Subgroups — C₁₀₉×#9677 #42127671

Order 27904.

552 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈×C₁₀₉	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₀₉	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×#97×C₁₀₉	—	2	1	no	—
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
11	#391×C₁₀₉	C₄	1	2	yes	—
12	#389×C₁₀₉	—	2	1	no	—
13	#389×C₁₀₉	—	2	1	no	—
14	#384×C₁₀₉	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₀₉	—	2	1	no	—
16	C₄×C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
18	#336×C₁₀₉	—	2	1	no	—
19	#336×C₁₀₉	—	2	1	no	—
20	Q₈⋊C₈×C₁₀₉	—	2	8	no	—
21	C₂²×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
25	C₄×Q₈×C₁₀₉	—	2	4	no	—
26	C₄×Q₈×C₁₀₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₀₉	—	2	2	no	—
34	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₁₀₉	—	2	4	no	—
42	C₄⋊C₈×C₁₀₉	—	2	2	no	—
43	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
44	#97×C₁₀₉	—	4	2	no	—
45	C₄×C₈×C₁₀₉	—	2	4	no	—
46	#95×C₁₀₉	—	2	2	no	—
47	#95×C₁₀₉	—	2	2	no	—
48	C₂×Q₈×C₁₀₉	—	2	2	no	—
49	C₂×Q₈×C₁₀₉	—	2	4	no	—
50	C₂×Q₈×C₁₀₉	—	2	4	no	—
51	C₂×Q₈×C₁₀₉	M₁₆	1	4	yes	—
52	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₀₉	—	2	4	no	—
62	C₂×C₈×C₁₀₉	—	2	4	no	—
63	C₂×C₈×C₁₀₉	—	4	2	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	2	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₄⋊C₄×C₁₀₉	—	2	4	no	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₄⋊C₄×C₁₀₉	—	2	4	no	—
70	C₄⋊C₄×C₁₀₉	—	4	2	no	—
71	C₄⋊C₄×C₁₀₉	—	4	2	no	—
72	C₄⋊C₄×C₁₀₉	—	4	2	no	—
73	C₄⋊C₄×C₁₀₉	—	4	2	no	—
74	C₄²×C₁₀₉	—	2	2	no	—
75	C₄²×C₁₀₉	—	2	2	no	—
76	C₄²×C₁₀₉	—	2	2	no	—
77	C₄²×C₁₀₉	—	2	2	no	—
78	C₄²×C₁₀₉	—	4	1	no	—
79	C₄²×C₁₀₉	—	4	4	no	—
80	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
81	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
82	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
83	C₄²×C₁₀₉	#41	1	2	yes	—
84	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₀₉	—	2	8	no	—
86	Q₈×C₁₀₉	—	4	4	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	4	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	4	2	no	—
102	C₂×C₄×C₁₀₉	—	4	2	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	#103	1	2	yes	—
105	C₂×C₄×C₁₀₉	#103	1	2	yes	—
106	C₂×C₄×C₁₀₉	#104	1	2	yes	—
107	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₀₉	#121	1	2	yes	—
109	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
111	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₀₉	#131	1	2	yes	—
113	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	#137	1	2	yes	—
116	C₈×C₁₀₉	—	4	4	no	—
117	C₈×C₁₀₉	—	4	4	no	—
118	C₂²×C₁₀₉	#390	1	1	yes	characteristic
119	C₂²×C₁₀₉	#407	1	1	yes	characteristic
120	C₂²×C₁₀₉	#408	1	1	yes	characteristic
121	C₂²×C₁₀₉	C₂×#103	1	1	yes	characteristic
122	C₂²×C₁₀₉	#419	1	1	yes	characteristic
123	C₂²×C₁₀₉	C₂×#104	1	1	yes	characteristic
124	C₂²×C₁₀₉	#421	1	1	yes	characteristic
125	C₄×C₁₀₉	—	2	1	no	—
126	C₄×C₁₀₉	—	2	1	no	—
127	C₄×C₁₀₉	—	2	2	no	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	2	2	no	—
130	C₄×C₁₀₉	—	4	4	no	—
131	C₄×C₁₀₉	—	4	1	no	—
132	C₄×C₁₀₉	—	8	2	no	—
