Subgroups — C₁₀₉×#29354 #42147348

Order 27904.

850 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2858	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2071	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#2023	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1987	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1982	C₂	1	2	yes	maximal
7	C₁₀₉×#1936	C₂	1	4	yes	maximal, complemented
8	C₁₀₉×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#124×C₁₀₉	—	2	1	no	—
13	C₂×#124×C₁₀₉	—	2	1	no	—
14	C₂×C₄×Q₈×C₁₀₉	—	2	1	no	—
15	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#102×C₁₀₉	C₂²	1	4	yes	—
17	C₂×#102×C₁₀₉	C₂²	1	4	yes	—
18	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
19	#391×C₁₀₉	—	2	1	no	—
20	#390×C₁₀₉	—	2	1	no	—
21	#390×C₁₀₉	—	2	1	no	—
22	#390×C₁₀₉	C₂²	1	2	yes	—
23	#390×C₁₀₉	C₂²	1	2	yes	—
24	#389×C₁₀₉	—	2	1	no	—
25	#389×C₁₀₉	C₂²	1	2	yes	—
26	#384×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
30	#340×C₁₀₉	C₂²	1	4	yes	—
31	#340×C₁₀₉	C₂²	1	4	yes	—
32	#336×C₁₀₉	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₀₉	—	2	1	no	—
34	C₂²×Q₈×C₁₀₉	—	2	1	no	—
35	C₂²×D₄×C₁₀₉	—	2	1	no	—
36	C₂²×D₄×C₁₀₉	—	2	1	no	—
37	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
40	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
41	#127×C₁₀₉	—	2	4	no	—
42	#124×C₁₀₉	—	4	4	no	—
43	#124×C₁₀₉	—	4	4	no	—
44	C₄×Q₈×C₁₀₉	—	4	4	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄²×C₁₀₉	—	2	2	no	—
58	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₁₀₉	—	2	4	no	—
66	#102×C₁₀₉	—	2	8	no	—
67	#102×C₁₀₉	—	4	4	no	—
68	#97×C₁₀₉	—	4	2	no	—
69	#95×C₁₀₉	—	2	2	no	—
70	#95×C₁₀₉	—	2	2	no	—
71	C₂⁴×C₁₀₉	—	2	2	no	—
72	C₂×Q₈×C₁₀₉	—	2	4	no	—
73	C₂×Q₈×C₁₀₉	—	4	2	no	—
74	C₂×Q₈×C₁₀₉	—	4	2	no	—
75	C₂×Q₈×C₁₀₉	—	4	2	no	—
76	C₂×Q₈×C₁₀₉	—	4	2	no	—
77	C₂×Q₈×C₁₀₉	—	4	2	no	—
78	C₂×D₄×C₁₀₉	—	2	4	no	—
79	C₂×D₄×C₁₀₉	—	2	4	no	—
80	C₂×D₄×C₁₀₉	—	2	2	no	—
81	C₂×D₄×C₁₀₉	—	2	2	no	—
82	C₂×D₄×C₁₀₉	—	4	2	no	—
83	C₂×D₄×C₁₀₉	—	4	2	no	—
84	C₂×D₄×C₁₀₉	—	4	2	no	—
85	C₂×D₄×C₁₀₉	—	4	2	no	—
86	C₂×D₄×C₁₀₉	—	4	2	no	—
87	C₂×D₄×C₁₀₉	—	4	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	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₁₀₉	—	2	4	no	—
99	C₂×C₈×C₁₀₉	—	2	4	no	—
100	C₂×C₈×C₁₀₉	—	2	4	no	—
101	C₄⋊C₄×C₁₀₉	—	2	4	no	—
102	C₄⋊C₄×C₁₀₉	—	2	2	no	—
103	C₄⋊C₄×C₁₀₉	—	2	4	no	—
104	C₄⋊C₄×C₁₀₉	—	2	2	no	—
105	C₄⋊C₄×C₁₀₉	—	2	2	no	—
106	C₄⋊C₄×C₁₀₉	—	4	2	no	—
107	C₄⋊C₄×C₁₀₉	—	4	4	no	—
108	C₄⋊C₄×C₁₀₉	—	4	2	no	—
109	C₄⋊C₄×C₁₀₉	—	4	2	no	—
110	C₄⋊C₄×C₁₀₉	#41	1	4	yes	—
111	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
112	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
113	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
114	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
115	C₄²×C₁₀₉	—	2	2	no	—
116	C₄²×C₁₀₉	—	2	2	no	—
117	C₄²×C₁₀₉	—	2	2	no	—
118	C₄²×C₁₀₉	—	2	2	no	—
119	C₄²×C₁₀₉	—	2	2	no	—
120	C₄²×C₁₀₉	—	2	1	no	—
121	C₄²×C₁₀₉	—	2	1	no	—
122	C₄²×C₁₀₉	—	4	4	no	—
123	C₄²×C₁₀₉	#41	1	4	yes	—
124	C₂³×C₁₀₉	—	4	2	no	—
125	C₂³×C₁₀₉	—	4	2	no	—
126	C₂³×C₁₀₉	—	4	2	no	—
127	C₂³×C₁₀₉	—	4	2	no	—
128	C₂³×C₁₀₉	—	4	2	no	—
129	C₂³×C₁₀₉	—	4	2	no	—
130	C₂³×C₁₀₉	—	4	2	no	—
131	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
132	Q₈×C₁₀₉	—	4	8	no	—
133	Q₈×C₁₀₉	—	8	4	no	—
134	D₄×C₁₀₉	—	4	4	no	—
135	D₄×C₁₀₉	—	4	4	no	—
136	D₄×C₁₀₉	—	8	4	no	—
137	D₄×C₁₀₉	—	8	4	no	—
138	C₂×C₄×C₁₀₉	—	2	1	no	—
