Subgroups — C₁₀₉×#22713 #42140707

Order 27904.

950 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2354	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1854	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1852	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1809	C₂	1	2	yes	maximal, complemented
10	C₄×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₁₀₉	—	2	1	no	—
14	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
17	#436×C₁₀₉	C₄	1	8	yes	complemented
18	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
21	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#390×C₁₀₉	—	2	1	no	—
26	#389×C₁₀₉	—	2	1	no	—
27	#388×C₁₀₉	—	2	2	no	—
28	#385×C₁₀₉	—	2	2	no	—
29	#384×C₁₀₉	C₂²	1	1	yes	characteristic
30	#384×C₁₀₉	C₂²	1	1	yes	characteristic
31	#380×C₁₀₉	C₂²	1	2	yes	—
32	#380×C₁₀₉	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	#340×C₁₀₉	C₂²	1	2	yes	—
37	#340×C₁₀₉	C₂²	1	2	yes	—
38	#340×C₁₀₉	C₂²	1	1	yes	characteristic
39	#340×C₁₀₉	C₂²	1	2	yes	—
40	#340×C₁₀₉	C₂²	1	1	yes	characteristic
41	#340×C₁₀₉	C₂²	1	2	yes	—
42	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₁₀₉	—	2	2	no	—
44	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₀₉	—	4	2	no	—
46	C₂×M₁₆×C₁₀₉	—	4	2	no	—
47	C₂×M₁₆×C₁₀₉	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
51	C₄×D₄×C₁₀₉	—	4	4	no	—
52	C₄×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₁₀₉	Q₈	1	4	yes	—
54	#117×C₁₀₉	—	2	4	no	—
55	#117×C₁₀₉	—	4	2	no	—
56	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
57	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
63	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
74	C₂×C₄²×C₁₀₉	—	2	2	no	—
75	C₂×C₄²×C₁₀₉	—	2	1	no	—
76	C₂×C₄²×C₁₀₉	—	2	1	no	—
77	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
78	#107×C₁₀₉	—	2	2	no	—
79	#107×C₁₀₉	—	2	2	no	—
80	#107×C₁₀₉	—	4	2	no	—
81	#106×C₁₀₉	—	4	2	no	—
82	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
83	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
84	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
85	#95×C₁₀₉	—	2	2	no	—
86	#95×C₁₀₉	—	2	1	no	—
87	#95×C₁₀₉	—	2	1	no	—
88	#95×C₁₀₉	—	2	2	no	—
89	C₂⁴×C₁₀₉	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₁₀₉	—	2	2	no	—
91	C₂×D₄×C₁₀₉	—	2	2	no	—
92	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
93	C₂²×C₄×C₁₀₉	—	2	2	no	—
94	C₂²×C₄×C₁₀₉	—	2	2	no	—
95	C₂²×C₄×C₁₀₉	—	2	2	no	—
96	C₂²×C₄×C₁₀₉	—	2	2	no	—
97	C₂²×C₄×C₁₀₉	—	2	1	no	—
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	C₂²×C₄×C₁₀₉	—	2	1	no	—
101	C₂²×C₄×C₁₀₉	—	2	2	no	—
102	C₂²×C₄×C₁₀₉	—	2	2	no	—
103	C₂²×C₄×C₁₀₉	—	4	4	no	—
104	C₂²×C₄×C₁₀₉	—	4	4	no	—
105	C₂²×C₄×C₁₀₉	—	8	2	no	—
106	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
111	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₁₀₉	—	4	8	no	—
113	C₂×C₈×C₁₀₉	—	2	1	no	—
114	C₂×C₈×C₁₀₉	—	2	1	no	—
115	C₂×C₈×C₁₀₉	—	2	2	no	—
116	C₂×C₈×C₁₀₉	—	2	2	no	—
117	C₂×C₈×C₁₀₉	—	2	1	no	—
118	C₂×C₈×C₁₀₉	—	2	1	no	—
119	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
122	C₄⋊C₄×C₁₀₉	—	2	1	no	—
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	2	no	—
127	C₄⋊C₄×C₁₀₉	—	2	2	no	—
128	C₄⋊C₄×C₁₀₉	—	2	1	no	—
129	C₄⋊C₄×C₁₀₉	—	2	2	no	—
130	C₄⋊C₄×C₁₀₉	—	2	1	no	—
131	C₄⋊C₄×C₁₀₉	—	2	2	no	—
132	C₄⋊C₄×C₁₀₉	—	4	4	no	—
