Subgroups — C₁₀₉×#19353 #42137347

Order 27904.

776 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	C₁₀₉×#3217	C₂	1	2	yes	maximal, complemented
2	C₂×#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2966	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1688	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1687	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1668	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1667	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1555	C₂	1	2	yes	maximal, complemented
10	C₂²×D₈×C₁₀₉	—	2	1	no	—
11	C₂²×D₈×C₁₀₉	—	2	1	no	—
12	C₂×#127×C₁₀₉	C₂²	1	2	yes	—
13	#529×C₁₀₉	C₂²	1	1	yes	characteristic
14	#496×C₁₀₉	—	2	2	no	—
15	#493×C₁₀₉	—	2	1	no	—
16	#493×C₁₀₉	—	2	1	no	—
17	#493×C₁₀₉	C₂²	1	2	yes	—
18	#493×C₁₀₉	C₂²	1	2	yes	—
19	#490×C₁₀₉	—	2	2	no	—
20	#460×C₁₀₉	C₂²	1	4	yes	complemented
21	#459×C₁₀₉	C₂²	1	4	yes	complemented
22	#450×C₁₀₉	—	2	2	no	—
23	#447×C₁₀₉	—	2	2	no	—
24	#446×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
25	C₈×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	#436×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
27	#435×C₁₀₉	—	2	1	no	—
28	C₈×D₄×C₁₀₉	—	2	1	no	—
29	C₈×D₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
30	#433×C₁₀₉	—	2	1	no	—
31	#432×C₁₀₉	C₂²	1	2	yes	—
32	#432×C₁₀₉	C₂²	1	1	yes	characteristic
33	#405×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	#335×C₁₀₉	C₂²	1	2	yes	complemented
36	#334×C₁₀₉	C₂²	1	2	yes	complemented
37	#331×C₁₀₉	C₂²	1	4	yes	complemented
38	#331×C₁₀₉	C₂²	1	4	yes	complemented
39	C₂²×D₄×C₁₀₉	—	2	2	no	—
40	C₂²×D₄×C₁₀₉	—	2	2	no	—
41	C₂²×D₄×C₁₀₉	—	2	2	no	—
42	C₂×SD₁₆×C₁₀₉	—	2	4	no	—
43	C₂×D₈×C₁₀₉	—	2	4	no	—
44	C₂×D₈×C₁₀₉	—	2	2	no	—
45	C₂×D₈×C₁₀₉	—	2	2	no	—
46	C₂×D₈×C₁₀₉	—	2	2	no	—
47	C₂×D₈×C₁₀₉	—	2	2	no	—
48	C₂×D₈×C₁₀₉	—	4	4	no	—
49	C₂×D₈×C₁₀₉	—	4	4	no	—
50	C₂×M₁₆×C₁₀₉	—	2	1	no	—
51	C₂²×C₈×C₁₀₉	—	2	1	no	—
52	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
54	#127×C₁₀₉	—	2	2	no	—
55	#127×C₁₀₉	—	2	2	no	—
56	#127×C₁₀₉	—	2	2	no	—
57	#127×C₁₀₉	C₂³	1	4	yes	—
58	#126×C₁₀₉	—	2	1	no	—
59	#125×C₁₀₉	D₄	1	1	yes	characteristic
60	#124×C₁₀₉	D₄	1	1	yes	characteristic
61	#123×C₁₀₉	—	2	1	no	—
62	C₂²⋊Q₈×C₁₀₉	D₄	1	1	yes	characteristic
63	#121×C₁₀₉	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₁₀₉	—	2	1	no	—
66	C₄×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	#117×C₁₀₉	—	2	1	no	—
68	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₁₀₉	—	2	1	no	—
70	C₄⋊C₈×C₁₀₉	—	2	1	no	—
71	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₁₀₉	C₂³	1	4	yes	—
74	#102×C₁₀₉	—	2	4	no	—
75	#102×C₁₀₉	—	2	4	no	—
76	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
77	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
78	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
79	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
80	#97×C₁₀₉	—	2	1	no	—
81	#97×C₁₀₉	—	2	1	no	—
82	#97×C₁₀₉	—	2	2	no	—
83	C₄×C₈×C₁₀₉	—	2	1	no	—
84	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
85	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂⁴×C₁₀₉	—	4	2	no	—
89	C₂×Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₁₀₉	—	2	1	no	—
91	C₂×D₄×C₁₀₉	—	2	4	no	—
92	C₂×D₄×C₁₀₉	—	2	4	no	—
93	C₂×D₄×C₁₀₉	—	2	4	no	—