139	C₂×C₄×C₁₀₉	—	2	1	no	—
140	C₂×C₄×C₁₀₉	—	2	1	no	—
141	C₂×C₄×C₁₀₉	—	2	1	no	—
142	C₂×C₄×C₁₀₉	—	2	1	no	—
143	C₂×C₄×C₁₀₉	—	2	1	no	—
144	C₂×C₄×C₁₀₉	—	2	4	no	—
145	C₂×C₄×C₁₀₉	—	2	1	no	—
146	C₂×C₄×C₁₀₉	—	2	1	no	—
147	C₂×C₄×C₁₀₉	—	2	1	no	—
148	C₂×C₄×C₁₀₉	—	2	1	no	—
149	C₂×C₄×C₁₀₉	—	2	1	no	—
150	C₂×C₄×C₁₀₉	—	2	1	no	—
151	C₂×C₄×C₁₀₉	—	4	2	no	—
152	C₂×C₄×C₁₀₉	—	4	2	no	—
153	C₂×C₄×C₁₀₉	—	4	2	no	—
154	C₂×C₄×C₁₀₉	—	4	2	no	—
155	C₂×C₄×C₁₀₉	—	4	2	no	—
156	C₂×C₄×C₁₀₉	—	4	2	no	—
157	C₂×C₄×C₁₀₉	—	4	2	no	—
158	C₂×C₄×C₁₀₉	—	4	2	no	—
159	C₂×C₄×C₁₀₉	#124	1	2	yes	—
160	C₂×C₄×C₁₀₉	#124	1	2	yes	—
161	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
163	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₄×C₁₀₉	#137	1	2	yes	—
165	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
166	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
167	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
169	C₈×C₁₀₉	—	4	4	no	—
170	C₂²×C₁₀₉	—	8	2	no	—
171	C₂²×C₁₀₉	—	8	2	no	—
172	C₂²×C₁₀₉	—	8	2	no	—
173	C₂²×C₁₀₉	—	8	2	no	—
174	C₂²×C₁₀₉	—	8	2	no	—
175	C₂²×C₁₀₉	—	8	2	no	—
176	C₂²×C₁₀₉	—	8	2	no	—
177	C₂²×C₁₀₉	#525	1	1	yes	characteristic
178	C₂²×C₁₀₉	C₂×#124	1	1	yes	characteristic
179	C₂²×C₁₀₉	#537	1	1	yes	characteristic
180	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×C₁₀₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₁₀₉	—	2	2	no	—
185	C₄×C₁₀₉	—	2	2	no	—
186	C₄×C₁₀₉	—	2	2	no	—
187	C₄×C₁₀₉	—	4	4	no	—
188	C₄×C₁₀₉	—	4	1	no	—
189	C₄×C₁₀₉	—	4	1	no	—
190	C₄×C₁₀₉	—	8	2	no	—
191	C₄×C₁₀₉	—	8	2	no	—
192	C₄×C₁₀₉	#489	1	4	yes	—
193	C₄×C₁₀₉	#556	1	2	yes	—
194	C₄×C₁₀₉	#559	1	2	yes	—
195	#29354	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₁₀₉	—	16	2	no	—
197	C₂×C₁₀₉	#2592	1	1	yes	characteristic
198	C₂×C₁₀₉	#3109	1	1	yes	characteristic
199	C₂×C₁₀₉	#3112	1	1	yes	characteristic
200	C₂×C₁₀₉	C₂×#489	1	1	yes	characteristic
201	C₂×C₁₀₉	#3138	1	1	yes	characteristic
202	C₂×C₁₀₉	#3253	1	1	yes	characteristic
203	C₂×C₁₀₉	#3254	1	1	yes	characteristic
204	#2858	C₂×C₁₀₉	1	1	yes	characteristic
205	C₄×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
206	#2071	C₂×C₁₀₉	1	2	yes	complemented
207	#2023	C₂×C₁₀₉	1	2	yes	complemented
208	#1987	C₂×C₁₀₉	1	2	yes	complemented
209	#1982	C₂×C₁₀₉	1	2	yes	—
210	#1936	C₂×C₁₀₉	1	4	yes	complemented
211	#1767	C₂×C₁₀₉	1	1	yes	characteristic, complemented
212	C₁₀₉	#29354	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
214	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
215	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
216	C₂×#124	—	2	1	no	—
217	C₂×#124	—	2	1	no	—
218	C₂×C₄×Q₈	—	2	1	no	—
219	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
220	C₂×#102	C₂²×C₁₀₉	1	4	yes	—
221	C₂×#102	C₂²×C₁₀₉	1	4	yes	—
222	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
223	#391	—	2	1	no	—
224	#390	—	2	1	no	—
225	#390	—	2	1	no	—
226	#390	C₂²×C₁₀₉	1	2	yes	—
227	#390	C₂²×C₁₀₉	1	2	yes	—
228	#389	—	2	1	no	—
229	#389	C₂²×C₁₀₉	1	2	yes	—
230	#384	C₂²×C₁₀₉	1	1	yes	characteristic
231	C₄×C₄⋊C₄	—	2	1	no	—
232	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
233	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
234	#340	C₂²×C₁₀₉	1	4	yes	—
235	#340	C₂²×C₁₀₉	1	4	yes	—
236	#336	C₂²×C₁₀₉	1	2	yes	—
237	C₂²×Q₈	—	2	1	no	—
238	C₂²×Q₈	—	2	1	no	—
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	—	2	1	no	—
241	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
