Subgroups — D₄×#106×C₁₀₉ #42140647

Order 27904.

1126 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#106×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	D₄×C₄⋊C₄×C₁₀₉	C₂	1	2	yes	maximal
4	C₁₀₉×#1948	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1853	C₂	1	4	yes	maximal, complemented
6	C₁₀₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#1807	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₀₉	—	2	2	no	—
11	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
13	C₈×D₄×C₁₀₉	C₄	1	8	yes	complemented
14	C₂×#106×C₁₀₉	—	2	8	no	—
15	C₂×#106×C₁₀₉	C₂²	1	4	yes	—
16	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
18	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	#389×C₁₀₉	—	2	2	no	—
24	#385×C₁₀₉	—	2	4	no	—
25	#384×C₁₀₉	C₂²	1	2	yes	—
26	#380×C₁₀₉	C₂²	1	4	yes	—
27	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
28	#340×C₁₀₉	C₂²	1	4	yes	—
29	#340×C₁₀₉	C₂²	1	2	yes	—
30	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₀₉	—	2	4	no	—
32	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₀₉	—	2	4	no	—
34	C₂²×C₈×C₁₀₉	—	2	4	no	—
35	C₂²×C₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
36	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄×D₄×C₁₀₉	—	4	8	no	—
40	C₄×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₁₀₉	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₀₉	—	4	4	no	—
47	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₀₉	—	2	2	no	—
54	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
55	#106×C₁₀₉	—	2	4	no	—
56	#106×C₁₀₉	—	2	2	no	—
57	#106×C₁₀₉	—	2	8	no	—
58	#106×C₁₀₉	—	4	2	no	—
59	#106×C₁₀₉	D₄	1	4	yes	direct product factor
60	C₄⋊C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
61	C₂²⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
63	#95×C₁₀₉	—	2	2	no	—
64	#95×C₁₀₉	—	2	4	no	—
65	C₂⁴×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
67	C₂×D₄×C₁₀₉	SD₁₆	1	8	yes	—
68	C₂²×C₄×C₁₀₉	—	2	2	no	—
69	C₂²×C₄×C₁₀₉	—	2	2	no	—
70	C₂²×C₄×C₁₀₉	—	2	2	no	—
71	C₂²×C₄×C₁₀₉	—	2	2	no	—
72	C₂²×C₄×C₁₀₉	—	2	2	no	—
73	C₂²×C₄×C₁₀₉	—	2	2	no	—
74	C₂²×C₄×C₁₀₉	—	2	4	no	—
75	C₂²×C₄×C₁₀₉	—	4	8	no	—
76	C₂²×C₄×C₁₀₉	—	4	8	no	—
77	C₂²×C₄×C₁₀₉	—	8	4	no	—
78	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₀₉	—	2	2	no	—
85	C₂×C₈×C₁₀₉	—	2	2	no	—
86	C₂×C₈×C₁₀₉	—	2	1	no	—
87	C₂×C₈×C₁₀₉	—	2	1	no	—
88	C₂×C₈×C₁₀₉	—	2	16	no	—
89	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₀₉	#41	1	2	yes	—
94	C₄⋊C₄×C₁₀₉	—	2	2	no	—
95	C₄⋊C₄×C₁₀₉	—	2	4	no	—
96	C₄⋊C₄×C₁₀₉	—	2	2	no	—
97	C₄⋊C₄×C₁₀₉	—	2	4	no	—
98	C₄⋊C₄×C₁₀₉	—	2	2	no	—
99	C₄⋊C₄×C₁₀₉	—	4	8	no	—
100	C₄⋊C₄×C₁₀₉	—	4	4	no	—
101	C₄⋊C₄×C₁₀₉	—	8	2	no	—
102	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
103	C₄⋊C₄×C₁₀₉	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
105	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
106	C₄²×C₁₀₉	—	4	4	no	—
107	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
109	C₂³×C₁₀₉	—	2	2	no	—
110	C₂³×C₁₀₉	—	2	2	no	—
111	C₂³×C₁₀₉	—	2	4	no	—
112	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
113	C₂³×C₁₀₉	C₂×SD₁₆	1	8	yes	—
114	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₁₀₉	#106	1	16	yes	direct product factor
116	C₂×C₄×C₁₀₉	—	2	2	no	—
117	C₂×C₄×C₁₀₉	—	2	2	no	—
118	C₂×C₄×C₁₀₉	—	2	2	no	—
119	C₂×C₄×C₁₀₉	—	2	4	no	—
120	C₂×C₄×C₁₀₉	—	2	4	no	—
121	C₂×C₄×C₁₀₉	—	2	4	no	—
122	C₂×C₄×C₁₀₉	—	2	4	no	—
123	C₂×C₄×C₁₀₉	—	2	4	no	—
124	C₂×C₄×C₁₀₉	—	4	2	no	—
125	C₂×C₄×C₁₀₉	—	4	2	no	—
126	C₂×C₄×C₁₀₉	—	4	4	no	—
127	C₂×C₄×C₁₀₉	—	4	2	no	—
128	C₂×C₄×C₁₀₉	—	4	2	no	—
129	C₂×C₄×C₁₀₉	—	4	4	no	—
130	C₂×C₄×C₁₀₉	—	4	4	no	—
131	C₂×C₄×C₁₀₉	—	4	2	no	—
132	C₂×C₄×C₁₀₉	—	8	16	no	—
133	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
