Subgroups — #28371984

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295155	C₂	1	2	yes	maximal, complemented
4	#20295154	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295150	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295048	C₂	1	1	yes	maximal, characteristic
8	#20295048	C₂	1	2	yes	maximal
9	#20294978	C₂	1	2	yes	maximal
10	C₄×#13182012	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	C₄	1	1	yes	characteristic
13	C₂×#13182012	C₂²	1	1	yes	characteristic
14	C₂×#13182012	C₂²	1	2	yes	—
15	C₂×#13182012	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₀₉	C₄	1	1	yes	characteristic
17	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₈	C₂²	1	2	yes	—
19	#16812664	C₄	1	2	yes	—
20	#16812664	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₀₉	C₄	1	2	yes	—
22	C₄⋊C₄×D₁₀₉	C₄	1	1	yes	characteristic
23	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
24	C₄²×D₁₀₉	C₂²	1	2	yes	—
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	C₄⋊#11724784	C₂²	1	2	yes	—
27	#16812628	C₂²	1	2	yes	—
28	#16812628	C₂²	1	2	yes	—
29	#16812628	C₂²	1	2	yes	—
30	#16812610	C₂²	1	2	yes	—
31	#16812609	C₂²	1	1	yes	characteristic
32	#16812609	C₂²	1	1	yes	characteristic
33	C₄×#11724784	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₉	C₂²	1	2	yes	—
35	C₈×Dic₁₀₉	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
39	#97×C₁₀₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₁₀₉	—	2	4	no	—
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
49	#13182012	—	2	2	no	—
50	#13182012	—	2	4	no	—
51	#13182012	Q₈	1	4	yes	complemented
52	C₂×#11724784	C₂³	1	2	yes	—
53	C₂×#11724784	C₂³	1	1	yes	characteristic
54	C₂×#11724784	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
57	#13182008	C₂×C₄	1	2	yes	—
58	#13182008	C₂×C₄	1	2	yes	—
59	#13182008	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀₉	C₂³	1	2	yes	—
65	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₀₉	C₂³	1	2	yes	—
74	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
86	C₄×D₁₀₉	—	2	2	no	—
87	C₄×D₁₀₉	—	2	4	no	—
88	C₄×D₁₀₉	—	4	1	no	—
89	C₄×D₁₀₉	—	4	2	no	—
90	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
91	C₄×D₁₀₉	#41	1	2	yes	—
92	Dic₂₁₈	—	2	4	no	—
93	Dic₂₁₈	M₁₆	1	4	yes	complemented
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	2	2	no	—
96	C₈×C₁₀₉	C₂×Q₈	1	2	yes	complemented
97	#11724784	—	2	2	no	—
98	#11724784	—	2	1	no	—
99	#11724784	C₂×Q₈	1	2	yes	complemented
100	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₀₉	—	2	1	no	—
102	C₂×D₁₀₉	—	2	1	no	—
103	C₂×D₁₀₉	C₄×Q₈	1	2	yes	—
104	C₄×C₁₀₉	—	2	2	no	—
105	C₄×C₁₀₉	—	2	1	no	—
106	C₄×C₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
108	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
110	Dic₁₀₉	—	2	1	no	—
111	Dic₁₀₉	—	2	2	no	—
112	Dic₁₀₉	—	2	1	no	—
113	Dic₁₀₉	C₄×Q₈	1	2	yes	—
114	Dic₁₀₉	C₂×M₁₆	1	4	yes	—
115	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×C₁₀₉	#568	1	1	yes	characteristic
118	D₁₀₉	—	2	2	no	—
119	Q₈×M₁₆	—	109	1	no	maximal, Sylow
120	C₁₀₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	109	1	no	—
122	#446	—	109	1	no	—
123	#446	—	109	2	no	—
124	#446	—	109	2	no	—
125	#446	D₁₀₉	1	1	yes	characteristic, complemented
126	C₈×Q₈	—	109	1	no	—
127	C₈×Q₈	—	109	1	no	—
128	#423	—	109	2	no	—
129	#423	—	109	1	no	—
130	C₄×M₁₆	—	109	2	no	—
131	C₄×M₁₆	—	109	1	no	—
132	C₂²×Q₈	—	109	1	no	—
133	C₂×M₁₆	—	109	1	no	—
