Subgroups — C₁₀₉×#22747 #42140741

Order 27904.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1950	C₂	1	4	yes	maximal
4	C₁₀₉×#1950	C₂	1	2	yes	maximal
5	C₁₀₉×#1838	C₂	1	2	yes	maximal
6	C₁₀₉×#1837	C₂	1	1	yes	maximal, characteristic
7	C₁₀₉×#1778	C₂	1	2	yes	maximal
8	C₄×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	#446×C₁₀₉	—	2	2	no	—
11	#446×C₁₀₉	—	2	4	no	—
12	#446×C₁₀₉	C₄	1	4	yes	—
13	#446×C₁₀₉	C₄	1	4	yes	—
14	C₈×Q₈×C₁₀₉	—	2	2	no	—
15	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
16	C₂×#108×C₁₀₉	C₂²	1	4	yes	—
17	#423×C₁₀₉	C₂²	1	2	yes	—
18	#423×C₁₀₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
21	C₄×M₁₆×C₁₀₉	C₂²	1	2	yes	—
22	C₄×M₁₆×C₁₀₉	C₂²	1	4	yes	—
23	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	#341×C₁₀₉	C₂²	1	2	yes	—
26	#341×C₁₀₉	C₂²	1	4	yes	—
27	#341×C₁₀₉	C₂²	1	1	yes	characteristic
28	#341×C₁₀₉	C₂²	1	1	yes	characteristic
29	#341×C₁₀₉	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₁₀₉	C₂³	1	4	yes	—
32	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
33	C₂×M₁₆×C₁₀₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₄×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
40	C₄×Q₈×C₁₀₉	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	2	yes	—
43	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
45	#108×C₁₀₉	—	2	4	no	—
46	#108×C₁₀₉	—	2	8	no	—
47	C₄⋊C₈×C₁₀₉	—	2	2	no	—
48	C₄⋊C₈×C₁₀₉	—	2	4	no	—
49	C₄⋊C₈×C₁₀₉	—	2	4	no	—
50	C₄⋊C₈×C₁₀₉	—	2	2	no	—
51	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₄⋊C₈×C₁₀₉	C₂×C₄	1	8	yes	—
54	#97×C₁₀₉	—	2	4	no	—
55	#97×C₁₀₉	—	2	2	no	—
56	#97×C₁₀₉	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₁₀₉	—	2	2	no	—
58	C₄×C₈×C₁₀₉	—	2	4	no	—
59	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
61	C₂×Q₈×C₁₀₉	—	4	2	no	—
62	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
63	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₀₉	—	2	2	no	—
70	M₁₆×C₁₀₉	—	2	2	no	—
71	M₁₆×C₁₀₉	—	2	4	no	—
72	M₁₆×C₁₀₉	—	2	4	no	—
73	M₁₆×C₁₀₉	—	2	2	no	—
74	M₁₆×C₁₀₉	#41	1	4	yes	—
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	2	2	no	—
77	C₂×C₈×C₁₀₉	—	2	4	no	—
78	C₂×C₈×C₁₀₉	—	2	2	no	—
79	C₂×C₈×C₁₀₉	—	2	2	no	—
80	C₂×C₈×C₁₀₉	—	2	4	no	—
81	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
93	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
95	Q₈×C₁₀₉	—	2	4	no	—
96	Q₈×C₁₀₉	—	4	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	4	1	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
110	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₁₀₉	—	2	2	no	—
112	C₈×C₁₀₉	—	2	4	no	—
113	C₈×C₁₀₉	—	2	2	no	—
114	C₈×C₁₀₉	—	2	4	no	—
115	C₈×C₁₀₉	—	4	2	no	—
116	C₈×C₁₀₉	—	4	4	no	—
117	C₂²×C₁₀₉	—	2	1	no	—
118	C₂²×C₁₀₉	—	2	1	no	—
119	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₁₀₉	#557	1	2	yes	—
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	—	2	4	no	—
123	C₄×C₁₀₉	—	2	1	no	—
124	C₄×C₁₀₉	—	2	1	no	—
125	C₄×C₁₀₉	—	2	4	no	—
126	C₄×C₁₀₉	#430	1	2	yes	—
127	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
128	C₄×C₁₀₉	#519	1	1	yes	characteristic
129	C₄×C₁₀₉	#548	1	2	yes	—
130	#22747	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₁₀₉	—	2	2	no	—
132	C₂×C₁₀₉	#2351	1	1	yes	characteristic
133	C₂×C₁₀₉	C₂×#430	1	1	yes	characteristic
134	C₂×C₁₀₉	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	C₂×C₁₀₉	1	2	yes	—