133	C₄⋊C₄×C₁₀₉	—	4	2	no	—
134	C₄⋊C₄×C₁₀₉	—	4	2	no	—
135	C₄⋊C₄×C₁₀₉	—	8	1	no	—
136	C₄⋊C₄×C₁₀₉	—	8	1	no	—
137	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
138	C₄⋊C₄×C₁₀₉	#41	1	2	yes	—
139	C₄⋊C₄×C₁₀₉	#41	1	2	yes	—
140	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
141	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
142	C₂²⋊C₄×C₁₀₉	—	8	2	no	—
143	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
144	C₄²×C₁₀₉	—	4	4	no	—
145	C₄²×C₁₀₉	—	4	2	no	—
146	C₄²×C₁₀₉	—	4	2	no	—
147	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
148	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
149	C₂³×C₁₀₉	—	2	2	no	—
150	C₂³×C₁₀₉	—	2	2	no	—
151	C₂³×C₁₀₉	—	2	2	no	—
152	C₂³×C₁₀₉	—	2	2	no	—
153	C₂³×C₁₀₉	—	2	2	no	—
154	C₂³×C₁₀₉	—	2	2	no	—
155	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
156	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
157	D₄×C₁₀₉	—	2	8	no	—
158	C₂×C₄×C₁₀₉	—	2	2	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	4	no	—
163	C₂×C₄×C₁₀₉	—	2	4	no	—
164	C₂×C₄×C₁₀₉	—	2	4	no	—
165	C₂×C₄×C₁₀₉	—	2	4	no	—
166	C₂×C₄×C₁₀₉	—	2	2	no	—
167	C₂×C₄×C₁₀₉	—	2	2	no	—
168	C₂×C₄×C₁₀₉	—	4	1	no	—
169	C₂×C₄×C₁₀₉	—	4	1	no	—
170	C₂×C₄×C₁₀₉	—	4	1	no	—
171	C₂×C₄×C₁₀₉	—	4	1	no	—
172	C₂×C₄×C₁₀₉	—	4	2	no	—
173	C₂×C₄×C₁₀₉	—	4	2	no	—
174	C₂×C₄×C₁₀₉	—	4	1	no	—
175	C₂×C₄×C₁₀₉	—	4	1	no	—
176	C₂×C₄×C₁₀₉	—	4	1	no	—
177	C₂×C₄×C₁₀₉	—	4	1	no	—
178	C₂×C₄×C₁₀₉	—	4	2	no	—
179	C₂×C₄×C₁₀₉	—	4	2	no	—
180	C₂×C₄×C₁₀₉	—	4	1	no	—
181	C₂×C₄×C₁₀₉	—	4	2	no	—
182	C₂×C₄×C₁₀₉	—	4	2	no	—
183	C₂×C₄×C₁₀₉	—	4	1	no	—
184	C₂×C₄×C₁₀₉	—	8	8	no	—
185	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
186	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
187	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
188	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₁₀₉	#137	1	2	yes	—
190	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
192	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
196	C₈×C₁₀₉	—	2	2	no	—
197	C₈×C₁₀₉	—	2	4	no	—
198	C₈×C₁₀₉	—	2	2	no	—
199	C₂²×C₁₀₉	—	2	4	no	—
200	C₂²×C₁₀₉	—	2	4	no	—
201	C₂²×C₁₀₉	—	2	4	no	—
202	C₂²×C₁₀₉	—	4	2	no	—
203	C₂²×C₁₀₉	—	4	2	no	—
204	C₂²×C₁₀₉	—	4	2	no	—
205	C₂²×C₁₀₉	—	4	2	no	—
206	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₁₀₉	#546	1	1	yes	characteristic
208	C₂²×C₁₀₉	#546	1	1	yes	characteristic
209	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₁₀₉	C₂×#137	1	1	yes	characteristic
211	C₂²×C₁₀₉	#576	1	1	yes	characteristic
212	C₂²×C₁₀₉	#578	1	1	yes	characteristic
213	C₄×C₁₀₉	—	2	4	no	—
214	C₄×C₁₀₉	—	2	2	no	—
215	C₄×C₁₀₉	—	4	2	no	—
216	C₄×C₁₀₉	—	4	2	no	—
217	C₄×C₁₀₉	—	8	1	no	—
218	C₄×C₁₀₉	—	8	2	no	—
219	C₄×C₁₀₉	—	8	1	no	—
220	C₄×C₁₀₉	—	8	2	no	—
221	C₄×C₁₀₉	#428	1	4	yes	—
222	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
223	C₄×C₁₀₉	#518	1	1	yes	characteristic
224	C₄×C₁₀₉	#554	1	1	yes	characteristic
225	C₄×C₁₀₉	#554	1	1	yes	characteristic
226	#22713	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₁₀₉	—	4	4	no	—
228	C₂×C₁₀₉	#2349	1	1	yes	characteristic
229	C₂×C₁₀₉	C₂×#428	1	1	yes	characteristic
230	C₂×C₁₀₉	#2913	1	1	yes	characteristic
231	C₂×C₁₀₉	#3325	1	1	yes	characteristic
232	C₂×C₁₀₉	#3328	1	1	yes	characteristic
233	C₂×C₁₀₉	#3330	1	1	yes	characteristic