242	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
243	C₄⋊Q₈	—	2	4	no	—
244	C₄⋊Q₈	—	2	4	no	—
245	#127	—	2	4	no	—
246	#124	—	4	4	no	—
247	#124	—	4	4	no	—
248	C₄×Q₈	—	4	4	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
263	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
269	C₄⋊C₈	—	2	4	no	—
270	#102	—	2	8	no	—
271	#102	—	4	4	no	—
272	#97	—	4	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	C₂⁴	—	2	2	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	4	2	no	—
281	C₂×Q₈	—	4	2	no	—
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	—	4	2	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₄⋊C₄	—	2	4	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	4	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	#41×C₁₀₉	1	4	yes	—
315	C₂²⋊C₄	—	4	4	no	—
316	C₂²⋊C₄	—	4	4	no	—
317	C₂²⋊C₄	—	4	4	no	—
318	C₂²⋊C₄	—	4	4	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	4	4	no	—
327	C₄²	#41×C₁₀₉	1	4	yes	—
328	C₂³	—	4	2	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
336	Q₈	—	4	8	no	—
337	Q₈	—	8	4	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	4	no	—
340	D₄	—	8	4	no	—
341	D₄	—	8	4	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	#124×C₁₀₉	1	2	yes	—
364	C₂×C₄	#124×C₁₀₉	1	2	yes	—
365	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
366	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
367	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
368	C₂×C₄	#137×C₁₀₉	1	2	yes	—
369	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
370	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
371	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
372	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
373	C₈	—	4	4	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	8	2	no	—
376	C₂²	—	8	2	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	8	2	no	—
380	C₂²	—	8	2	no	—
381	C₂²	#525×C₁₀₉	1	1	yes	characteristic
382	C₂²	C₂×#124×C₁₀₉	1	1	yes	characteristic
383	C₂²	#537×C₁₀₉	1	1	yes	characteristic
384	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
385	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
386	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
387	C₂²	C₂×#137×C₁₀₉	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	4	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	#489×C₁₀₉	1	4	yes	—
397	C₄	#556×C₁₀₉	1	2	yes	—
398	C₄	#559×C₁₀₉	1	2	yes	—
399	C₂	—	16	2	no	—
400	C₂	C₁₀₉×#2592	1	1	yes	characteristic
401	C₂	C₁₀₉×#3109	1	1	yes	characteristic
402	C₂	C₁₀₉×#3112	1	1	yes	characteristic
403	C₂	C₂×#489×C₁₀₉	1	1	yes	characteristic
404	C₂	C₁₀₉×#3138	1	1	yes	characteristic
405	C₂	C₁₀₉×#3253	1	1	yes	characteristic
406	C₂	C₁₀₉×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊Q₈×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2071	C₂	semidirect
4. C₁₀₉×#2023	C₂	semidirect
5. C₁₀₉×#1987	C₂	semidirect
7. C₁₀₉×#1936	C₂	semidirect
8. C₁₀₉×#1767	C₂	semidirect
195. #29354	C₁₀₉	direct (N × H)
205. C₄×C₄⋊Q₈	C₂×C₁₀₉	semidirect
206. #2071	C₂×C₁₀₉	semidirect
207. #2023	C₂×C₁₀₉	semidirect
208. #1987	C₂×C₁₀₉	semidirect
210. #1936	C₂×C₁₀₉	semidirect
211. #1767	C₂×C₁₀₉	semidirect
212. C₁₀₉	#29354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2858	2
C₄×C₄⋊Q₈×C₁₀₉	2
C₁₀₉×#2071	2
C₁₀₉×#2023	2
C₁₀₉×#1987	2
C₁₀₉×#1982	2
C₁₀₉×#1936	2
C₁₀₉×#1767	2
#29354	109

Subgroups — C109×#29354 #42147348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#29354 #42147348