Subgroups — C₁₀₉×#19301 #42137295

Order 27904.

732 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#3039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#3038	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#2965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1702	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1701	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₁₀₉	—	2	1	no	—
11	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×#41×C₁₀₉	C₂²	1	1	yes	characteristic
15	#486×C₁₀₉	—	2	1	no	—
16	#486×C₁₀₉	—	2	1	no	—
17	#486×C₁₀₉	C₂²	1	4	yes	—
18	#462×C₁₀₉	—	2	1	no	—
19	#462×C₁₀₉	C₂²	1	2	yes	complemented
20	#461×C₁₀₉	—	2	1	no	—
21	#461×C₁₀₉	C₂²	1	2	yes	complemented
22	#460×C₁₀₉	—	2	1	no	—
23	#460×C₁₀₉	C₂²	1	2	yes	—
24	#459×C₁₀₉	—	2	1	no	—
25	#459×C₁₀₉	C₂²	1	2	yes	—
26	#446×C₁₀₉	C₂²	1	2	yes	complemented
27	#436×C₁₀₉	C₂²	1	2	yes	complemented
28	#435×C₁₀₉	—	2	2	no	—
29	#431×C₁₀₉	—	2	1	no	—
30	#431×C₁₀₉	C₂²	1	1	yes	characteristic
31	#423×C₁₀₉	C₂²	1	1	yes	characteristic
32	#423×C₁₀₉	C₂²	1	1	yes	characteristic
33	#419×C₁₀₉	—	2	1	no	—
34	#418×C₁₀₉	—	2	1	no	—
35	C₂×#102×C₁₀₉	—	2	2	no	—
36	C₄×M₁₆×C₁₀₉	—	2	1	no	—
37	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	#334×C₁₀₉	C₂²	1	4	yes	complemented
39	#332×C₁₀₉	C₂²	1	4	yes	complemented
40	#331×C₁₀₉	C₂²	1	4	yes	—
41	C₂×#41×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₁₀₉	—	2	1	no	—
43	C₂²×D₄×C₁₀₉	—	2	1	no	—
44	C₂²×D₄×C₁₀₉	—	2	1	no	—
45	C₂²×D₄×C₁₀₉	—	2	1	no	—
46	#137×C₁₀₉	—	4	4	no	—
47	AGL(1,ℤ/8)×C₁₀₉	—	4	4	no	—
48	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
49	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
50	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
51	C₂×D₈×C₁₀₉	—	2	2	no	—
52	C₂×M₁₆×C₁₀₉	—	2	1	no	—
53	C₂×M₁₆×C₁₀₉	—	2	1	no	—
54	C₂×M₁₆×C₁₀₉	—	2	2	no	—
55	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
56	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
57	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
58	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
59	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
60	#127×C₁₀₉	—	2	1	no	—
61	#127×C₁₀₉	—	2	1	no	—
62	#127×C₁₀₉	—	2	1	no	—
63	#127×C₁₀₉	C₂³	1	2	yes	—
64	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₁₀₉	—	2	1	no	—
67	C₄×D₄×C₁₀₉	—	2	1	no	—
68	C₄×D₄×C₁₀₉	D₄	1	1	yes	characteristic
69	C₄×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	#117×C₁₀₉	—	2	1	no	—
71	#117×C₁₀₉	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂×C₄²×C₁₀₉	—	2	1	no	—
75	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₁₀₉	—	2	2	no	—
77	C₄⋊C₈×C₁₀₉	—	2	1	no	—
78	C₄⋊C₈×C₁₀₉	—	2	1	no	—
79	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
80	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
81	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
82	#103×C₁₀₉	—	2	2	no	—
83	#103×C₁₀₉	—	2	2	no	—
84	#102×C₁₀₉	—	2	4	no	—
85	#102×C₁₀₉	—	2	4	no	—
86	#102×C₁₀₉	—	2	2	no	—
87	#102×C₁₀₉	—	2	2	no	—
88	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
89	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
90	#97×C₁₀₉	—	2	2	no	—
91	C₄×C₈×C₁₀₉	—	2	1	no	—
92	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
93	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
94	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
95	C₂⁴×C₁₀₉	—	4	1	no	—
