Subgroups — C₁₀₉×#32684 #42150678

Order 27904.

648 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2567	C₂	1	2	yes	maximal
2	C₁₀₉×#2087	C₂	1	2	yes	maximal
3	C₁₀₉×#2057	C₂	1	4	yes	maximal
4	C₁₀₉×#1984	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1942	C₂	1	2	yes	maximal
6	C₁₀₉×#1921	C₂	1	4	yes	maximal
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
10	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#395×C₁₀₉	—	2	2	no	—
15	#395×C₁₀₉	—	2	2	no	—
16	#391×C₁₀₉	—	2	2	no	—
17	#389×C₁₀₉	C₂²	1	4	yes	—
18	#389×C₁₀₉	C₂²	1	4	yes	—
19	#384×C₁₀₉	—	2	2	no	—
20	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	#340×C₁₀₉	C₂²	1	2	yes	—
22	#340×C₁₀₉	C₂²	1	4	yes	—
23	#340×C₁₀₉	C₂²	1	4	yes	—
24	#336×C₁₀₉	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₀₉	—	2	2	no	—
26	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
30	C₄⋊Q₈×C₁₀₉	—	4	2	no	—
31	C₄×Q₈×C₁₀₉	—	4	1	no	—
32	C₄×Q₈×C₁₀₉	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₀₉	—	2	2	no	—
41	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
43	#107×C₁₀₉	—	4	2	no	—
44	#107×C₁₀₉	—	4	2	no	—
45	#107×C₁₀₉	Q₈	1	8	yes	complemented
46	C₄⋊C₈×C₁₀₉	—	4	1	no	—
47	C₄×C₈×C₁₀₉	—	4	1	no	—
48	#95×C₁₀₉	—	2	2	no	—
49	#95×C₁₀₉	—	2	4	no	—
50	#95×C₁₀₉	—	2	4	no	—
51	C₂×Q₈×C₁₀₉	—	4	1	no	—
52	C₂×Q₈×C₁₀₉	—	4	1	no	—
53	C₂×Q₈×C₁₀₉	—	4	4	no	—
54	C₂×Q₈×C₁₀₉	—	8	4	no	—
55	C₂×Q₈×C₁₀₉	Q₁₆	1	4	yes	—
56	C₂²×C₄×C₁₀₉	—	2	4	no	—
57	C₂²×C₄×C₁₀₉	—	2	4	no	—
58	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	—	2	2	no	—
64	C₂×C₈×C₁₀₉	—	2	2	no	—
65	C₂×C₈×C₁₀₉	—	2	2	no	—
66	C₂×C₈×C₁₀₉	—	2	4	no	—
67	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
68	C₄⋊C₄×C₁₀₉	—	2	4	no	—
69	C₄⋊C₄×C₁₀₉	—	2	4	no	—
70	C₄⋊C₄×C₁₀₉	—	2	4	no	—
71	C₄⋊C₄×C₁₀₉	—	4	2	no	—
72	C₄⋊C₄×C₁₀₉	—	4	1	no	—
73	C₄⋊C₄×C₁₀₉	—	4	4	no	—
74	C₄⋊C₄×C₁₀₉	—	8	4	no	—
75	C₄⋊C₄×C₁₀₉	—	8	4	no	—
76	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	—	2	2	no	—
80	C₄²×C₁₀₉	—	2	2	no	—
81	C₄²×C₁₀₉	—	4	4	no	—
82	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	4	4	no	—
84	Q₈×C₁₀₉	—	8	8	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	1	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	8	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	4	no	—
94	C₂×C₄×C₁₀₉	—	4	4	no	—
95	C₂×C₄×C₁₀₉	—	4	4	no	—
96	C₂×C₄×C₁₀₉	—	4	4	no	—
97	C₂×C₄×C₁₀₉	—	4	4	no	—
98	C₂×C₄×C₁₀₉	—	4	4	no	—
99	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	4	yes	—
101	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
105	C₈×C₁₀₉	—	2	4	no	—
106	C₈×C₁₀₉	—	4	2	no	—
107	C₂²×C₁₀₉	#534	1	1	yes	characteristic
108	C₂²×C₁₀₉	D₄×Q₈	1	2	yes	—
109	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
110	C₂²×C₁₀₉	#575	1	1	yes	characteristic
111	C₂²×C₁₀₉	#576	1	2	yes	—
112	C₄×C₁₀₉	—	2	4	no	—
113	C₄×C₁₀₉	—	2	2	no	—
114	C₄×C₁₀₉	—	4	8	no	—
115	C₄×C₁₀₉	—	4	1	no	—
116	C₄×C₁₀₉	—	4	2	no	—
117	C₄×C₁₀₉	—	8	4	no	—
118	C₄×C₁₀₉	#544	1	2	yes	—
119	C₄×C₁₀₉	Q₈²	1	1	yes	characteristic
120	C₄×C₁₀₉	#563	1	1	yes	characteristic
121	#32684	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₁₀₉	#2734	1	1	yes	characteristic