234	C₂×C₁₀₉	#3333	1	1	yes	characteristic
235	C₂×#436	C₂×C₁₀₉	1	1	yes	characteristic
236	#2354	C₂×C₁₀₉	1	1	yes	characteristic
237	D₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
238	#1948	C₂×C₁₀₉	1	1	yes	characteristic, complemented
239	#1947	C₂×C₁₀₉	1	1	yes	characteristic, complemented
240	#1854	C₂×C₁₀₉	1	2	yes	complemented
241	#1852	C₂×C₁₀₉	1	2	yes	complemented
242	#1834	C₂×C₁₀₉	1	1	yes	characteristic, complemented
243	#1809	C₂×C₁₀₉	1	2	yes	complemented
244	C₄×#107	C₂×C₁₀₉	1	1	yes	characteristic, complemented
245	#1736	C₂×C₁₀₉	1	2	yes	complemented
246	C₁₀₉	#22713	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
248	C₂×C₄×D₄	—	2	1	no	—
249	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
250	C₂×#117	C₂²×C₁₀₉	1	2	yes	—
251	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
252	#436	C₄×C₁₀₉	1	8	yes	complemented
253	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
254	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
255	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
256	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
258	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
259	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
260	#390	—	2	1	no	—
261	#389	—	2	1	no	—
262	#388	—	2	2	no	—
263	#385	—	2	2	no	—
264	#384	C₂²×C₁₀₉	1	1	yes	characteristic
265	#384	C₂²×C₁₀₉	1	1	yes	characteristic
266	#380	C₂²×C₁₀₉	1	2	yes	—
267	#380	C₂²×C₁₀₉	1	2	yes	—
268	C₄×C₄⋊C₄	—	2	1	no	—
269	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
270	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
271	#340	C₂²×C₁₀₉	1	1	yes	characteristic
272	#340	C₂²×C₁₀₉	1	2	yes	—
273	#340	C₂²×C₁₀₉	1	2	yes	—
274	#340	C₂²×C₁₀₉	1	2	yes	—
275	#340	C₂²×C₁₀₉	1	2	yes	—
276	#340	C₂²×C₁₀₉	1	1	yes	characteristic
277	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
280	C₂×M₁₆	—	4	2	no	—
281	C₂×M₁₆	—	4	2	no	—
282	C₂×M₁₆	C₂×C₄×C₁₀₉	1	8	yes	complemented
283	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
285	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
286	C₄×D₄	—	4	4	no	—
287	C₄×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
288	C₄×D₄	Q₈×C₁₀₉	1	4	yes	—
289	#117	—	2	4	no	—
290	#117	—	4	2	no	—
291	C₂×C₄⋊C₄	—	2	4	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	4	2	no	—
298	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
313	#107	—	2	2	no	—
314	#107	—	2	2	no	—
315	#107	—	4	2	no	—
316	#106	—	4	2	no	—
317	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
318	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
319	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
320	#95	—	2	2	no	—
321	#95	—	2	1	no	—
322	#95	—	2	1	no	—
323	#95	—	2	2	no	—
324	C₂⁴	C₂×D₄×C₁₀₉	1	2	yes	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	4	4	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	8	2	no	—
341	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
342	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
345	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
347	M₁₆	—	4	8	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
355	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
356	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	8	1	no	—
371	C₄⋊C₄	—	8	1	no	—
