₄×C₁₀₉	1	8	yes	—
287	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	4	4	no	—
295	C₂²×C₄	—	8	4	no	—
296	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
297	C₂²×C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
298	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
299	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
300	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
301	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
302	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
303	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
304	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
305	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
306	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
307	M₁₆	—	2	8	no	—
308	M₁₆	—	2	16	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	16	no	—
312	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	16	yes	complemented
313	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
314	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
315	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
316	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
317	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
318	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
319	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	4	4	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₂²⋊C₄	—	8	8	no	—
327	C₄²	—	4	4	no	—
328	C₂³	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	4	no	—
333	C₂³	—	2	4	no	—
334	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
335	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
336	C₂³	#121×C₁₀₉	1	2	yes	—
337	C₂³	#121×C₁₀₉	1	2	yes	—
338	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
339	Q₈	—	2	8	no	—
340	D₄	—	2	16	no	—
341	D₄	—	2	8	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	4	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	8	8	no	—
354	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
355	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
356	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
357	C₂×C₄	#117×C₁₀₉	1	4	yes	—
358	C₂×C₄	#121×C₁₀₉	1	2	yes	—
359	C₂×C₄	#121×C₁₀₉	1	2	yes	—
360	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
361	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	—
362	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
363	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
365	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
366	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
368	C₈	—	2	4	no	—
369	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
370	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	8	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	4	no	—
380	C₂²	#380×C₁₀₉	1	4	yes	—
381	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
382	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
383	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
384	C₂²	#576×C₁₀₉	1	2	yes	—
385	C₂²	#578×C₁₀₉	1	2	yes	—
386	C₄	—	2	4	no	—
387	C₄	—	8	4	no	—
388	C₄	—	8	4	no	—
389	C₄	#380×C₁₀₉	1	4	yes	—
390	C₄	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
391	C₄	C₂×#117×C₁₀₉	1	1	yes	characteristic
392	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
393	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
394	C₂	—	2	4	no	—
395	C₂	—	4	4	no	—
396	C₂	C₂×#380×C₁₀₉	1	1	yes	characteristic
397	C₂	C₁₀₉×#2913	1	2	yes	—
398	C₂	C₁₀₉×#3058	1	1	yes	characteristic
399	C₂	C₁₀₉×#3059	1	1	yes	characteristic
400	C₂	C₁₀₉×#3061	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×#427×C₁₀₉	C₂	semidirect
3. C₂²×#107×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1809	C₂	semidirect
6. C₁₀₉×#1808	C₂	semidirect
7. C₁₀₉×#1736	C₂	semidirect
11. C₂×#131×C₁₀₉	C₄	semidirect
47. C₂×M₁₆×C₁₀₉	C₂×C₄	semidirect
51. C₂²×C₈×C₁₀₉	C₂×C₄	semidirect
111. C₂×C₈×C₁₀₉	C₂²⋊C₄	semidirect
193. #20663	C₁₀₉	direct (N × H)
202. C₂×#427	C₂×C₁₀₉	semidirect
203. C₂²×#107	C₂×C₁₀₉	semidirect
205. #1809	C₂×C₁₀₉	semidirect
206. #1808	C₂×C₁₀₉	semidirect
207. #1736	C₂×C₁₀₉	semidirect
208. C₁₀₉	#20663	direct (N × H)
212. C₂×#131	C₄×C₁₀₉	semidirect
248. C₂×M₁₆	C₂×C₄×C₁₀₉	semidirect
252. C₂²×C₈	C₂×C₄×C₁₀₉	semidirect
312. C₂×C₈	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×#427×C₁₀₉	2
C₂²×#107×C₁₀₉	2
C₁₀₉×#2311	2
C₁₀₉×#1809	2
C₁₀₉×#1808	2
C₁₀₉×#1736	2
#20663	109

Subgroups — C109×#20663 #42138657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20663 #42138657