Subgroups — C₁₀₉×#18759 #42136753

Order 27904.

1138 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#3230	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#3077	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1649	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1637	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1478	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₁₀₉	—	2	1	no	—
11	D₄×Q₈×C₁₀₉	—	2	2	no	—
12	D₄×Q₈×C₁₀₉	—	2	2	no	—
13	D₄×Q₈×C₁₀₉	—	2	2	no	—
14	D₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
15	#533×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
20	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	#500×C₁₀₉	C₂²	1	2	yes	complemented
24	#487×C₁₀₉	C₂²	1	2	yes	complemented
25	#484×C₁₀₉	—	2	2	no	—
26	#484×C₁₀₉	—	2	1	no	—
27	#478×C₁₀₉	—	2	2	no	—
28	#477×C₁₀₉	—	2	2	no	—
29	#477×C₁₀₉	C₂²	1	2	yes	complemented
30	#476×C₁₀₉	C₂²	1	2	yes	—
31	#474×C₁₀₉	C₂²	1	2	yes	—
32	#462×C₁₀₉	C₂²	1	2	yes	complemented
33	#451×C₁₀₉	—	2	1	no	—
34	C₄×Q₁₆×C₁₀₉	—	2	2	no	—
35	#431×C₁₀₉	C₂²	1	1	yes	characteristic
36	#431×C₁₀₉	C₂²	1	1	yes	characteristic
37	#428×C₁₀₉	—	2	1	no	—
38	#423×C₁₀₉	C₂²	1	1	yes	characteristic
39	#418×C₁₀₉	—	2	1	no	—
40	#332×C₁₀₉	C₂²	1	4	yes	complemented
41	Q₈⋊C₈×C₁₀₉	C₂²	1	4	yes	complemented
42	D₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
43	C₂²×Q₈×C₁₀₉	—	2	2	no	—
44	C₂²×Q₈×C₁₀₉	—	2	4	no	—
45	C₂²×Q₈×C₁₀₉	—	2	4	no	—
46	C₂²×Q₈×C₁₀₉	—	2	1	no	—
47	C₂²×Q₈×C₁₀₉	—	4	8	no	—
48	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₁₀₉	—	2	2	no	—
51	C₂³×C₄×C₁₀₉	—	2	1	no	—
52	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
53	C₂×M₁₆×C₁₀₉	—	2	1	no	—
54	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
55	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
56	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
57	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
58	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
59	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₁₀₉	C₂³	1	2	yes	—
62	#125×C₁₀₉	—	2	1	no	—
63	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
64	C₂²⋊Q₈×C₁₀₉	—	2	8	no	—
65	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
66	C₂²⋊Q₈×C₁₀₉	—	2	8	no	—
67	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
68	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
69	C₄×Q₈×C₁₀₉	—	2	1	no	—
70	C₄×Q₈×C₁₀₉	—	2	1	no	—
71	C₄×Q₈×C₁₀₉	—	2	1	no	—
72	C₄×Q₈×C₁₀₉	—	2	1	no	—
73	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
74	C₄×Q₈×C₁₀₉	C₂³	1	2	yes	—
75	C₄×D₄×C₁₀₉	—	2	4	no	—
76	C₄×D₄×C₁₀₉	—	2	4	no	—
77	C₄×D₄×C₁₀₉	—	2	1	no	—
78	C₄×D₄×C₁₀₉	—	2	1	no	—
79	C₄×D₄×C₁₀₉	D₄	1	2	yes	—
80	C₄×D₄×C₁₀₉	C₂³	1	2	yes	—
81	#117×C₁₀₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄²×C₁₀₉	—	2	1	no	—
91	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
92	#107×C₁₀₉	—	2	4	no	—
93	#107×C₁₀₉	—	2	2	no	—