96	#41×C₁₀₉	—	2	2	no	—
97	#41×C₁₀₉	—	2	2	no	—
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	—	4	1	no	—
100	C₂×Q₈×C₁₀₉	—	4	2	no	—
101	C₂×Q₈×C₁₀₉	—	4	2	no	—
102	C₂×Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₁₀₉	—	2	1	no	—
104	C₂×D₄×C₁₀₉	—	2	2	no	—
105	C₂×D₄×C₁₀₉	—	2	2	no	—
106	C₂×D₄×C₁₀₉	—	2	2	no	—
107	C₂×D₄×C₁₀₉	—	4	4	no	—
108	C₂×D₄×C₁₀₉	—	4	1	no	—
109	C₂×D₄×C₁₀₉	—	4	1	no	—
110	C₂×D₄×C₁₀₉	—	4	1	no	—
111	C₂×D₄×C₁₀₉	—	4	2	no	—
112	C₂×D₄×C₁₀₉	—	4	2	no	—
113	C₂×D₄×C₁₀₉	—	8	2	no	—
114	C₂×D₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₀₉	—	2	1	no	—
116	C₂²×C₄×C₁₀₉	—	2	1	no	—
117	C₂²×C₄×C₁₀₉	—	2	1	no	—
118	C₂²×C₄×C₁₀₉	—	4	1	no	—
119	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
122	Q₁₆×C₁₀₉	—	4	4	no	—
123	SD₁₆×C₁₀₉	—	4	4	no	—
124	SD₁₆×C₁₀₉	—	4	4	no	—
125	D₈×C₁₀₉	—	4	4	no	—
126	M₁₆×C₁₀₉	—	4	4	no	—
127	M₁₆×C₁₀₉	—	4	2	no	—
128	M₁₆×C₁₀₉	—	4	2	no	—
129	C₂×C₈×C₁₀₉	—	2	2	no	—
130	C₂×C₈×C₁₀₉	—	2	2	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	—	4	2	no	—
133	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₁₀₉	—	2	1	no	—
136	C₄⋊C₄×C₁₀₉	—	2	1	no	—
137	C₄⋊C₄×C₁₀₉	—	2	2	no	—
138	C₄⋊C₄×C₁₀₉	—	2	2	no	—
139	C₄⋊C₄×C₁₀₉	—	4	1	no	—
140	C₄⋊C₄×C₁₀₉	—	4	1	no	—
141	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
144	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
145	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
146	C₄²×C₁₀₉	—	2	1	no	—
147	C₄²×C₁₀₉	—	2	1	no	—
148	C₄²×C₁₀₉	—	2	1	no	—
149	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₁₀₉	#41	1	1	yes	characteristic
152	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
153	C₂³×C₁₀₉	—	2	1	no	—
154	C₂³×C₁₀₉	—	4	2	no	—
155	C₂³×C₁₀₉	—	8	2	no	—
156	C₂³×C₁₀₉	—	8	2	no	—
157	C₂³×C₁₀₉	—	8	2	no	—
158	C₂³×C₁₀₉	#121	1	1	yes	characteristic
159	Q₈×C₁₀₉	—	2	1	no	—
160	Q₈×C₁₀₉	—	2	1	no	—
161	Q₈×C₁₀₉	—	4	2	no	—
162	Q₈×C₁₀₉	—	4	2	no	—
163	Q₈×C₁₀₉	—	8	2	no	—
164	D₄×C₁₀₉	—	2	1	no	—
165	D₄×C₁₀₉	—	2	1	no	—
166	D₄×C₁₀₉	—	4	2	no	—
167	D₄×C₁₀₉	—	4	2	no	—
168	D₄×C₁₀₉	—	4	2	no	—
169	D₄×C₁₀₉	—	4	4	no	—
170	D₄×C₁₀₉	—	8	2	no	—
171	D₄×C₁₀₉	—	8	2	no	—
172	D₄×C₁₀₉	—	8	2	no	—
173	D₄×C₁₀₉	—	8	2	no	—
174	C₂×C₄×C₁₀₉	—	2	2	no	—
175	C₂×C₄×C₁₀₉	—	2	2	no	—
176	C₂×C₄×C₁₀₉	—	2	1	no	—
177	C₂×C₄×C₁₀₉	—	2	1	no	—
178	C₂×C₄×C₁₀₉	—	2	1	no	—
179	C₂×C₄×C₁₀₉	—	4	2	no	—
180	C₂×C₄×C₁₀₉	—	4	2	no	—
181	C₂×C₄×C₁₀₉	—	4	2	no	—
182	C₂×C₄×C₁₀₉	—	4	2	no	—
183	C₂×C₄×C₁₀₉	—	8	2	no	—
184	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
185	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
186	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
187	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
192	C₈×C₁₀₉	—	2	2	no	—
193	C₈×C₁₀₉	—	2	2	no	—
194	C₈×C₁₀₉	—	4	2	no	—
195	C₈×C₁₀₉	—	4	2	no	—
196	C₈×C₁₀₉	—	4	2	no	—
197	C₂²×C₁₀₉	—	2	1	no	—
198	C₂²×C₁₀₉	—	2	1	no	—
199	C₂²×C₁₀₉	—	4	1	no	—
200	C₂²×C₁₀₉	—	4	1	no	—
201	C₂²×C₁₀₉	—	4	1	no	—
202	C₂²×C₁₀₉	—	8	2	no	—
203	C₂²×C₁₀₉	—	8	2	no	—
204	C₂²×C₁₀₉	—	8	2	no	—
205	C₂²×C₁₀₉	—	16	4	no	—
206	C₂²×C₁₀₉	#469	1	2	yes	—
207	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
208	C₄×C₁₀₉	—	2	1	no	—
209	C₄×C₁₀₉	—	2	1	no	—
210	C₄×C₁₀₉	—	4	1	no	—
211	C₄×C₁₀₉	—	4	1	no	—
212	C₄×C₁₀₉	—	4	1	no	—
213	C₄×C₁₀₉	—	8	2	no	—
214	C₄×C₁₀₉	#469	1	2	yes	—
215	C₄×C₁₀₉	#471	1	2	yes	—
216	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₄×C₁₀₉	#575	1	2	yes	—
