Subgroups — C₁₀₉×#3555 #28360809

Order 13952.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563×C₁₀₉	C₂	1	4	yes	maximal, complemented
2	Q₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#557×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#556×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#556×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#556×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#552×C₁₀₉	C₂	1	2	yes	maximal, complemented
10	#544×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	#542×C₁₀₉	C₂	1	2	yes	maximal
12	#541×C₁₀₉	C₂	1	2	yes	maximal, complemented
13	#541×C₁₀₉	C₂	1	2	yes	maximal, complemented
14	#541×C₁₀₉	C₂	1	4	yes	maximal, complemented
15	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
17	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
24	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
25	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
27	#126×C₁₀₉	C₂²	1	2	yes	—
28	#126×C₁₀₉	C₂²	1	4	yes	—
29	#126×C₁₀₉	C₂²	1	4	yes	—
30	#126×C₁₀₉	C₂²	1	4	yes	—
31	#126×C₁₀₉	C₂²	1	4	yes	—
32	#126×C₁₀₉	C₂²	1	4	yes	—
33	#126×C₁₀₉	C₂²	1	4	yes	—
34	#125×C₁₀₉	C₂²	1	1	yes	characteristic
35	#125×C₁₀₉	C₂²	1	2	yes	—
36	#125×C₁₀₉	C₂²	1	2	yes	—
37	#125×C₁₀₉	C₂²	1	4	yes	—
38	#125×C₁₀₉	C₂²	1	4	yes	—
39	#125×C₁₀₉	C₂²	1	4	yes	—
40	#125×C₁₀₉	C₂²	1	2	yes	—
41	#125×C₁₀₉	C₂²	1	4	yes	—
42	#125×C₁₀₉	C₂²	1	4	yes	—
43	#125×C₁₀₉	C₂²	1	2	yes	—
44	#124×C₁₀₉	C₂²	1	1	yes	characteristic
45	#124×C₁₀₉	C₂²	1	2	yes	—
46	#124×C₁₀₉	C₂²	1	4	yes	—
47	#123×C₁₀₉	C₂²	1	2	yes	—
48	#123×C₁₀₉	C₂²	1	4	yes	—
49	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
50	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
51	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
56	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
57	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
58	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
59	C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
60	#121×C₁₀₉	C₂²	1	1	yes	characteristic
61	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
62	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
63	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
64	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
65	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
66	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
67	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
68	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
69	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
70	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
71	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
72	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
73	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
74	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
75	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
76	#117×C₁₀₉	C₂²	1	2	yes	—
77	#117×C₁₀₉	C₂²	1	2	yes	—
78	#117×C₁₀₉	C₂²	1	2	yes	—
79	#117×C₁₀₉	C₂²	1	4	yes	—
80	#117×C₁₀₉	C₂²	1	2	yes	—
81	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
82	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
83	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
87	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
88	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
89	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
90	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
95	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
97	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
99	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
100	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
104	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
105	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
107	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
108	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
110	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
111	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
113	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
114	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
115	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
116	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
117	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
118	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
119	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
120	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
121	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
122	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
123	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
124	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
125	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
126	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
137	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
138	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
139	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
140	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
141	C₄²×C₁₀₉	C₂³	1	4	yes	—
142	C₄²×C₁₀₉	C₂³	1	2	yes	—
143	C₄²×C₁₀₉	C₂³	1	2	yes	—
144	C₄²×C₁₀₉	C₂³	1	2	yes	—
145	C₄²×C₁₀₉	C₂³	1	4	yes	—
146	C₄²×C₁₀₉	C₂³	1	4	yes	—
147	C₄²×C₁₀₉	C₂³	1	4	yes	—
148	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
149	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
150	Q₈×C₁₀₉	—	2	2	no	—
151	Q₈×C₁₀₉	—	2	2	no	—
152	Q₈×C₁₀₉	—	2	2	no	—
153	Q₈×C₁₀₉	—	2	4	no	—
154	Q₈×C₁₀₉	—	4	2	no	—
155	Q₈×C₁₀₉	—	4	2	no	—
156	Q₈×C₁₀₉	—	4	4	no	—
157	D₄×C₁₀₉	—	2	2	no	—
158	D₄×C₁₀₉	—	4	2	no	—
159	C₂×C₄×C₁₀₉	—	2	2	no	—
160	C₂×C₄×C₁₀₉	—	2	4	no	—
161	C₂×C₄×C₁₀₉	—	2	2	no	—
162	C₂×C₄×C₁₀₉	—	4	4	no	—
163	C₂×C₄×C₁₀₉	—	4	1	no	—
164	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