137	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₁₀₉	—	2	4	no	—
144	C₈×C₁₀₉	—	2	2	no	—
145	C₈×C₁₀₉	C₄×D₄	1	4	yes	complemented
146	C₂²×C₁₀₉	—	2	8	no	—
147	C₂²×C₁₀₉	—	2	4	no	—
148	C₂²×C₁₀₉	—	2	4	no	—
149	C₂²×C₁₀₉	—	2	8	no	—
150	C₂²×C₁₀₉	C₂×#106	1	8	yes	complemented
151	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₀₉	D₄²	1	1	yes	characteristic
153	C₂²×C₁₀₉	#547	1	1	yes	characteristic
154	C₂²×C₁₀₉	C₂²×SD₁₆	1	2	yes	—
155	C₂²×C₁₀₉	#577	1	2	yes	—
156	C₄×C₁₀₉	—	2	4	no	—
157	C₄×C₁₀₉	—	2	2	no	—
158	C₄×C₁₀₉	—	4	4	no	—
159	C₄×C₁₀₉	—	8	2	no	—
160	C₄×C₁₀₉	—	8	4	no	—
161	C₄×C₁₀₉	C₂×#106	1	4	yes	complemented
162	C₄×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₀₉	#520	1	1	yes	characteristic
164	C₄×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
165	C₄×C₁₀₉	#554	1	1	yes	characteristic
166	D₄×#106	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₁₀₉	—	2	8	no	—
168	C₂×C₁₀₉	D₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₁₀₉	C₂²×#106	1	1	yes	characteristic
170	C₂×C₁₀₉	#2912	1	1	yes	characteristic
171	C₂×C₁₀₉	D₄×SD₁₆	1	2	yes	—
172	C₂×C₁₀₉	#3335	1	2	yes	—
173	C₂×C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic
174	C₂²×#106	C₂×C₁₀₉	1	2	yes	complemented
175	D₄×C₄⋊C₄	C₂×C₁₀₉	1	2	yes	—
176	#1948	C₂×C₁₀₉	1	2	yes	complemented
177	#1853	C₂×C₁₀₉	1	4	yes	complemented
178	#1844	C₂×C₁₀₉	1	1	yes	characteristic, complemented
179	#1807	C₂×C₁₀₉	1	2	yes	complemented
180	C₄×#106	C₂×C₁₀₉	1	1	yes	characteristic, complemented
181	C₁₀₉	D₄×#106	1	1	yes	characteristic, direct product factor, Sylow
182	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
183	C₂×C₄×D₄	—	2	2	no	—
184	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
186	C₈×D₄	C₄×C₁₀₉	1	8	yes	complemented
187	C₂×#106	—	2	8	no	—
188	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
189	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
190	C₂×#106	C₂²×C₁₀₉	1	4	yes	—
191	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
192	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
194	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
196	#389	—	2	2	no	—
197	#385	—	2	4	no	—
198	#384	C₂²×C₁₀₉	1	2	yes	—
199	#380	C₂²×C₁₀₉	1	4	yes	—
200	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
201	#340	C₂²×C₁₀₉	1	4	yes	—
202	#340	C₂²×C₁₀₉	1	2	yes	—
203	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
204	C₂³×C₄	—	2	4	no	—
205	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
209	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
210	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
212	C₄×D₄	—	4	8	no	—
213	C₄×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
214	C₄×D₄	Q₈×C₁₀₉	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	4	4	no	—
220	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
222	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	4	no	—
225	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
228	#106	—	2	4	no	—
229	#106	—	2	2	no	—
230	#106	—	2	8	no	—
231	#106	—	4	2	no	—
232	#106	D₄×C₁₀₉	1	4	yes	direct product factor
233	C₄⋊C₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
234	C₂²⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
235	C₄×C₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
236	#95	—	2	2	no	—
237	#95	—	2	4	no	—
238	C₂⁴	C₂×D₄×C₁₀₉	1	2	yes	—
239	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
240	C₂×D₄	SD₁₆×C₁₀₉	1	8	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	—	4	8	no	—
250	C₂²×C₄	—	8	4	no	—
251	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
256	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	16	no	—
262	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