94	#107×C₁₀₉	—	2	2	no	—
95	#106×C₁₀₉	—	2	2	no	—
96	C₄⋊C₈×C₁₀₉	—	2	1	no	—
97	C₄⋊C₈×C₁₀₉	—	2	1	no	—
98	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
99	C₄⋊C₈×C₁₀₉	C₂³	1	2	yes	—
100	#103×C₁₀₉	—	2	2	no	—
101	#103×C₁₀₉	—	2	2	no	—
102	#103×C₁₀₉	—	2	4	no	—
103	#102×C₁₀₉	—	2	2	no	—
104	#102×C₁₀₉	—	2	2	no	—
105	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
106	C₂²⋊C₈×C₁₀₉	—	2	1	no	—
107	C₄×C₈×C₁₀₉	—	2	1	no	—
108	C₄×C₈×C₁₀₉	C₂³	1	2	yes	—
109	C₂⁴×C₁₀₉	—	2	1	no	—
110	C₂×Q₈×C₁₀₉	—	2	4	no	—
111	C₂×Q₈×C₁₀₉	—	2	8	no	—
112	C₂×Q₈×C₁₀₉	—	2	1	no	—
113	C₂×Q₈×C₁₀₉	—	2	2	no	—
114	C₂×Q₈×C₁₀₉	—	2	2	no	—
115	C₂×Q₈×C₁₀₉	—	2	1	no	—
116	C₂×Q₈×C₁₀₉	—	2	1	no	—
117	C₂×Q₈×C₁₀₉	—	4	8	no	—
118	C₂×Q₈×C₁₀₉	—	4	4	no	—
119	C₂×Q₈×C₁₀₉	—	4	16	no	—
120	C₂×Q₈×C₁₀₉	—	4	8	no	—
121	C₂×Q₈×C₁₀₉	—	4	8	no	—
122	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₁₀₉	—	2	1	no	—
124	C₂×D₄×C₁₀₉	—	2	2	no	—
125	C₂×D₄×C₁₀₉	Q₁₆	1	4	yes	complemented
126	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₁₀₉	—	2	1	no	—
128	C₂²×C₄×C₁₀₉	—	2	8	no	—
129	C₂²×C₄×C₁₀₉	—	2	1	no	—
130	C₂²×C₄×C₁₀₉	—	2	2	no	—
131	C₂²×C₄×C₁₀₉	—	2	1	no	—
132	C₂²×C₄×C₁₀₉	—	2	2	no	—
133	C₂²×C₄×C₁₀₉	—	2	2	no	—
134	C₂²×C₄×C₁₀₉	—	4	8	no	—
135	C₂²×C₄×C₁₀₉	—	4	4	no	—
136	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
139	Q₁₆×C₁₀₉	—	4	4	no	—
140	M₁₆×C₁₀₉	—	4	2	no	—
141	C₂×C₈×C₁₀₉	—	2	2	no	—
142	C₂×C₈×C₁₀₉	—	2	2	no	—
143	C₂×C₈×C₁₀₉	—	2	2	no	—
144	C₄⋊C₄×C₁₀₉	—	2	8	no	—
145	C₄⋊C₄×C₁₀₉	—	2	2	no	—
146	C₄⋊C₄×C₁₀₉	—	2	2	no	—
147	C₄⋊C₄×C₁₀₉	—	2	1	no	—
148	C₄⋊C₄×C₁₀₉	—	2	1	no	—
149	C₄⋊C₄×C₁₀₉	—	2	2	no	—
150	C₄⋊C₄×C₁₀₉	—	2	2	no	—
151	C₄⋊C₄×C₁₀₉	—	2	2	no	—
152	C₄⋊C₄×C₁₀₉	—	2	1	no	—
153	C₄⋊C₄×C₁₀₉	—	2	1	no	—
154	C₄⋊C₄×C₁₀₉	—	2	1	no	—
155	C₄⋊C₄×C₁₀₉	—	2	1	no	—
156	C₄⋊C₄×C₁₀₉	—	4	1	no	—
157	C₄⋊C₄×C₁₀₉	—	4	1	no	—
158	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₁₀₉	C₂×D₄	1	4	yes	—
160	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
161	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
162	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
163	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
164	C₂²⋊C₄×C₁₀₉	—	4	1	no	—
165	C₄²×C₁₀₉	—	2	2	no	—
166	C₄²×C₁₀₉	—	2	2	no	—
167	C₄²×C₁₀₉	—	4	1	no	—
168	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₁₀₉	C₂⁴	1	1	yes	characteristic
172	C₂³×C₁₀₉	—	2	2	no	—
173	C₂³×C₁₀₉	—	2	4	no	—
174	C₂³×C₁₀₉	—	4	2	no	—
175	C₂³×C₁₀₉	—	4	2	no	—
176	C₂³×C₁₀₉	#120	1	1	yes	characteristic
177	Q₈×C₁₀₉	—	2	4	no	—
178	Q₈×C₁₀₉	—	2	2	no	—
179	Q₈×C₁₀₉	—	4	16	no	—
180	Q₈×C₁₀₉	—	4	4	no	—
181	Q₈×C₁₀₉	—	4	2	no	—
182	Q₈×C₁₀₉	—	4	4	no	—
183	Q₈×C₁₀₉	—	4	2	no	—
184	Q₈×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
185	D₄×C₁₀₉	—	2	2	no	—
186	D₄×C₁₀₉	—	2	4	no	—
187	D₄×C₁₀₉	C₂×Q₁₆	1	4	yes	—
188	C₂×C₄×C₁₀₉	—	2	4	no	—
189	C₂×C₄×C₁₀₉	—	2	2	no	—
190	C₂×C₄×C₁₀₉	—	2	2	no	—
191	C₂×C₄×C₁₀₉	—	2	2	no	—
192	C₂×C₄×C₁₀₉	—	2	4	no	—