94	C₂×D₄×C₁₀₉	—	4	4	no	—
95	C₂×D₄×C₁₀₉	—	4	4	no	—
96	C₂×D₄×C₁₀₉	—	4	2	no	—
97	C₂×D₄×C₁₀₉	—	4	2	no	—
98	C₂×D₄×C₁₀₉	—	4	2	no	—
99	C₂×D₄×C₁₀₉	—	8	4	no	—
100	C₂×D₄×C₁₀₉	—	8	4	no	—
101	C₂×D₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₀₉	—	2	1	no	—
103	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
106	SD₁₆×C₁₀₉	—	8	4	no	—
107	D₈×C₁₀₉	—	4	4	no	—
108	D₈×C₁₀₉	—	4	4	no	—
109	D₈×C₁₀₉	—	4	4	no	—
110	D₈×C₁₀₉	—	4	4	no	—
111	D₈×C₁₀₉	—	8	4	no	—
112	M₁₆×C₁₀₉	—	4	2	no	—
113	C₂×C₈×C₁₀₉	—	2	2	no	—
114	C₂×C₈×C₁₀₉	—	2	2	no	—
115	C₂×C₈×C₁₀₉	—	2	1	no	—
116	C₂×C₈×C₁₀₉	—	2	1	no	—
117	C₂×C₈×C₁₀₉	—	2	4	no	—
118	C₂×C₈×C₁₀₉	—	4	2	no	—
119	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄×C₁₀₉	—	2	1	no	—
124	C₄⋊C₄×C₁₀₉	—	2	1	no	—
125	C₄⋊C₄×C₁₀₉	—	2	1	no	—
126	C₄⋊C₄×C₁₀₉	—	2	1	no	—
127	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
130	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
131	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
132	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
133	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
134	C₄²×C₁₀₉	—	2	1	no	—
135	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₁₀₉	#41	1	1	yes	characteristic
138	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
139	C₂³×C₁₀₉	—	2	1	no	—
140	C₂³×C₁₀₉	—	4	4	no	—
141	C₂³×C₁₀₉	—	8	4	no	—
142	C₂³×C₁₀₉	—	8	4	no	—
143	C₂³×C₁₀₉	—	8	4	no	—
144	C₂³×C₁₀₉	#121	1	1	yes	characteristic
145	Q₈×C₁₀₉	—	2	2	no	—
146	D₄×C₁₀₉	—	2	2	no	—
147	D₄×C₁₀₉	—	4	2	no	—
148	D₄×C₁₀₉	—	4	4	no	—
149	D₄×C₁₀₉	—	4	4	no	—
150	D₄×C₁₀₉	—	4	4	no	—
151	D₄×C₁₀₉	—	4	4	no	—
152	D₄×C₁₀₉	—	8	4	no	—
153	D₄×C₁₀₉	—	8	4	no	—
154	D₄×C₁₀₉	—	8	4	no	—
155	D₄×C₁₀₉	—	8	4	no	—
156	C₂×C₄×C₁₀₉	—	2	2	no	—
157	C₂×C₄×C₁₀₉	—	2	1	no	—
158	C₂×C₄×C₁₀₉	—	2	1	no	—
159	C₂×C₄×C₁₀₉	—	2	2	no	—
160	C₂×C₄×C₁₀₉	—	2	1	no	—
161	C₂×C₄×C₁₀₉	—	2	1	no	—
162	C₂×C₄×C₁₀₉	—	2	1	no	—
163	C₂×C₄×C₁₀₉	—	4	1	no	—
164	C₂×C₄×C₁₀₉	—	4	1	no	—
165	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
167	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
168	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
171	C₈×C₁₀₉	—	2	1	no	—
172	C₈×C₁₀₉	—	2	1	no	—
173	C₈×C₁₀₉	—	4	1	no	—
174	C₈×C₁₀₉	—	4	1	no	—
175	C₈×C₁₀₉	—	4	4	no	—
176	C₈×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
177	C₈×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
178	C₂²×C₁₀₉	—	2	1	no	—
179	C₂²×C₁₀₉	—	2	1	no	—
180	C₂²×C₁₀₉	—	4	1	no	—
181	C₂²×C₁₀₉	—	4	1	no	—
182	C₂²×C₁₀₉	—	4	1	no	—
183	C₂²×C₁₀₉	—	8	4	no	—
184	C₂²×C₁₀₉	—	8	4	no	—
185	C₂²×C₁₀₉	—	8	4	no	—
186	C₂²×C₁₀₉	—	16	8	no	—
187	C₂²×C₁₀₉	#472	1	2	yes	—
188	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
189	C₄×C₁₀₉	—	2	1	no	—
190	C₄×C₁₀₉	—	2	1	no	—
191	C₄×C₁₀₉	—	2	1	no	—
192	C₄×C₁₀₉	—	2	1	no	—
193	C₄×C₁₀₉	—	4	1	no	—
194	C₄×C₁₀₉	—	4	1	no	—
195	C₄×C₁₀₉	—	4	1	no	—
196	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₁₀₉	#576	1	2	yes	—
199	#19353	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₁₀₉	—	2	2	no	—
201	C₂×C₁₀₉	—	8	1	no	—
202	C₂×C₁₀₉	—	16	4	no	—
203	C₂×C₁₀₉	#3038	1	1	yes	characteristic
204	C₂×C₁₀₉	#3058	1	1	yes	characteristic
205	C₂×C₁₀₉	C₂×#472	1	1	yes	characteristic
206	#3217	C₂×C₁₀₉	1	2	yes	complemented
207	C₂×#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
