Subgroups — C₁₀₉×#3518 #28360772

Order 13952.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#556×C₁₀₉	C₂	1	6	yes	maximal, complemented
3	#556×C₁₀₉	C₂	1	3	yes	maximal, complemented
4	#555×C₁₀₉	C₂	1	3	yes	maximal, complemented
5	D₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#548×C₁₀₉	C₂	1	2	yes	maximal, complemented
7	#542×C₁₀₉	C₂	1	6	yes	maximal, complemented
8	#533×C₁₀₉	C₂	1	3	yes	maximal, complemented
9	#529×C₁₀₉	C₂	1	3	yes	maximal, complemented
10	#519×C₁₀₉	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
15	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
17	#126×C₁₀₉	C₂²	1	6	yes	—
18	#126×C₁₀₉	C₂²	1	12	yes	—
19	#125×C₁₀₉	C₂²	1	3	yes	—
20	#124×C₁₀₉	C₂²	1	1	yes	characteristic
21	#124×C₁₀₉	C₂²	1	6	yes	—
22	#124×C₁₀₉	C₂²	1	12	yes	—
23	#123×C₁₀₉	C₂²	1	6	yes	—
24	C₂²⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
25	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
26	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
27	C₂²⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
28	#121×C₁₀₉	C₂²	1	3	yes	—
29	#121×C₁₀₉	C₂²	1	6	yes	—
30	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	—
33	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
34	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	—
35	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
36	C₄×Q₈×C₁₀₉	C₂²	1	6	yes	—
37	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
38	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
39	C₄×D₄×C₁₀₉	C₂²	1	6	yes	—
40	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
41	#117×C₁₀₉	C₂²	1	3	yes	—
42	#117×C₁₀₉	C₂²	1	6	yes	—
43	#117×C₁₀₉	C₂²	1	6	yes	—
44	#117×C₁₀₉	C₂²	1	6	yes	—
45	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
46	#41×C₁₀₉	—	2	4	no	—
47	C₂×Q₈×C₁₀₉	—	2	8	no	—
48	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
50	C₂×Q₈×C₁₀₉	C₂³	1	6	yes	—
51	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
54	C₂×D₄×C₁₀₉	C₂³	1	6	yes	—
55	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
56	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
57	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
58	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
59	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
60	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
61	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
62	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
63	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
64	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
65	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
66	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
67	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
68	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
69	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
70	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
72	C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
73	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
74	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
75	C₂²⋊C₄×C₁₀₉	C₂³	1	6	yes	—
76	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄²×C₁₀₉	C₂³	1	3	yes	—
78	C₄²×C₁₀₉	C₂³	1	3	yes	—
79	C₄²×C₁₀₉	C₂³	1	3	yes	—
80	C₄²×C₁₀₉	C₂³	1	12	yes	—
81	C₄²×C₁₀₉	C₂³	1	6	yes	—
82	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
83	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
84	Q₈×C₁₀₉	—	2	12	no	—
85	Q₈×C₁₀₉	—	2	2	no	—
86	Q₈×C₁₀₉	—	4	6	no	—
87	Q₈×C₁₀₉	#41	1	4	yes	—
88	D₄×C₁₀₉	—	2	6	no	—
89	D₄×C₁₀₉	—	2	2	no	—
90	D₄×C₁₀₉	—	4	6	no	—
91	C₂×C₄×C₁₀₉	—	2	12	no	—
92	C₂×C₄×C₁₀₉	—	2	6	no	—
93	C₂×C₄×C₁₀₉	—	2	6	no	—
94	C₂×C₄×C₁₀₉	—	4	3	no	—
95	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
100	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
101	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	—	2	3	no	—
106	C₂²×C₁₀₉	—	2	3	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	3	no	—
109	C₂²×C₁₀₉	—	2	2	no	—
110	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
111	C₄×C₁₀₉	—	2	1	no	—
112	C₄×C₁₀₉	—	2	6	no	—
113	C₄×C₁₀₉	—	2	3	no	—
114	C₄×C₁₀₉	—	2	2	no	—
115	C₄×C₁₀₉	—	2	1	no	—
116	C₄×C₁₀₉	—	2	3	no	—
117	C₄×C₁₀₉	—	2	6	no	—
118	C₄×C₁₀₉	C₂×#41	1	6	yes	—
119	C₄×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
120	C₂×C₁₀₉	—	4	2	no	—
121	C₂×C₁₀₉	—	4	3	no	—
122	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
123	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₁₀₉	#585	1	1	yes	characteristic
125	#3518	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₁₀₉	#3518	1	1	yes	characteristic, direct product factor, Sylow
127	Q₈²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
128	#556	C₂×C₁₀₉	1	6	yes	complemented
129	#556	C₂×C₁₀₉	1	3	yes	complemented
130	#555	C₂×C₁₀₉	1	3	yes	complemented
131	D₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
132	#548	C₂×C₁₀₉	1	2	yes	complemented
133	#542	C₂×C₁₀₉	1	6	yes	complemented
134	#533	C₂×C₁₀₉	1	3	yes	complemented
135	#529	C₂×C₁₀₉	1	3	yes	complemented
136	#519	C₂×C₁₀₉	1	2	yes	complemented