193	C₂×C₄×C₁₀₉	—	2	2	no	—
194	C₂×C₄×C₁₀₉	—	4	1	no	—
195	C₂×C₄×C₁₀₉	—	4	16	no	—
196	C₂×C₄×C₁₀₉	—	4	4	no	—
197	C₂×C₄×C₁₀₉	—	4	2	no	—
198	C₂×C₄×C₁₀₉	—	4	4	no	—
199	C₂×C₄×C₁₀₉	—	4	2	no	—
200	C₂×C₄×C₁₀₉	—	4	4	no	—
201	C₂×C₄×C₁₀₉	—	4	1	no	—
202	C₂×C₄×C₁₀₉	—	4	1	no	—
203	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
204	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
205	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
206	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
207	C₂×C₄×C₁₀₉	#137	1	2	yes	—
208	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
211	C₈×C₁₀₉	—	4	2	no	—
212	C₈×C₁₀₉	—	4	2	no	—
213	C₈×C₁₀₉	—	4	2	no	—
214	C₂²×C₁₀₉	—	2	1	no	—
215	C₂²×C₁₀₉	—	2	1	no	—
216	C₂²×C₁₀₉	—	2	4	no	—
217	C₂²×C₁₀₉	—	4	8	no	—
218	C₂²×C₁₀₉	—	4	2	no	—
219	C₂²×C₁₀₉	—	4	2	no	—
220	C₂²×C₁₀₉	#451	1	2	yes	—
221	C₂²×C₁₀₉	C₂×#120	1	1	yes	characteristic
222	C₄×C₁₀₉	—	2	4	no	—
223	C₄×C₁₀₉	—	2	1	no	—
224	C₄×C₁₀₉	—	2	1	no	—
225	C₄×C₁₀₉	—	2	1	no	—
226	C₄×C₁₀₉	—	4	4	no	—
227	C₄×C₁₀₉	—	4	2	no	—
228	C₄×C₁₀₉	—	4	2	no	—
229	C₄×C₁₀₉	—	8	1	no	—
230	C₄×C₁₀₉	—	8	1	no	—
231	C₄×C₁₀₉	#451	1	2	yes	—
232	C₄×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
233	C₄×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
234	C₄×C₁₀₉	C₂×#137	1	1	yes	characteristic
235	#18759	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₁₀₉	—	2	2	no	—
237	C₂×C₁₀₉	—	4	4	no	—
238	C₂×C₁₀₉	C₂×#451	1	1	yes	characteristic
239	C₂×C₁₀₉	#3003	1	1	yes	characteristic
240	C₂×C₁₀₉	#3020	1	1	yes	characteristic
241	C₂×D₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
242	#3230	C₂×C₁₀₉	1	1	yes	characteristic, complemented
243	#3104	C₂×C₁₀₉	1	1	yes	characteristic, complemented
244	#3077	C₂×C₁₀₉	1	1	yes	characteristic
245	#1649	C₂×C₁₀₉	1	4	yes	complemented
246	#1637	C₂×C₁₀₉	1	4	yes	complemented
247	#1549	C₂×C₁₀₉	1	1	yes	characteristic, complemented
248	#1487	C₂×C₁₀₉	1	1	yes	characteristic, complemented
249	#1478	C₂×C₁₀₉	1	1	yes	characteristic
250	C₁₀₉	#18759	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³×Q₈	—	2	1	no	—
252	D₄×Q₈	—	2	2	no	—
253	D₄×Q₈	—	2	2	no	—
254	D₄×Q₈	—	2	2	no	—
255	D₄×Q₈	C₂²×C₁₀₉	1	4	yes	—
256	#533	C₂²×C₁₀₉	1	1	yes	characteristic
257	C₂×C₄⋊Q₈	—	2	1	no	—
258	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
259	C₂×C₂²⋊Q₈	—	2	1	no	—
260	C₂×C₂²⋊Q₈	—	2	2	no	—
261	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
262	C₂×C₄×D₄	—	2	1	no	—
263	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
264	#500	C₂²×C₁₀₉	1	2	yes	complemented
265	#487	C₂²×C₁₀₉	1	2	yes	complemented
266	#484	—	2	1	no	—
267	#484	—	2	2	no	—
268	#478	—	2	2	no	—
269	#477	—	2	2	no	—
270	#477	C₂²×C₁₀₉	1	2	yes	complemented
271	#476	C₂²×C₁₀₉	1	2	yes	—
272	#474	C₂²×C₁₀₉	1	2	yes	—
273	#462	C₂²×C₁₀₉	1	2	yes	complemented
274	#451	—	2	1	no	—
275	C₄×Q₁₆	—	2	2	no	—
276	#431	C₂²×C₁₀₉	1	1	yes	characteristic