133	C₄×C₁₀₉	#435	1	4	yes	—
134	C₄×C₁₀₉	#461	1	2	yes	—
135	C₄×C₁₀₉	#462	1	2	yes	—
136	#9677	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₁₀₉	#1487	1	1	yes	characteristic
138	C₂×C₁₀₉	#1491	1	1	yes	characteristic
139	C₂×C₁₀₉	#1493	1	1	yes	characteristic
140	C₂×C₁₀₉	#1497	1	1	yes	characteristic
141	C₂×C₁₀₉	#1955	1	1	yes	characteristic
142	C₂×C₁₀₉	#1980	1	1	yes	characteristic
143	C₂×C₁₀₉	#1981	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
145	#1767	C₂×C₁₀₉	1	1	yes	characteristic
146	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
147	C₂×Q₈⋊C₈	C₂×C₁₀₉	1	4	yes	—
148	C₁₀₉	#9677	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×C₄⋊Q₈	C₄×C₁₀₉	1	1	yes	characteristic
150	C₂×C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
153	C₂×#97	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
155	#391	C₄×C₁₀₉	1	2	yes	—
156	#389	—	2	1	no	—
157	#389	—	2	1	no	—
158	#384	C₄×C₁₀₉	1	1	yes	characteristic
159	C₄×C₄⋊C₄	—	2	1	no	—
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
162	#336	—	2	1	no	—
163	#336	—	2	1	no	—
164	Q₈⋊C₈	—	2	8	no	—
165	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	4	no	—
169	C₄×Q₈	—	2	4	no	—
170	C₄×Q₈	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
188	#97	—	4	2	no	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	M₁₆×C₁₀₉	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	4	no	—
224	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
225	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
226	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
227	C₄²	#41×C₁₀₉	1	2	yes	—
228	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#103×C₁₀₉	1	2	yes	—
249	C₂×C₄	#103×C₁₀₉	1	2	yes	—
250	C₂×C₄	#104×C₁₀₉	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
252	C₂×C₄	#121×C₁₀₉	1	2	yes	—
253	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
254	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
256	C₂×C₄	#131×C₁₀₉	1	2	yes	—
257	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
259	C₂×C₄	#137×C₁₀₉	1	2	yes	—
260	C₈	—	4	4	no	—
261	C₈	—	4	4	no	—
262	C₂²	#390×C₁₀₉	1	1	yes	characteristic
263	C₂²	#407×C₁₀₉	1	1	yes	characteristic
264	C₂²	#408×C₁₀₉	1	1	yes	characteristic
265	C₂²	C₂×#103×C₁₀₉	1	1	yes	characteristic
266	C₂²	#419×C₁₀₉	1	1	yes	characteristic
267	C₂²	C₂×#104×C₁₀₉	1	1	yes	characteristic
268	C₂²	#421×C₁₀₉	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	1	no	—
276	C₄	—	8	2	no	—
277	C₄	#435×C₁₀₉	1	4	yes	—
278	C₄	#461×C₁₀₉	1	2	yes	—
279	C₄	#462×C₁₀₉	1	2	yes	—
280	C₂	C₁₀₉×#1487	1	1	yes	characteristic
281	C₂	C₁₀₉×#1491	1	1	yes	characteristic
282	C₂	C₁₀₉×#1493	1	1	yes	characteristic
283	C₂	C₁₀₉×#1497	1	1	yes	characteristic
284	C₂	C₁₀₉×#1955	1	1	yes	characteristic
285	C₂	C₁₀₉×#1980	1	1	yes	characteristic
286	C₂	C₁₀₉×#1981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
136. #9677	C₁₀₉	direct (N × H)
148. C₁₀₉	#9677	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₀₉	2
C₁₀₉×#1767	2
C₄×C₄⋊C₈×C₁₀₉	2
C₂×Q₈⋊C₈×C₁₀₉	2
#9677	109

Subgroups — C109×#9677 #42127671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9677 #42127671