Subgroups — C₁₀₉×#32670 #42150664

Order 27904.

712 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2568	C₂	1	2	yes	maximal
2	C₁₀₉×#2056	C₂	1	2	yes	maximal
3	C₁₀₉×#2016	C₂	1	4	yes	maximal
4	C₁₀₉×#1984	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1969	C₂	1	2	yes	maximal
6	C₁₀₉×#1928	C₂	1	4	yes	maximal
7	C₂²×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#103×C₁₀₉	C₂²	1	4	yes	—
14	C₂×#103×C₁₀₉	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	#400×C₁₀₉	—	2	2	no	—
17	#400×C₁₀₉	—	2	2	no	—
18	#391×C₁₀₉	—	2	2	no	—
19	#391×C₁₀₉	C₂²	1	4	yes	—
20	#391×C₁₀₉	C₂²	1	4	yes	—
21	#382×C₁₀₉	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	#340×C₁₀₉	C₂²	1	2	yes	—
24	#336×C₁₀₉	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×Q₈×C₁₀₉	C₂³	1	4	yes	—
27	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
28	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
29	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
30	C₂×Q₁₆×C₁₀₉	—	4	4	no	—
31	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
32	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
35	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
36	C₄×Q₈×C₁₀₉	—	4	1	no	—
37	C₄×Q₈×C₁₀₉	—	4	1	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₀₉	—	2	2	no	—
44	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
46	#107×C₁₀₉	—	4	2	no	—
47	C₄⋊C₈×C₁₀₉	—	4	1	no	—
48	#103×C₁₀₉	—	2	8	no	—
49	#103×C₁₀₉	—	4	4	no	—
50	C₄×C₈×C₁₀₉	—	4	1	no	—
51	#95×C₁₀₉	—	2	2	no	—
52	#95×C₁₀₉	—	2	4	no	—
53	#95×C₁₀₉	—	2	4	no	—
54	#95×C₁₀₉	—	2	4	no	—
55	#95×C₁₀₉	—	2	4	no	—
56	C₂×Q₈×C₁₀₉	—	2	4	no	—
57	C₂×Q₈×C₁₀₉	—	2	4	no	—
58	C₂×Q₈×C₁₀₉	—	2	4	no	—
59	C₂×Q₈×C₁₀₉	—	2	4	no	—
60	C₂×Q₈×C₁₀₉	—	2	4	no	—
61	C₂×Q₈×C₁₀₉	—	2	4	no	—
62	C₂×Q₈×C₁₀₉	—	4	1	no	—
63	C₂×Q₈×C₁₀₉	—	4	1	no	—
64	C₂×Q₈×C₁₀₉	Q₁₆	1	4	yes	complemented
65	C₂²×C₄×C₁₀₉	—	2	4	no	—
66	C₂²×C₄×C₁₀₉	—	2	4	no	—
67	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₁₀₉	—	4	8	no	—
73	Q₁₆×C₁₀₉	—	4	8	no	—
74	C₂×C₈×C₁₀₉	—	2	2	no	—
75	C₂×C₈×C₁₀₉	—	2	2	no	—
76	C₂×C₈×C₁₀₉	—	2	4	no	—
77	C₂×C₈×C₁₀₉	—	2	2	no	—
78	C₂×C₈×C₁₀₉	#41	1	4	yes	—
79	C₄⋊C₄×C₁₀₉	—	2	4	no	—
80	C₄⋊C₄×C₁₀₉	—	2	4	no	—
81	C₄⋊C₄×C₁₀₉	—	4	2	no	—
82	C₄⋊C₄×C₁₀₉	—	4	1	no	—
83	C₄⋊C₄×C₁₀₉	—	4	4	no	—
84	C₄⋊C₄×C₁₀₉	—	4	4	no	—
85	C₄²×C₁₀₉	—	2	1	no	—
86	C₄²×C₁₀₉	—	2	1	no	—
87	C₄²×C₁₀₉	—	2	2	no	—
88	C₄²×C₁₀₉	—	2	2	no	—
89	C₄²×C₁₀₉	—	4	4	no	—
90	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₁₀₉	—	4	4	no	—
92	Q₈×C₁₀₉	—	4	4	no	—
93	Q₈×C₁₀₉	—	4	4	no	—
94	Q₈×C₁₀₉	—	4	4	no	—
95	Q₈×C₁₀₉	—	4	4	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	2	no	—
101	C₂×C₄×C₁₀₉	—	2	1	no	—
102	C₂×C₄×C₁₀₉	—	2	2	no	—
103	C₂×C₄×C₁₀₉	—	2	4	no	—
104	C₂×C₄×C₁₀₉	—	4	4	no	—
105	C₂×C₄×C₁₀₉	—	4	4	no	—
106	C₂×C₄×C₁₀₉	—	4	4	no	—
107	C₂×C₄×C₁₀₉	—	4	4	no	—
108	C₂×C₄×C₁₀₉	—	4	4	no	—
109	C₂×C₄×C₁₀₉	—	4	4	no	—
110	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₁₀₉	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
115	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
116	C₈×C₁₀₉	—	2	4	no	—
117	C₈×C₁₀₉	—	4	2	no	—
118	C₂²×C₁₀₉	#534	1	1	yes	characteristic
119	C₂²×C₁₀₉	#550	1	2	yes	—
120	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
121	C₂²×C₁₀₉	#575	1	1	yes	characteristic
122	C₂²×C₁₀₉	#578	1	2	yes	—
123	C₄×C₁₀₉	—	2	4	no	—
124	C₄×C₁₀₉	—	2	2	no	—
125	C₄×C₁₀₉	—	4	2	no	—
126	C₄×C₁₀₉	—	4	1	no	—
127	C₄×C₁₀₉	—	8	4	no	—
128	C₄×C₁₀₉	—	8	4	no	—
129	C₄×C₁₀₉	#538	1	2	yes	—
130	C₄×C₁₀₉	#555	1	1	yes	characteristic
131	C₄×C₁₀₉	H(3,𝔽₄)	1	1	yes	characteristic
