Subgroups — C₁₀₉×#32682 #42150676

Order 27904.

688 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2546	C₂	1	2	yes	maximal
2	C₁₀₉×#2086	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×#1918	C₂	1	4	yes	maximal
7	C₂×C₄⋊Q₈×C₁₀₉	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#398×C₁₀₉	—	2	2	no	—
16	#398×C₁₀₉	—	2	2	no	—
17	#391×C₁₀₉	—	2	2	no	—
18	#389×C₁₀₉	C₂²	1	4	yes	—
19	#389×C₁₀₉	C₂²	1	4	yes	—
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	8	no	—
32	C₄×Q₈×C₁₀₉	—	4	1	no	—
33	C₄×Q₈×C₁₀₉	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₀₉	—	2	2	no	—
42	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	#107×C₁₀₉	—	4	2	no	—
45	#107×C₁₀₉	—	4	2	no	—
46	#107×C₁₀₉	Q₈	1	8	yes	complemented
47	C₄⋊C₈×C₁₀₉	—	4	1	no	—
48	C₄×C₈×C₁₀₉	—	4	1	no	—
49	#95×C₁₀₉	—	2	2	no	—
50	#95×C₁₀₉	—	2	4	no	—
51	#95×C₁₀₉	—	2	4	no	—
52	C₂×Q₈×C₁₀₉	—	4	1	no	—
53	C₂×Q₈×C₁₀₉	—	4	1	no	—
54	C₂×Q₈×C₁₀₉	—	4	4	no	—
55	C₂×Q₈×C₁₀₉	—	4	4	no	—
56	C₂×Q₈×C₁₀₉	—	4	4	no	—
57	C₂×Q₈×C₁₀₉	D₈	1	4	yes	—
58	C₂²×C₄×C₁₀₉	—	2	4	no	—
59	C₂²×C₄×C₁₀₉	—	2	4	no	—
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₁₀₉	—	2	2	no	—
66	C₂×C₈×C₁₀₉	—	2	2	no	—
67	C₂×C₈×C₁₀₉	—	2	2	no	—
68	C₂×C₈×C₁₀₉	—	2	4	no	—
69	C₂×C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
70	C₄⋊C₄×C₁₀₉	—	2	4	no	—
71	C₄⋊C₄×C₁₀₉	—	2	4	no	—
72	C₄⋊C₄×C₁₀₉	—	2	4	no	—
73	C₄⋊C₄×C₁₀₉	—	4	1	no	—
74	C₄⋊C₄×C₁₀₉	—	4	2	no	—
75	C₄⋊C₄×C₁₀₉	—	4	4	no	—
76	C₄⋊C₄×C₁₀₉	—	4	8	no	—
77	C₄⋊C₄×C₁₀₉	—	4	8	no	—
78	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
79	C₄²×C₁₀₉	—	2	1	no	—
80	C₄²×C₁₀₉	—	2	1	no	—
81	C₄²×C₁₀₉	—	2	2	no	—
82	C₄²×C₁₀₉	—	2	2	no	—
83	C₄²×C₁₀₉	—	4	4	no	—
84	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₀₉	—	4	4	no	—
86	Q₈×C₁₀₉	—	8	8	no	—
87	C₂×C₄×C₁₀₉	—	2	1	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	2	1	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	8	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	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	—	4	4	no	—
100	C₂×C₄×C₁₀₉	—	4	4	no	—
101	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
102	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
107	C₈×C₁₀₉	—	2	4	no	—
108	C₈×C₁₀₉	—	4	2	no	—
109	C₂²×C₁₀₉	#534	1	1	yes	characteristic
110	C₂²×C₁₀₉	D₄×Q₈	1	2	yes	—
111	C₂²×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
112	C₂²×C₁₀₉	#575	1	1	yes	characteristic
113	C₂²×C₁₀₉	#578	1	2	yes	—
114	C₄×C₁₀₉	—	2	2	no	—
115	C₄×C₁₀₉	—	2	4	no	—
116	C₄×C₁₀₉	—	4	8	no	—
117	C₄×C₁₀₉	—	4	2	no	—
118	C₄×C₁₀₉	—	4	1	no	—
119	C₄×C₁₀₉	—	8	4	no	—
120	C₄×C₁₀₉	#544	1	2	yes	—
121	C₄×C₁₀₉	Q₈²	1	1	yes	characteristic
122	C₄×C₁₀₉	#563	1	1	yes	characteristic
123	#32682	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₁₀₉	#2734	1	1	yes	characteristic
125	C₂×C₁₀₉	#3186	1	1	yes	characteristic
126	C₂×C₁₀₉	#3202	1	1	yes	characteristic
127	C₂×C₁₀₉	Q₈×D₈	1	2	yes	—
128	C₂×C₁₀₉	#3390	1	2	yes	—
129	#2546	C₂×C₁₀₉	1	2	yes	—
130	#2086	C₂×C₁₀₉	1	2	yes	—