372	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
373	C₄⋊C₄	#41×C₁₀₉	1	2	yes	—
374	C₄⋊C₄	#41×C₁₀₉	1	2	yes	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	8	yes	—
379	C₄²	—	4	4	no	—
380	C₄²	—	4	2	no	—
381	C₄²	—	4	2	no	—
382	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
383	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
391	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
392	D₄	—	2	8	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	8	8	no	—
420	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
421	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
422	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
423	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
424	C₂×C₄	#137×C₁₀₉	1	2	yes	—
425	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
426	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
429	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
430	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	2	4	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
442	C₂²	#546×C₁₀₉	1	1	yes	characteristic
443	C₂²	#546×C₁₀₉	1	1	yes	characteristic
444	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
445	C₂²	C₂×#137×C₁₀₉	1	1	yes	characteristic
446	C₂²	#576×C₁₀₉	1	1	yes	characteristic
447	C₂²	#578×C₁₀₉	1	1	yes	characteristic
448	C₄	—	2	4	no	—
449	C₄	—	2	2	no	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	8	2	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	1	no	—
455	C₄	—	8	2	no	—
456	C₄	#428×C₁₀₉	1	4	yes	—
457	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
458	C₄	#518×C₁₀₉	1	1	yes	characteristic
459	C₄	#554×C₁₀₉	1	1	yes	characteristic
460	C₄	#554×C₁₀₉	1	1	yes	characteristic
461	C₂	—	4	4	no	—
462	C₂	C₁₀₉×#2349	1	1	yes	characteristic
463	C₂	C₂×#428×C₁₀₉	1	1	yes	characteristic
464	C₂	C₁₀₉×#2913	1	1	yes	characteristic
465	C₂	C₁₀₉×#3325	1	1	yes	characteristic
466	C₂	C₁₀₉×#3328	1	1	yes	characteristic
467	C₂	C₁₀₉×#3330	1	1	yes	characteristic
468	C₂	C₁₀₉×#3333	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1948	C₂	semidirect
5. C₁₀₉×#1947	C₂	semidirect
6. C₁₀₉×#1854	C₂	semidirect
7. C₁₀₉×#1852	C₂	semidirect
8. C₁₀₉×#1834	C₂	semidirect
9. C₁₀₉×#1809	C₂	semidirect
10. C₄×#107×C₁₀₉	C₂	semidirect
11. C₁₀₉×#1736	C₂	semidirect
17. #436×C₁₀₉	C₄	semidirect
47. C₂×M₁₆×C₁₀₉	C₂×C₄	semidirect
82. C₄⋊C₈×C₁₀₉	C₂×C₄	semidirect
83. C₂²⋊C₈×C₁₀₉	C₂×C₄	semidirect
84. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
226. #22713	C₁₀₉	direct (N × H)
238. #1948	C₂×C₁₀₉	semidirect
239. #1947	C₂×C₁₀₉	semidirect
240. #1854	C₂×C₁₀₉	semidirect
241. #1852	C₂×C₁₀₉	semidirect
242. #1834	C₂×C₁₀₉	semidirect
243. #1809	C₂×C₁₀₉	semidirect
244. C₄×#107	C₂×C₁₀₉	semidirect
245. #1736	C₂×C₁₀₉	semidirect
246. C₁₀₉	#22713	direct (N × H)
252. #436	C₄×C₁₀₉	semidirect
282. C₂×M₁₆	C₂×C₄×C₁₀₉	semidirect
317. C₄⋊C₈	C₂×C₄×C₁₀₉	semidirect
318. C₂²⋊C₈	C₂×C₄×C₁₀₉	semidirect
319. C₄×C₈	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₀₉	2
C₁₀₉×#2354	2
D₄×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1948	2
C₁₀₉×#1947	2
C₁₀₉×#1854	2
C₁₀₉×#1852	2
C₁₀₉×#1834	2
C₁₀₉×#1809	2
C₄×#107×C₁₀₉	2
C₁₀₉×#1736	2
#22713	109

Subgroups — C109×#22713 #42140707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22713 #42140707