134	C₂×M₁₆	—	109	2	no	—
135	C₂×M₁₆	—	109	1	no	—
136	C₄×Q₈	—	109	1	no	—
137	C₄×Q₈	—	109	1	no	—
138	C₄×Q₈	—	109	2	no	—
139	C₄×Q₈	—	109	1	no	—
140	C₄×Q₈	—	109	2	no	—
141	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	109	2	no	—
143	C₂×C₄⋊C₄	—	109	1	no	—
144	C₂×C₄²	—	109	2	no	—
145	C₂×C₄²	—	109	1	no	—
146	C₄⋊C₈	—	109	2	no	—
147	C₄⋊C₈	—	109	2	no	—
148	C₄⋊C₈	—	109	2	no	—
149	C₄⋊C₈	—	109	1	no	—
150	C₄⋊C₈	—	109	2	no	—
151	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
153	#97	—	109	1	no	—
154	#97	—	109	1	no	—
155	#97	—	109	2	no	—
156	#97	C₂×D₁₀₉	1	2	yes	—
157	C₄×C₈	—	109	2	no	—
158	C₄×C₈	—	109	2	no	—
159	C₄×C₈	—	109	1	no	—
160	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₂×Q₈	—	109	1	no	—
162	C₂×Q₈	—	109	2	no	—
163	C₂×Q₈	—	218	4	no	—
164	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
165	C₂²×C₄	—	109	1	no	—
166	C₂²×C₄	—	109	2	no	—
167	C₂²×C₄	—	109	1	no	—
168	C₂²×C₄	—	109	2	no	—
169	C₂²×C₄	—	109	1	no	—
170	M₁₆	—	109	4	no	—
171	M₁₆	—	218	4	no	—
172	M₁₆	—	218	2	no	—
173	C₂×C₈	—	109	1	no	—
174	C₂×C₈	—	109	2	no	—
175	C₂×C₈	—	109	1	no	—
176	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
177	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
179	C₄⋊C₄	—	109	2	no	—
180	C₄⋊C₄	—	109	2	no	—
181	C₄⋊C₄	—	109	2	no	—
182	C₄⋊C₄	—	109	2	no	—
183	C₄⋊C₄	—	109	1	no	—
184	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
185	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
186	C₄²	—	109	2	no	—
187	C₄²	—	109	2	no	—
188	C₄²	—	109	1	no	—
189	C₄²	—	109	1	no	—
190	C₄²	—	109	2	no	—
191	C₄²	—	109	1	no	—
192	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
193	C₄²	C₂²×D₁₀₉	1	2	yes	—
194	C₂³	—	109	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	109	4	no	—
197	Q₈	—	218	4	no	—
198	C₂×C₄	—	109	1	no	—
199	C₂×C₄	—	109	2	no	—
200	C₂×C₄	—	109	2	no	—
201	C₂×C₄	—	109	2	no	—
202	C₂×C₄	—	109	2	no	—
203	C₂×C₄	—	109	1	no	—
204	C₂×C₄	—	109	1	no	—
205	C₂×C₄	—	436	1	no	—
206	C₂×C₄	—	436	2	no	—
207	C₂×C₄	—	218	4	no	—
208	C₂×C₄	—	218	2	no	—
209	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	109	2	no	—
217	C₈	—	218	2	no	—
218	C₈	—	218	1	no	—
219	C₈	Q₈×D₁₀₉	1	2	yes	complemented
220	C₂²	—	109	2	no	—
221	C₂²	—	218	1	no	—
222	C₂²	—	218	1	no	—
223	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	109	2	no	—
228	C₄	—	109	4	no	—
229	C₄	—	218	1	no	—
230	C₄	—	218	2	no	—
231	C₄	—	218	1	no	—
232	C₄	M₁₆×D₁₀₉	1	2	yes	—
233	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
234	C₄	#41×D₁₀₉	1	1	yes	characteristic
235	C₂	—	218	2	no	—
236	C₂	C₄×Q₈×D₁₀₉	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₀₉	1	1	yes	characteristic
238	C₂	#20295791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#446×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#446	D₁₀₉
hypocenter	C₁₀₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20295157	C₂	semidirect
3. #20295155	C₂	semidirect
4. #20295154	C₂	semidirect
5. C₈×Dic₂₁₈	C₂	semidirect
6. #20295150	C₂	semidirect
11. #446×C₁₀₉	C₂	semidirect
51. #13182012	Q₈	semidirect
93. Dic₂₁₈	M₁₆	semidirect
96. C₈×C₁₀₉	C₂×Q₈	semidirect
99. #11724784	C₂×Q₈	semidirect
120. C₁₀₉	Q₈×M₁₆	semidirect
125. #446	D₁₀₉	semidirect
219. C₈	Q₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₀₉	2
#20295157	2
#20295155	2
#20295154	2
C₈×Dic₂₁₈	2
#20295150	2
#20295048	2
#20294978	2
C₄×#13182012	2
#446×C₁₀₉	2
Q₈×M₁₆	109