208	#2966	C₂×C₁₀₉	1	1	yes	characteristic, complemented
209	#1688	C₂×C₁₀₉	1	2	yes	complemented
210	#1687	C₂×C₁₀₉	1	2	yes	complemented
211	#1668	C₂×C₁₀₉	1	2	yes	complemented
212	#1667	C₂×C₁₀₉	1	2	yes	complemented
213	#1568	C₂×C₁₀₉	1	1	yes	characteristic, complemented
214	#1555	C₂×C₁₀₉	1	2	yes	complemented
215	C₁₀₉	#19353	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×D₈	—	2	1	no	—
217	C₂²×D₈	—	2	1	no	—
218	C₂×#127	C₂²×C₁₀₉	1	2	yes	—
219	#529	C₂²×C₁₀₉	1	1	yes	characteristic
220	#496	—	2	2	no	—
221	#493	—	2	1	no	—
222	#493	—	2	1	no	—
223	#493	C₂²×C₁₀₉	1	2	yes	—
224	#493	C₂²×C₁₀₉	1	2	yes	—
225	#490	—	2	2	no	—
226	#460	C₂²×C₁₀₉	1	4	yes	complemented
227	#459	C₂²×C₁₀₉	1	4	yes	complemented
228	#450	—	2	2	no	—
229	#447	—	2	2	no	—
230	#446	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
231	C₈×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
232	#436	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
233	#435	—	2	1	no	—
234	C₈×D₄	—	2	1	no	—
235	C₈×D₄	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
236	#433	—	2	1	no	—
237	#432	C₂²×C₁₀₉	1	1	yes	characteristic
238	#432	C₂²×C₁₀₉	1	2	yes	—
239	#405	—	2	1	no	—
240	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
241	#335	C₂²×C₁₀₉	1	2	yes	complemented
242	#334	C₂²×C₁₀₉	1	2	yes	complemented
243	#331	C₂²×C₁₀₉	1	4	yes	complemented
244	#331	C₂²×C₁₀₉	1	4	yes	complemented
245	C₂²×D₄	—	2	2	no	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	—	2	2	no	—
248	C₂×SD₁₆	—	2	4	no	—
249	C₂×D₈	—	2	4	no	—
250	C₂×D₈	—	2	2	no	—
251	C₂×D₈	—	2	2	no	—
252	C₂×D₈	—	2	2	no	—
253	C₂×D₈	—	2	2	no	—
254	C₂×D₈	—	4	4	no	—
255	C₂×D₈	—	4	4	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
259	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
260	#127	—	2	2	no	—
261	#127	—	2	2	no	—
262	#127	—	2	2	no	—
263	#127	C₂³×C₁₀₉	1	4	yes	—
264	#126	—	2	1	no	—
265	#125	D₄×C₁₀₉	1	1	yes	characteristic
266	#124	D₄×C₁₀₉	1	1	yes	characteristic
267	#123	—	2	1	no	—
268	C₂²⋊Q₈	D₄×C₁₀₉	1	1	yes	characteristic
269	#121	D₄×C₁₀₉	1	1	yes	characteristic
270	C₄×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
271	C₄×D₄	—	2	1	no	—
272	C₄×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
273	#117	—	2	1	no	—
274	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
278	C₄⋊C₈	C₂³×C₁₀₉	1	1	yes	characteristic
279	C₄⋊C₈	C₂³×C₁₀₉	1	4	yes	—
280	#102	—	2	4	no	—
281	#102	—	2	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	2	1	no	—
285	C₂²⋊C₈	—	2	1	no	—
286	#97	—	2	1	no	—
287	#97	—	2	1	no	—
288	#97	—	2	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
291	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
292	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
293	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
294	C₂⁴	—	4	2	no	—
295	C₂×Q₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	2	4	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	4	4	no	—
301	C₂×D₄	—	4	4	no	—
302	C₂×D₄	—	4	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	8	4	no	—
306	C₂×D₄	—	8	4	no	—
307	C₂×D₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
311	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
312	SD₁₆	—	8	4	no	—
313	D₈	—	4	4	no	—
314	D₈	—	4	4	no	—