137	C₄×#41	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
140	C₄⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
141	C₄⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
142	C₄⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
143	#126	C₂²×C₁₀₉	1	6	yes	—
144	#126	C₂²×C₁₀₉	1	12	yes	—
145	#125	C₂²×C₁₀₉	1	3	yes	—
146	#124	C₂²×C₁₀₉	1	1	yes	characteristic
147	#124	C₂²×C₁₀₉	1	6	yes	—
148	#124	C₂²×C₁₀₉	1	12	yes	—
149	#123	C₂²×C₁₀₉	1	6	yes	—
150	C₂²⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
151	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
152	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
153	C₂²⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
154	#121	C₂²×C₁₀₉	1	3	yes	—
155	#121	C₂²×C₁₀₉	1	6	yes	—
156	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
157	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
158	C₄×Q₈	C₂²×C₁₀₉	1	6	yes	—
159	C₄×Q₈	C₂²×C₁₀₉	1	2	yes	—
160	C₄×Q₈	C₂²×C₁₀₉	1	6	yes	—
161	C₄×Q₈	C₂²×C₁₀₉	1	3	yes	—
162	C₄×Q₈	C₂²×C₁₀₉	1	6	yes	—
163	C₄×D₄	C₂²×C₁₀₉	1	3	yes	—
164	C₄×D₄	C₂²×C₁₀₉	1	3	yes	—
165	C₄×D₄	C₂²×C₁₀₉	1	6	yes	—
166	C₄×D₄	C₂²×C₁₀₉	1	3	yes	—
167	#117	C₂²×C₁₀₉	1	3	yes	—
168	#117	C₂²×C₁₀₉	1	6	yes	—
169	#117	C₂²×C₁₀₉	1	6	yes	—
170	#117	C₂²×C₁₀₉	1	6	yes	—
171	C₂×C₄²	C₂²×C₁₀₉	1	3	yes	—
172	#41	—	2	4	no	—
173	C₂×Q₈	—	2	8	no	—
174	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
175	C₂×Q₈	C₂³×C₁₀₉	1	6	yes	—
176	C₂×Q₈	C₂³×C₁₀₉	1	6	yes	—
177	C₂×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₁₀₉	1	3	yes	—
180	C₂×D₄	C₂³×C₁₀₉	1	6	yes	—
181	C₂²×C₄	C₂³×C₁₀₉	1	3	yes	—
182	C₂²×C₄	C₂³×C₁₀₉	1	3	yes	—
183	C₂²×C₄	C₂³×C₁₀₉	1	6	yes	—
184	C₂²×C₄	C₂³×C₁₀₉	1	3	yes	—
185	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
186	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
187	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
188	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
189	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
190	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
191	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
192	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
193	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
194	C₄⋊C₄	C₂³×C₁₀₉	1	6	yes	—
195	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
196	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
197	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
198	C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
199	C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
200	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
201	C₂²⋊C₄	C₂³×C₁₀₉	1	6	yes	—
202	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
203	C₄²	C₂³×C₁₀₉	1	3	yes	—
204	C₄²	C₂³×C₁₀₉	1	3	yes	—
205	C₄²	C₂³×C₁₀₉	1	3	yes	—
206	C₄²	C₂³×C₁₀₉	1	12	yes	—
207	C₄²	C₂³×C₁₀₉	1	6	yes	—
208	C₂³	C₂⁴×C₁₀₉	1	2	yes	—
209	C₂³	C₂⁴×C₁₀₉	1	3	yes	—
210	Q₈	—	2	12	no	—
211	Q₈	—	2	2	no	—
212	Q₈	—	4	6	no	—
213	Q₈	#41×C₁₀₉	1	4	yes	—
214	D₄	—	2	6	no	—
215	D₄	—	2	2	no	—
216	D₄	—	4	6	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	4	3	no	—
221	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
222	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
224	C₂×C₄	C₂⁴×C₁₀₉	1	2	yes	—
225	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
226	C₂×C₄	C₂⁴×C₁₀₉	1	6	yes	—
227	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
228	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
229	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
230	C₂²	—	2	2	no	—
231	C₂²	—	2	3	no	—
232	C₂²	—	2	3	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	3	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	6	no	—
239	C₄	—	2	3	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	3	no	—
243	C₄	—	2	6	no	—
244	C₄	C₂×#41×C₁₀₉	1	6	yes	—
245	C₄	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
246	C₂	—	4	2	no	—
247	C₂	—	4	3	no	—
248	C₂	C₂²×#41×C₁₀₉	1	1	yes	characteristic
249	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
250	C₂	#585×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₁₀₉	C₂	semidirect
2. #556×C₁₀₉	C₂	semidirect
3. #556×C₁₀₉	C₂	semidirect
4. #555×C₁₀₉	C₂	semidirect
5. D₄×Q₈×C₁₀₉	C₂	semidirect
6. #548×C₁₀₉	C₂	semidirect
7. #542×C₁₀₉	C₂	semidirect
8. #533×C₁₀₉	C₂	semidirect
9. #529×C₁₀₉	C₂	semidirect
10. #519×C₁₀₉	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
125. #3518	C₁₀₉	direct (N × H)
126. C₁₀₉	#3518	direct (N × H)
127. Q₈²	C₂×C₁₀₉	semidirect
128. #556	C₂×C₁₀₉	semidirect
129. #556	C₂×C₁₀₉	semidirect
130. #555	C₂×C₁₀₉	semidirect
131. D₄×Q₈	C₂×C₁₀₉	semidirect
132. #548	C₂×C₁₀₉	semidirect
133. #542	C₂×C₁₀₉	semidirect
134. #533	C₂×C₁₀₉	semidirect
135. #529	C₂×C₁₀₉	semidirect
136. #519	C₂×C₁₀₉	semidirect
137. C₄×#41	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₁₀₉	2
#556×C₁₀₉	2
#555×C₁₀₉	2
D₄×Q₈×C₁₀₉	2
#548×C₁₀₉	2
#542×C₁₀₉	2
#533×C₁₀₉	2
#529×C₁₀₉	2
#519×C₁₀₉	2
C₄×#41×C₁₀₉	2
#3518	109

Subgroups — C109×#3518 #28360772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3518 #28360772