219	#19301	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₁₀₉	—	2	2	no	—
221	C₂×C₁₀₉	—	8	1	no	—
222	C₂×C₁₀₉	—	16	2	no	—
223	C₂×C₁₀₉	#3040	1	1	yes	characteristic
224	C₂×C₁₀₉	C₂×#469	1	1	yes	characteristic
225	C₂×C₁₀₉	C₂×#471	1	1	yes	characteristic
226	C₂×#486	C₂×C₁₀₉	1	1	yes	characteristic, complemented
227	#3039	C₂×C₁₀₉	1	1	yes	characteristic, complemented
228	#3038	C₂×C₁₀₉	1	1	yes	characteristic
229	#2965	C₂×C₁₀₉	1	1	yes	characteristic, complemented
230	#1702	C₂×C₁₀₉	1	4	yes	complemented
231	#1701	C₂×C₁₀₉	1	4	yes	complemented
232	#1566	C₂×C₁₀₉	1	1	yes	characteristic, complemented
233	#1546	C₂×C₁₀₉	1	1	yes	characteristic, complemented
234	#1545	C₂×C₁₀₉	1	1	yes	characteristic, complemented
235	C₁₀₉	#19301	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×#137	—	2	1	no	—
237	C₂×AGL(1,ℤ/8)	—	2	1	no	—
238	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
239	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
240	C₄×#41	C₂²×C₁₀₉	1	1	yes	characteristic
241	#486	—	2	1	no	—
242	#486	—	2	1	no	—
243	#486	C₂²×C₁₀₉	1	4	yes	—
244	#462	—	2	1	no	—
245	#462	C₂²×C₁₀₉	1	2	yes	complemented
246	#461	—	2	1	no	—
247	#461	C₂²×C₁₀₉	1	2	yes	complemented
248	#460	—	2	1	no	—
249	#460	C₂²×C₁₀₉	1	2	yes	—
250	#459	—	2	1	no	—
251	#459	C₂²×C₁₀₉	1	2	yes	—
252	#446	C₂²×C₁₀₉	1	2	yes	complemented
253	#436	C₂²×C₁₀₉	1	2	yes	complemented
254	#435	—	2	2	no	—
255	#431	—	2	1	no	—
256	#431	C₂²×C₁₀₉	1	1	yes	characteristic
257	#423	C₂²×C₁₀₉	1	1	yes	characteristic
258	#423	C₂²×C₁₀₉	1	1	yes	characteristic
259	#419	—	2	1	no	—
260	#418	—	2	1	no	—
261	C₂×#102	—	2	2	no	—
262	C₄×M₁₆	—	2	1	no	—
263	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
264	#334	C₂²×C₁₀₉	1	4	yes	complemented
265	#332	C₂²×C₁₀₉	1	4	yes	complemented
266	#331	C₂²×C₁₀₉	1	4	yes	—
267	C₂×#41	D₄×C₁₀₉	1	1	yes	characteristic
268	C₂²×Q₈	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂²×D₄	—	2	1	no	—
272	#137	—	4	4	no	—
273	AGL(1,ℤ/8)	—	4	4	no	—
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×SD₁₆	—	2	2	no	—
277	C₂×D₈	—	2	2	no	—
278	C₂×M₁₆	—	2	1	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
282	C₄⋊Q₈	—	2	1	no	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
286	#127	—	2	1	no	—
287	#127	—	2	1	no	—
288	#127	—	2	1	no	—
289	#127	C₂³×C₁₀₉	1	2	yes	—
290	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
291	C₄×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
292	C₄×D₄	—	2	1	no	—
293	C₄×D₄	—	2	1	no	—
294	C₄×D₄	D₄×C₁₀₉	1	1	yes	characteristic
295	C₄×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
296	#117	—	2	1	no	—
297	#117	D₄×C₁₀₉	1	1	yes	characteristic
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
302	C₄⋊C₈	—	2	2	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
306	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
307	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#102	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	—	2	2	no	—
316	#97	—	2	2	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
319	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
320	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
321	C₂⁴	—	4	1	no	—
322	#41	—	2	2	no	—
323	#41	—	2	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	4	1	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	1	no	—
336	C₂×D₄	—	4	1	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	2	no	—