277	#431	C₂²×C₁₀₉	1	1	yes	characteristic
278	#428	—	2	1	no	—
279	#423	C₂²×C₁₀₉	1	1	yes	characteristic
280	#418	—	2	1	no	—
281	#332	C₂²×C₁₀₉	1	4	yes	complemented
282	Q₈⋊C₈	C₂²×C₁₀₉	1	4	yes	complemented
283	D₄⋊C₈	C₂²×C₁₀₉	1	4	yes	—
284	C₂²×Q₈	—	2	2	no	—
285	C₂²×Q₈	—	2	4	no	—
286	C₂²×Q₈	—	2	4	no	—
287	C₂²×Q₈	—	2	1	no	—
288	C₂²×Q₈	—	4	8	no	—
289	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
290	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
291	C₂³×C₄	—	2	2	no	—
292	C₂³×C₄	—	2	1	no	—
293	C₂×Q₁₆	—	2	2	no	—
294	C₂×M₁₆	—	2	1	no	—
295	C₄⋊Q₈	—	2	4	no	—
296	C₄⋊Q₈	—	2	4	no	—
297	C₄⋊Q₈	—	2	1	no	—
298	C₄⋊Q₈	—	2	1	no	—
299	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
300	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
301	C₄⋊Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₁₀₉	1	2	yes	—
303	#125	—	2	1	no	—
304	C₂²⋊Q₈	—	2	4	no	—
305	C₂²⋊Q₈	—	2	8	no	—
306	C₂²⋊Q₈	—	2	4	no	—
307	C₂²⋊Q₈	—	2	8	no	—
308	C₂²⋊Q₈	—	2	1	no	—
309	C₂²⋊Q₈	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	—	2	1	no	—
312	C₄×Q₈	—	2	1	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
315	C₄×Q₈	C₂³×C₁₀₉	1	2	yes	—
316	C₄×D₄	—	2	4	no	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	D₄×C₁₀₉	1	2	yes	—
321	C₄×D₄	C₂³×C₁₀₉	1	2	yes	—
322	#117	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
328	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
333	#107	—	2	2	no	—
334	#107	—	2	2	no	—
335	#107	—	2	4	no	—
336	#106	—	2	2	no	—
337	C₄⋊C₈	—	2	1	no	—
338	C₄⋊C₈	—	2	1	no	—
339	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
340	C₄⋊C₈	C₂³×C₁₀₉	1	2	yes	—
341	#103	—	2	2	no	—
342	#103	—	2	2	no	—
343	#103	—	2	4	no	—
344	#102	—	2	2	no	—
345	#102	—	2	2	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	C₂²⋊C₈	—	2	1	no	—
348	C₄×C₈	—	2	1	no	—
349	C₄×C₈	C₂³×C₁₀₉	1	2	yes	—
350	C₂⁴	—	2	1	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	2	8	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	4	8	no	—
359	C₂×Q₈	—	4	4	no	—
360	C₂×Q₈	—	4	16	no	—
361	C₂×Q₈	—	4	8	no	—
362	C₂×Q₈	—	4	8	no	—
363	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	Q₁₆×C₁₀₉	1	4	yes	complemented
367	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	8	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	4	8	no	—
376	C₂²×C₄	—	4	4	no	—
377	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
380	Q₁₆	—	4	4	no	—
381	M₁₆	—	4	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₄⋊C₄	—	2	8	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	4	1	no	—
398	C₄⋊C₄	—	4	1	no	—
399	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
400	C₄⋊C₄	C₂×D₄×C₁₀₉	1	4	yes	—
401	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
402	C₂²⋊C₄	—	2	8	no	—
403	C₂²⋊C₄	—	2	2	no	—
404	C₂²⋊C₄	—	2	2	no	—
405	C₂²⋊C₄	—	4	1	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	4	1	no	—
409	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