123	C₂×C₁₀₉	#3189	1	1	yes	characteristic
124	C₂×C₁₀₉	#3199	1	1	yes	characteristic
125	C₂×C₁₀₉	Q₈×Q₁₆	1	2	yes	—
126	C₂×C₁₀₉	#3389	1	2	yes	—
127	#2567	C₂×C₁₀₉	1	2	yes	—
128	#2087	C₂×C₁₀₉	1	2	yes	—
129	#2057	C₂×C₁₀₉	1	4	yes	—
130	#1984	C₂×C₁₀₉	1	1	yes	characteristic
131	#1942	C₂×C₁₀₉	1	2	yes	—
132	#1921	C₂×C₁₀₉	1	4	yes	—
133	C₁₀₉	#32684	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
136	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
137	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
138	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
141	#395	—	2	2	no	—
142	#395	—	2	2	no	—
143	#391	—	2	2	no	—
144	#389	C₂²×C₁₀₉	1	4	yes	—
145	#389	C₂²×C₁₀₉	1	4	yes	—
146	#384	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
148	#340	C₂²×C₁₀₉	1	4	yes	—
149	#340	C₂²×C₁₀₉	1	4	yes	—
150	#340	C₂²×C₁₀₉	1	2	yes	—
151	#336	C₂²×C₁₀₉	1	4	yes	—
152	C₂²×Q₈	—	2	2	no	—
153	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₄⋊Q₈	—	4	2	no	—
158	C₄×Q₈	—	4	1	no	—
159	C₄×Q₈	—	4	1	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
164	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
169	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
170	#107	—	4	2	no	—
171	#107	—	4	2	no	—
172	#107	Q₈×C₁₀₉	1	8	yes	complemented
173	C₄⋊C₈	—	4	1	no	—
174	C₄×C₈	—	4	1	no	—
175	#95	—	2	2	no	—
176	#95	—	2	4	no	—
177	#95	—	2	4	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×Q₈	—	8	4	no	—
182	C₂×Q₈	Q₁₆×C₁₀₉	1	4	yes	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	8	4	no	—
202	C₄⋊C₄	—	8	4	no	—
203	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	8	yes	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	4	4	no	—
209	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	Q₈	—	8	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
227	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	4	yes	—
228	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
229	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
230	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
231	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₂²	#534×C₁₀₉	1	1	yes	characteristic
235	C₂²	D₄×Q₈×C₁₀₉	1	2	yes	—
236	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
237	C₂²	#575×C₁₀₉	1	1	yes	characteristic
238	C₂²	#576×C₁₀₉	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	8	no	—
244	C₄	—	8	4	no	—
245	C₄	#544×C₁₀₉	1	2	yes	—
246	C₄	Q₈²×C₁₀₉	1	1	yes	characteristic
247	C₄	#563×C₁₀₉	1	1	yes	characteristic
248	C₂	C₁₀₉×#2734	1	1	yes	characteristic
249	C₂	C₁₀₉×#3189	1	1	yes	characteristic
250	C₂	C₁₀₉×#3199	1	1	yes	characteristic
251	C₂	Q₈×Q₁₆×C₁₀₉	1	2	yes	—
252	C₂	C₁₀₉×#3389	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
45. #107×C₁₀₉	Q₈	semidirect
121. #32684	C₁₀₉	direct (N × H)
133. C₁₀₉	#32684	direct (N × H)
172. #107	Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2567	2
C₁₀₉×#2087	2
C₁₀₉×#2057	2
C₁₀₉×#1984	2
C₁₀₉×#1942	2
C₁₀₉×#1921	2
#32684	109

Subgroups — C109×#32684 #42150678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32684 #42150678