315	D₈	—	4	4	no	—
316	D₈	—	4	4	no	—
317	D₈	—	8	4	no	—
318	M₁₆	—	4	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	4	2	no	—
325	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
334	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
335	C₂²⋊C₄	—	2	1	no	—
336	C₂²⋊C₄	—	2	1	no	—
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	2	1	no	—
339	C₂²⋊C₄	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
342	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
343	C₄²	#41×C₁₀₉	1	1	yes	characteristic
344	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	4	4	no	—
347	C₂³	—	8	4	no	—
348	C₂³	—	8	4	no	—
349	C₂³	—	8	4	no	—
350	C₂³	#121×C₁₀₉	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	D₄	—	2	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	4	no	—
355	D₄	—	4	4	no	—
356	D₄	—	4	4	no	—
357	D₄	—	4	4	no	—
358	D₄	—	8	4	no	—
359	D₄	—	8	4	no	—
360	D₄	—	8	4	no	—
361	D₄	—	8	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
372	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
373	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
375	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	4	no	—
382	C₈	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
383	C₈	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	8	4	no	—
390	C₂²	—	8	4	no	—
391	C₂²	—	8	4	no	—
392	C₂²	—	16	8	no	—
393	C₂²	#472×C₁₀₉	1	2	yes	—
394	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
403	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
404	C₄	#576×C₁₀₉	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	8	1	no	—
407	C₂	—	16	4	no	—
408	C₂	C₁₀₉×#3038	1	1	yes	characteristic
409	C₂	C₁₀₉×#3058	1	1	yes	characteristic
410	C₂	C₂×#472×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×#3217	C₂	semidirect
2. C₂×#493×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2966	C₂	semidirect
4. C₁₀₉×#1688	C₂	semidirect
5. C₁₀₉×#1687	C₂	semidirect
6. C₁₀₉×#1668	C₂	semidirect
7. C₁₀₉×#1667	C₂	semidirect
8. C₁₀₉×#1568	C₂	semidirect
9. C₁₀₉×#1555	C₂	semidirect
20. #460×C₁₀₉	C₂²	semidirect
21. #459×C₁₀₉	C₂²	semidirect
24. #446×C₁₀₉	C₂²	semidirect
25. C₈×Q₈×C₁₀₉	C₂²	semidirect
26. #436×C₁₀₉	C₂²	semidirect
29. C₈×D₄×C₁₀₉	C₂²	semidirect
35. #335×C₁₀₉	C₂²	semidirect
36. #334×C₁₀₉	C₂²	semidirect
37. #331×C₁₀₉	C₂²	semidirect
38. #331×C₁₀₉	C₂²	semidirect
199. #19353	C₁₀₉	direct (N × H)
206. #3217	C₂×C₁₀₉	semidirect
207. C₂×#493	C₂×C₁₀₉	semidirect
208. #2966	C₂×C₁₀₉	semidirect
209. #1688	C₂×C₁₀₉	semidirect
210. #1687	C₂×C₁₀₉	semidirect
211. #1668	C₂×C₁₀₉	semidirect
212. #1667	C₂×C₁₀₉	semidirect
213. #1568	C₂×C₁₀₉	semidirect
214. #1555	C₂×C₁₀₉	semidirect
215. C₁₀₉	#19353	direct (N × H)
226. #460	C₂²×C₁₀₉	semidirect
227. #459	C₂²×C₁₀₉	semidirect
230. #446	C₂²×C₁₀₉	semidirect
231. C₈×Q₈	C₂²×C₁₀₉	semidirect
232. #436	C₂²×C₁₀₉	semidirect
235. C₈×D₄	C₂²×C₁₀₉	semidirect
241. #335	C₂²×C₁₀₉	semidirect
242. #334	C₂²×C₁₀₉	semidirect
243. #331	C₂²×C₁₀₉	semidirect
244. #331	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#3217	2
C₂×#493×C₁₀₉	2
C₁₀₉×#2966	2
C₁₀₉×#1688	2
C₁₀₉×#1687	2
C₁₀₉×#1668	2
C₁₀₉×#1667	2
C₁₀₉×#1568	2
C₁₀₉×#1555	2
#19353	109

Subgroups — C109×#19353 #42137347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19353 #42137347