136	C₂×#446	C₂×C₁₀₉	1	1	yes	characteristic
137	#1950	C₂×C₁₀₉	1	4	yes	—
138	#1950	C₂×C₁₀₉	1	2	yes	—
139	#1838	C₂×C₁₀₉	1	2	yes	—
140	#1837	C₂×C₁₀₉	1	1	yes	characteristic
141	#1778	C₂×C₁₀₉	1	2	yes	—
142	C₄×#108	C₂×C₁₀₉	1	1	yes	characteristic
143	C₁₀₉	#22747	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
145	#446	—	2	2	no	—
146	#446	—	2	4	no	—
147	#446	C₄×C₁₀₉	1	4	yes	—
148	#446	C₄×C₁₀₉	1	4	yes	—
149	C₈×Q₈	—	2	2	no	—
150	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
151	C₂×#108	C₂²×C₁₀₉	1	4	yes	—
152	#423	C₂²×C₁₀₉	1	2	yes	—
153	#423	C₂²×C₁₀₉	1	4	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
156	C₄×M₁₆	C₂²×C₁₀₉	1	2	yes	—
157	C₄×M₁₆	C₂²×C₁₀₉	1	4	yes	—
158	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
160	#341	C₂²×C₁₀₉	1	2	yes	—
161	#341	C₂²×C₁₀₉	1	2	yes	—
162	#341	C₂²×C₁₀₉	1	1	yes	characteristic
163	#341	C₂²×C₁₀₉	1	4	yes	—
164	#341	C₂²×C₁₀₉	1	1	yes	characteristic
165	C₂²×Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
166	C₂×M₁₆	C₂³×C₁₀₉	1	4	yes	—
167	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
168	C₂×M₁₆	C₂³×C₁₀₉	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
172	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
173	C₄×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
174	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
175	C₄×Q₈	D₄×C₁₀₉	1	2	yes	—
176	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
180	#108	—	2	4	no	—
181	#108	—	2	8	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
187	C₄⋊C₈	C₂×C₄×C₁₀₉	1	2	yes	—
188	C₄⋊C₈	C₂×C₄×C₁₀₉	1	8	yes	—
189	#97	—	2	4	no	—
190	#97	—	2	2	no	—
191	#97	C₂×C₄×C₁₀₉	1	8	yes	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	4	no	—
194	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
195	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
198	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	#41×C₁₀₉	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₁₀₉	1	4	yes	—
219	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
220	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
221	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
222	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
223	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
226	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
228	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
229	C₂³	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
245	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	4	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
255	C₂²	#557×C₁₀₉	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#430×C₁₀₉	1	2	yes	—
262	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
263	C₄	#519×C₁₀₉	1	1	yes	characteristic
264	C₄	#548×C₁₀₉	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	C₁₀₉×#2351	1	1	yes	characteristic
267	C₂	C₂×#430×C₁₀₉	1	1	yes	characteristic
268	C₂	C₁₀₉×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
130. #22747	C₁₀₉	direct (N × H)
143. C₁₀₉	#22747	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₁₀₉	2
C₂×#446×C₁₀₉	2
C₁₀₉×#1950	2
C₁₀₉×#1838	2
C₁₀₉×#1837	2
C₁₀₉×#1778	2
C₄×#108×C₁₀₉	2
#22747	109

Subgroups — C109×#22747 #42140741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22747 #42140741