263	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
264	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
266	C₂×C₈	#41×C₁₀₉	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	8	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
276	C₄⋊C₄	C₂×D₄×C₁₀₉	1	4	yes	—
277	C₂²⋊C₄	—	4	8	no	—
278	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	8	yes	—
279	C₄²	—	4	4	no	—
280	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
281	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	4	no	—
285	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
286	C₂³	C₂×SD₁₆×C₁₀₉	1	8	yes	—
287	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
288	D₄	#106×C₁₀₉	1	16	yes	direct product factor
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	16	no	—
306	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
308	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
309	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	4	yes	—
310	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
312	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
314	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
315	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
316	C₈	—	2	4	no	—
317	C₈	—	2	2	no	—
318	C₈	C₄×D₄×C₁₀₉	1	4	yes	complemented
319	C₂²	—	2	8	no	—
320	C₂²	—	2	8	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	C₂×#106×C₁₀₉	1	8	yes	complemented
324	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
325	C₂²	D₄²×C₁₀₉	1	1	yes	characteristic
326	C₂²	#547×C₁₀₉	1	1	yes	characteristic
327	C₂²	C₂²×SD₁₆×C₁₀₉	1	2	yes	—
328	C₂²	#577×C₁₀₉	1	2	yes	—
329	C₄	—	2	4	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	4	no	—
333	C₄	—	8	2	no	—
334	C₄	C₂×#106×C₁₀₉	1	4	yes	complemented
335	C₄	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
336	C₄	#520×C₁₀₉	1	1	yes	characteristic
337	C₄	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
338	C₄	#554×C₁₀₉	1	1	yes	characteristic
339	C₂	—	2	8	no	—
340	C₂	D₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
341	C₂	C₂²×#106×C₁₀₉	1	1	yes	characteristic
342	C₂	C₁₀₉×#2912	1	1	yes	characteristic
343	C₂	D₄×SD₁₆×C₁₀₉	1	2	yes	—
344	C₂	C₁₀₉×#3335	1	2	yes	—

	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
2. C₂²×#106×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1948	C₂	semidirect
5. C₁₀₉×#1853	C₂	semidirect
6. C₁₀₉×#1844	C₂	semidirect
7. C₁₀₉×#1807	C₂	semidirect
8. C₄×#106×C₁₀₉	C₂	semidirect
13. C₈×D₄×C₁₀₉	C₄	semidirect
35. C₂²×C₈×C₁₀₉	C₂×C₄	semidirect
59. #106×C₁₀₉	D₄	direct (N × H)
60. C₄⋊C₈×C₁₀₉	C₂×C₄	semidirect
61. C₂²⋊C₈×C₁₀₉	C₂×C₄	semidirect
62. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
115. D₄×C₁₀₉	#106	direct (N × H)
145. C₈×C₁₀₉	C₄×D₄	semidirect
150. C₂²×C₁₀₉	C₂×#106	semidirect
161. C₄×C₁₀₉	C₂×#106	semidirect
166. D₄×#106	C₁₀₉	direct (N × H)
174. C₂²×#106	C₂×C₁₀₉	semidirect
176. #1948	C₂×C₁₀₉	semidirect
177. #1853	C₂×C₁₀₉	semidirect
178. #1844	C₂×C₁₀₉	semidirect
179. #1807	C₂×C₁₀₉	semidirect
180. C₄×#106	C₂×C₁₀₉	semidirect
181. C₁₀₉	D₄×#106	direct (N × H)
186. C₈×D₄	C₄×C₁₀₉	semidirect
208. C₂²×C₈	C₂×C₄×C₁₀₉	semidirect
232. #106	D₄×C₁₀₉	direct (N × H)
233. C₄⋊C₈	C₂×C₄×C₁₀₉	semidirect
234. C₂²⋊C₈	C₂×C₄×C₁₀₉	semidirect
235. C₄×C₈	C₂×C₄×C₁₀₉	semidirect
288. D₄	#106×C₁₀₉	direct (N × H)
318. C₈	C₄×D₄×C₁₀₉	semidirect
323. C₂²	C₂×#106×C₁₀₉	semidirect
334. C₄	C₂×#106×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₀₉	2
C₂²×#106×C₁₀₉	2
D₄×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1948	2
C₁₀₉×#1853	2
C₁₀₉×#1844	2
C₁₀₉×#1807	2
C₄×#106×C₁₀₉	2
D₄×#106	109

Subgroups — D4×#106×C109 #42140647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#106×C₁₀₉ #42140647