131	#2057	C₂×C₁₀₉	1	4	yes	—
132	#1984	C₂×C₁₀₉	1	1	yes	characteristic
133	#1942	C₂×C₁₀₉	1	2	yes	—
134	#1918	C₂×C₁₀₉	1	4	yes	—
135	C₁₀₉	#32682	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×C₄⋊Q₈	—	2	2	no	—
137	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
138	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
140	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
141	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
142	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
144	#398	—	2	2	no	—
145	#398	—	2	2	no	—
146	#391	—	2	2	no	—
147	#389	C₂²×C₁₀₉	1	4	yes	—
148	#389	C₂²×C₁₀₉	1	4	yes	—
149	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
150	#340	C₂²×C₁₀₉	1	4	yes	—
151	#340	C₂²×C₁₀₉	1	4	yes	—
152	#340	C₂²×C₁₀₉	1	2	yes	—
153	#336	C₂²×C₁₀₉	1	4	yes	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
157	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
158	C₄⋊Q₈	—	4	2	no	—
159	C₄⋊Q₈	—	4	2	no	—
160	C₄⋊Q₈	—	4	8	no	—
161	C₄×Q₈	—	4	1	no	—
162	C₄×Q₈	—	4	1	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
167	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
173	#107	—	4	2	no	—
174	#107	—	4	2	no	—
175	#107	Q₈×C₁₀₉	1	8	yes	complemented
176	C₄⋊C₈	—	4	1	no	—
177	C₄×C₈	—	4	1	no	—
178	#95	—	2	2	no	—
179	#95	—	2	4	no	—
180	#95	—	2	4	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	D₈×C₁₀₉	1	4	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂×Q₈×C₁₀₉	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	1	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	8	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	8	yes	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	4	no	—
213	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	8	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₂×D₈×C₁₀₉	1	4	yes	—
231	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
232	C₂×C₄	C₂²×Q₈×C₁₀₉	1	2	yes	—
233	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	4	2	no	—
238	C₂²	#534×C₁₀₉	1	1	yes	characteristic
239	C₂²	D₄×Q₈×C₁₀₉	1	2	yes	—
240	C₂²	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
241	C₂²	#575×C₁₀₉	1	1	yes	characteristic
242	C₂²	#578×C₁₀₉	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	8	no	—
248	C₄	—	8	4	no	—
249	C₄	#544×C₁₀₉	1	2	yes	—
250	C₄	Q₈²×C₁₀₉	1	1	yes	characteristic
251	C₄	#563×C₁₀₉	1	1	yes	characteristic
252	C₂	C₁₀₉×#2734	1	1	yes	characteristic
253	C₂	C₁₀₉×#3186	1	1	yes	characteristic
254	C₂	C₁₀₉×#3202	1	1	yes	characteristic
255	C₂	Q₈×D₈×C₁₀₉	1	2	yes	—
256	C₂	C₁₀₉×#3390	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
46. #107×C₁₀₉	Q₈	semidirect
123. #32682	C₁₀₉	direct (N × H)
135. C₁₀₉	#32682	direct (N × H)
175. #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×#2546	2
C₁₀₉×#2086	2
C₁₀₉×#2057	2
C₁₀₉×#1984	2
C₁₀₉×#1942	2
C₁₀₉×#1918	2
#32682	109

Subgroups — C109×#32682 #42150676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32682 #42150676