339	C₂×D₄	—	8	2	no	—
340	C₂×D₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
347	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
348	Q₁₆	—	4	4	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	—	4	4	no	—
351	D₈	—	4	4	no	—
352	M₁₆	—	4	4	no	—
353	M₁₆	—	4	2	no	—
354	M₁₆	—	4	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	2	no	—
359	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	4	1	no	—
366	C₄⋊C₄	—	4	1	no	—
367	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
368	C₄⋊C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	1	no	—
375	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
376	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
377	C₄²	#41×C₁₀₉	1	1	yes	characteristic
378	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
379	C₂³	—	2	1	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	8	2	no	—
382	C₂³	—	8	2	no	—
383	C₂³	—	8	2	no	—
384	C₂³	#121×C₁₀₉	1	1	yes	characteristic
385	Q₈	—	2	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	4	2	no	—
389	Q₈	—	8	2	no	—
390	D₄	—	2	1	no	—
391	D₄	—	2	1	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	4	no	—
396	D₄	—	8	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	8	2	no	—
399	D₄	—	8	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
411	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
412	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
413	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
414	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₈	—	4	2	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	16	4	no	—
432	C₂²	#469×C₁₀₉	1	2	yes	—
433	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	8	2	no	—
440	C₄	#469×C₁₀₉	1	2	yes	—
441	C₄	#471×C₁₀₉	1	2	yes	—
442	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
443	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
444	C₄	#575×C₁₀₉	1	2	yes	—
445	C₂	—	2	2	no	—
446	C₂	—	8	1	no	—
447	C₂	—	16	2	no	—
448	C₂	C₁₀₉×#3040	1	1	yes	characteristic
449	C₂	C₂×#469×C₁₀₉	1	1	yes	characteristic
450	C₂	C₂×#471×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
prime radical	C₁₀₉×#3038	C₂
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. C₂×#486×C₁₀₉	C₂	semidirect
2. C₁₀₉×#3039	C₂	semidirect
4. C₁₀₉×#2965	C₂	semidirect
5. C₁₀₉×#1702	C₂	semidirect
6. C₁₀₉×#1701	C₂	semidirect
7. C₁₀₉×#1566	C₂	semidirect
8. C₁₀₉×#1546	C₂	semidirect
9. C₁₀₉×#1545	C₂	semidirect
19. #462×C₁₀₉	C₂²	semidirect
21. #461×C₁₀₉	C₂²	semidirect
26. #446×C₁₀₉	C₂²	semidirect
27. #436×C₁₀₉	C₂²	semidirect
38. #334×C₁₀₉	C₂²	semidirect
39. #332×C₁₀₉	C₂²	semidirect
219. #19301	C₁₀₉	direct (N × H)
226. C₂×#486	C₂×C₁₀₉	semidirect
227. #3039	C₂×C₁₀₉	semidirect
229. #2965	C₂×C₁₀₉	semidirect
230. #1702	C₂×C₁₀₉	semidirect
231. #1701	C₂×C₁₀₉	semidirect
232. #1566	C₂×C₁₀₉	semidirect
233. #1546	C₂×C₁₀₉	semidirect
234. #1545	C₂×C₁₀₉	semidirect
235. C₁₀₉	#19301	direct (N × H)
245. #462	C₂²×C₁₀₉	semidirect
247. #461	C₂²×C₁₀₉	semidirect
252. #446	C₂²×C₁₀₉	semidirect
253. #436	C₂²×C₁₀₉	semidirect
264. #334	C₂²×C₁₀₉	semidirect
265. #332	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₁₀₉	2
C₁₀₉×#3039	2
C₁₀₉×#3038	2
C₁₀₉×#2965	2
C₁₀₉×#1702	2
C₁₀₉×#1701	2
C₁₀₉×#1566	2
C₁₀₉×#1546	2
C₁₀₉×#1545	2
#19301	109

Subgroups — C109×#19301 #42137295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#19301 #42137295