410	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
411	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
412	C₄²	C₂⁴×C₁₀₉	1	1	yes	characteristic
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	#120×C₁₀₉	1	1	yes	characteristic
418	Q₈	—	2	4	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	4	16	no	—
421	Q₈	—	4	4	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	4	no	—
424	Q₈	—	4	2	no	—
425	Q₈	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
426	D₄	—	2	2	no	—
427	D₄	—	2	4	no	—
428	D₄	C₂×Q₁₆×C₁₀₉	1	4	yes	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	16	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
445	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
446	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
447	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
448	C₂×C₄	#137×C₁₀₉	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
452	C₈	—	4	2	no	—
453	C₈	—	4	2	no	—
454	C₈	—	4	2	no	—
455	C₂²	—	2	1	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	4	no	—
458	C₂²	—	4	8	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	#451×C₁₀₉	1	2	yes	—
462	C₂²	C₂×#120×C₁₀₉	1	1	yes	characteristic
463	C₄	—	2	4	no	—
464	C₄	—	2	1	no	—
465	C₄	—	2	1	no	—
466	C₄	—	2	1	no	—
467	C₄	—	4	4	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	1	no	—
472	C₄	#451×C₁₀₉	1	2	yes	—
473	C₄	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
474	C₄	C₂×AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
475	C₄	C₂×#137×C₁₀₉	1	1	yes	characteristic
476	C₂	—	2	2	no	—
477	C₂	—	4	4	no	—
478	C₂	C₂×#451×C₁₀₉	1	1	yes	characteristic
479	C₂	C₁₀₉×#3003	1	1	yes	characteristic
480	C₂	C₁₀₉×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	D₄
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
2. C₁₀₉×#3230	C₂	semidirect
3. C₁₀₉×#3104	C₂	semidirect
5. C₁₀₉×#1649	C₂	semidirect
6. C₁₀₉×#1637	C₂	semidirect
7. C₁₀₉×#1549	C₂	semidirect
8. C₁₀₉×#1487	C₂	semidirect
23. #500×C₁₀₉	C₂²	semidirect
24. #487×C₁₀₉	C₂²	semidirect
29. #477×C₁₀₉	C₂²	semidirect
32. #462×C₁₀₉	C₂²	semidirect
40. #332×C₁₀₉	C₂²	semidirect
41. Q₈⋊C₈×C₁₀₉	C₂²	semidirect
125. C₂×D₄×C₁₀₉	Q₁₆	semidirect
235. #18759	C₁₀₉	direct (N × H)
242. #3230	C₂×C₁₀₉	semidirect
243. #3104	C₂×C₁₀₉	semidirect
245. #1649	C₂×C₁₀₉	semidirect
246. #1637	C₂×C₁₀₉	semidirect
247. #1549	C₂×C₁₀₉	semidirect
248. #1487	C₂×C₁₀₉	semidirect
250. C₁₀₉	#18759	direct (N × H)
264. #500	C₂²×C₁₀₉	semidirect
265. #487	C₂²×C₁₀₉	semidirect
270. #477	C₂²×C₁₀₉	semidirect
273. #462	C₂²×C₁₀₉	semidirect
281. #332	C₂²×C₁₀₉	semidirect
282. Q₈⋊C₈	C₂²×C₁₀₉	semidirect
366. C₂×D₄	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₁₀₉	2
C₁₀₉×#3230	2
C₁₀₉×#3104	2
C₁₀₉×#3077	2
C₁₀₉×#1649	2
C₁₀₉×#1637	2
C₁₀₉×#1549	2
C₁₀₉×#1487	2
C₁₀₉×#1478	2
#18759	109

Subgroups — C109×#18759 #42136753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#18759 #42136753