165	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
166	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
168	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
169	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
170	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
171	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
172	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
173	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
175	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
176	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
177	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
179	C₂²×C₁₀₉	—	2	1	no	—
180	C₂²×C₁₀₉	—	2	1	no	—
181	C₂²×C₁₀₉	—	2	2	no	—
182	C₂²×C₁₀₉	—	2	2	no	—
183	C₂²×C₁₀₉	—	2	2	no	—
184	C₂²×C₁₀₉	—	2	1	no	—
185	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
186	C₄×C₁₀₉	—	2	1	no	—
187	C₄×C₁₀₉	—	2	2	no	—
188	C₄×C₁₀₉	—	2	1	no	—
189	C₄×C₁₀₉	—	2	2	no	—
190	C₄×C₁₀₉	—	2	2	no	—
191	C₄×C₁₀₉	—	2	4	no	—
192	C₄×C₁₀₉	—	2	2	no	—
193	C₄×C₁₀₉	—	2	2	no	—
194	C₄×C₁₀₉	—	2	4	no	—
195	C₄×C₁₀₉	—	2	1	no	—
196	C₄×C₁₀₉	—	2	1	no	—
197	C₄×C₁₀₉	—	2	2	no	—
198	C₄×C₁₀₉	—	2	2	no	—
199	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
200	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
201	C₂×C₁₀₉	—	4	1	no	—
202	C₂×C₁₀₉	—	4	2	no	—
203	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×C₁₀₉	#585	1	1	yes	characteristic
206	#3555	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₁₀₉	#3555	1	1	yes	characteristic, direct product factor, Sylow
208	#563	C₂×C₁₀₉	1	4	yes	complemented
209	Q₈²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
210	#557	C₂×C₁₀₉	1	1	yes	characteristic, complemented
211	#557	C₂×C₁₀₉	1	2	yes	complemented
212	#556	C₂×C₁₀₉	1	2	yes	complemented
213	#556	C₂×C₁₀₉	1	2	yes	complemented
214	#556	C₂×C₁₀₉	1	1	yes	characteristic, complemented
215	#554	C₂×C₁₀₉	1	1	yes	characteristic, complemented
216	#552	C₂×C₁₀₉	1	2	yes	complemented
217	#544	C₂×C₁₀₉	1	2	yes	complemented
218	#542	C₂×C₁₀₉	1	2	yes	—
219	#541	C₂×C₁₀₉	1	2	yes	complemented
220	#541	C₂×C₁₀₉	1	2	yes	complemented
221	#541	C₂×C₁₀₉	1	4	yes	complemented
222	#533	C₂×C₁₀₉	1	1	yes	characteristic, complemented
223	#533	C₂×C₁₀₉	1	1	yes	characteristic, complemented
224	#529	C₂×C₁₀₉	1	1	yes	characteristic
225	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
226	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
227	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
228	C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
229	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
230	C₄⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
231	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
232	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
233	C₄⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
234	#126	C₂²×C₁₀₉	1	2	yes	—
235	#126	C₂²×C₁₀₉	1	4	yes	—
236	#126	C₂²×C₁₀₉	1	4	yes	—
237	#126	C₂²×C₁₀₉	1	4	yes	—
238	#126	C₂²×C₁₀₉	1	4	yes	—
239	#126	C₂²×C₁₀₉	1	4	yes	—
240	#126	C₂²×C₁₀₉	1	4	yes	—
241	#125	C₂²×C₁₀₉	1	1	yes	characteristic
242	#125	C₂²×C₁₀₉	1	2	yes	—
243	#125	C₂²×C₁₀₉	1	2	yes	—
244	#125	C₂²×C₁₀₉	1	4	yes	—
245	#125	C₂²×C₁₀₉	1	4	yes	—
246	#125	C₂²×C₁₀₉	1	4	yes	—
247	#125	C₂²×C₁₀₉	1	2	yes	—
248	#125	C₂²×C₁₀₉	1	4	yes	—
249	#125	C₂²×C₁₀₉	1	4	yes	—
250	#125	C₂²×C₁₀₉	1	2	yes	—
251	#124	C₂²×C₁₀₉	1	1	yes	characteristic
252	#124	C₂²×C₁₀₉	1	2	yes	—
253	#124	C₂²×C₁₀₉	1	4	yes	—
254	#123	C₂²×C₁₀₉	1	2	yes	—
255	#123	C₂²×C₁₀₉	1	4	yes	—
256	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
257	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
258	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
259	C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
260	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
261	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
262	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
263	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
264	C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
265	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
266	C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
267	#121	C₂²×C₁₀₉	1	1	yes	characteristic
268	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
269	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
270	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
271	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
272	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
273	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
274	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
275	C₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
276	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
277	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
278	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
279	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
280	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
281	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
282	C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
283	#117	C₂²×C₁₀₉	1	2	yes	—
284	#117	C₂²×C₁₀₉	1	2	yes	—
285	#117	C₂²×C₁₀₉	1	2	yes	—
286	#117	C₂²×C₁₀₉	1	4	yes	—
287	#117	C₂²×C₁₀₉	1	2	yes	—
288	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
289	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
290	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
291	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
292	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
293	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
294	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
295	C₂×Q₈	C₂³×C₁₀₉	1	2	yes	—
296	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
297	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
298	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