132	#32670	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₁₀₉	#2733	1	1	yes	characteristic
134	C₂×C₁₀₉	#3189	1	1	yes	characteristic
135	C₂×C₁₀₉	#3202	1	1	yes	characteristic
136	C₂×C₁₀₉	#3395	1	2	yes	—
137	C₂×C₁₀₉	#3402	1	2	yes	—
138	#2568	C₂×C₁₀₉	1	2	yes	—
139	#2056	C₂×C₁₀₉	1	2	yes	—
140	#2016	C₂×C₁₀₉	1	4	yes	—
141	#1984	C₂×C₁₀₉	1	1	yes	characteristic
142	#1969	C₂×C₁₀₉	1	2	yes	—
143	#1928	C₂×C₁₀₉	1	4	yes	—
144	C₁₀₉	#32670	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
146	C₂²×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
147	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
148	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
149	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
151	C₂×#103	C₂²×C₁₀₉	1	4	yes	—
152	C₂×#103	C₂²×C₁₀₉	1	4	yes	—
153	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
154	#400	—	2	2	no	—
155	#400	—	2	2	no	—
156	#391	—	2	2	no	—
157	#391	C₂²×C₁₀₉	1	4	yes	—
158	#391	C₂²×C₁₀₉	1	4	yes	—
159	#382	—	2	2	no	—
160	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
161	#340	C₂²×C₁₀₉	1	2	yes	—
162	#336	C₂²×C₁₀₉	1	4	yes	—
163	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
164	C₂²×Q₈	C₂³×C₁₀₉	1	4	yes	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	4	2	no	—
167	C₂×Q₁₆	—	4	2	no	—
168	C₂×Q₁₆	—	4	4	no	—
169	C₂×Q₁₆	—	4	2	no	—
170	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
172	C₄⋊Q₈	—	4	2	no	—
173	C₄⋊Q₈	—	4	2	no	—
174	C₄×Q₈	—	4	1	no	—
175	C₄×Q₈	—	4	1	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
179	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
184	#107	—	4	2	no	—
185	C₄⋊C₈	—	4	1	no	—
186	#103	—	2	8	no	—
187	#103	—	4	4	no	—
188	C₄×C₈	—	4	1	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	4	no	—
192	#95	—	2	4	no	—
193	#95	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	Q₁₆×C₁₀₉	1	4	yes	complemented
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
210	Q₁₆	—	4	8	no	—
211	Q₁₆	—	4	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	#41×C₁₀₉	1	4	yes	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	4	4	no	—
228	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	4	4	no	—
233	Q₈	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
249	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	4	yes	—
250	C₂×C₄	C₂×#41×C₁₀₉	1	2	yes	—
251	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
252	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
253	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
254	C₈	—	2	4	no	—
255	C₈	—	4	2	no	—
256	C₂²	#534×C₁₀₉	1	1	yes	characteristic
257	C₂²	#550×C₁₀₉	1	2	yes	—
258	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
259	C₂²	#575×C₁₀₉	1	1	yes	characteristic
260	C₂²	#578×C₁₀₉	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	4	no	—
266	C₄	—	8	4	no	—
267	C₄	#538×C₁₀₉	1	2	yes	—
268	C₄	#555×C₁₀₉	1	1	yes	characteristic
269	C₄	H(3,𝔽₄)×C₁₀₉	1	1	yes	characteristic
270	C₂	C₁₀₉×#2733	1	1	yes	characteristic
271	C₂	C₁₀₉×#3189	1	1	yes	characteristic
272	C₂	C₁₀₉×#3202	1	1	yes	characteristic
273	C₂	C₁₀₉×#3395	1	2	yes	—
274	C₂	C₁₀₉×#3402	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
64. C₂×Q₈×C₁₀₉	Q₁₆	semidirect
132. #32670	C₁₀₉	direct (N × H)
144. C₁₀₉	#32670	direct (N × H)
202. C₂×Q₈	Q₁₆×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2568	2
C₁₀₉×#2056	2
C₁₀₉×#2016	2
C₁₀₉×#1984	2
C₁₀₉×#1969	2
C₁₀₉×#1928	2
#32670	109

Subgroups — C109×#32670 #42150664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32670 #42150664