299	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
300	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
301	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
302	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
303	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
304	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
305	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
306	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
307	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
308	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
309	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
310	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
311	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
312	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
313	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
314	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
315	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
316	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
317	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
318	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
319	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
320	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
321	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
322	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
323	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
324	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
325	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
326	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
327	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
328	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
329	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
330	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
331	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
332	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
333	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
334	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
335	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
336	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
337	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
338	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
339	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
340	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
341	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
342	C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
343	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
344	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
345	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
346	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
347	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
348	C₄²	C₂³×C₁₀₉	1	4	yes	—
349	C₄²	C₂³×C₁₀₉	1	2	yes	—
350	C₄²	C₂³×C₁₀₉	1	2	yes	—
351	C₄²	C₂³×C₁₀₉	1	2	yes	—
352	C₄²	C₂³×C₁₀₉	1	4	yes	—
353	C₄²	C₂³×C₁₀₉	1	4	yes	—
354	C₄²	C₂³×C₁₀₉	1	4	yes	—
355	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
356	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
357	Q₈	—	2	2	no	—
358	Q₈	—	2	2	no	—
359	Q₈	—	2	2	no	—
360	Q₈	—	2	4	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	4	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
372	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
373	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
374	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
375	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
376	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
377	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
378	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
379	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
380	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
381	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
382	C₂×C₄	C₂⁴×C₁₀₉	1	4	yes	—
383	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
384	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
385	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	1	no	—
392	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	4	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
407	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	4	2	no	—
410	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
411	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
412	C₂	#585×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	#121×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #563×C₁₀₉	C₂	semidirect
2. Q₈²×C₁₀₉	C₂	semidirect
3. #557×C₁₀₉	C₂	semidirect
4. #557×C₁₀₉	C₂	semidirect
5. #556×C₁₀₉	C₂	semidirect
6. #556×C₁₀₉	C₂	semidirect
7. #556×C₁₀₉	C₂	semidirect
8. #554×C₁₀₉	C₂	semidirect
9. #552×C₁₀₉	C₂	semidirect
10. #544×C₁₀₉	C₂	semidirect
12. #541×C₁₀₉	C₂	semidirect
13. #541×C₁₀₉	C₂	semidirect
14. #541×C₁₀₉	C₂	semidirect
15. #533×C₁₀₉	C₂	semidirect
16. #533×C₁₀₉	C₂	semidirect
206. #3555	C₁₀₉	direct (N × H)
207. C₁₀₉	#3555	direct (N × H)
208. #563	C₂×C₁₀₉	semidirect
209. Q₈²	C₂×C₁₀₉	semidirect
210. #557	C₂×C₁₀₉	semidirect
211. #557	C₂×C₁₀₉	semidirect
212. #556	C₂×C₁₀₉	semidirect
213. #556	C₂×C₁₀₉	semidirect
214. #556	C₂×C₁₀₉	semidirect
215. #554	C₂×C₁₀₉	semidirect
216. #552	C₂×C₁₀₉	semidirect
217. #544	C₂×C₁₀₉	semidirect
219. #541	C₂×C₁₀₉	semidirect
220. #541	C₂×C₁₀₉	semidirect
221. #541	C₂×C₁₀₉	semidirect
222. #533	C₂×C₁₀₉	semidirect
223. #533	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#563×C₁₀₉	2
Q₈²×C₁₀₉	2
#557×C₁₀₉	2
#556×C₁₀₉	2
#554×C₁₀₉	2
#552×C₁₀₉	2
#544×C₁₀₉	2
#542×C₁₀₉	2
#541×C₁₀₉	2
#533×C₁₀₉	2
#529×C₁₀₉	2
#3555	109

Subgroups — C109×#3555 #28360809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3555 #28360809