Subgroups — #28376614

Order 13952.

1068 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295752	C₂	1	2	yes	maximal, complemented
3	#20295736	C₂	1	3	yes	maximal, complemented
4	#20295732	C₂	1	6	yes	maximal, complemented
5	#20295691	C₂	1	3	yes	maximal, complemented
6	#20295582	C₂	1	6	yes	maximal, complemented
7	#20295581	C₂	1	3	yes	maximal, complemented
8	C₄×#13182030	C₂	1	2	yes	maximal
9	C₄×#13182030	C₂	1	1	yes	maximal, characteristic
10	C₄×Q₈×D₁₀₉	C₂	1	3	yes	maximal, complemented
11	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182030	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₀₉	C₂²	1	3	yes	—
14	#16812699	C₂²	1	3	yes	—
15	#16812698	C₂²	1	6	yes	—
16	#16812698	C₂²	1	6	yes	—
17	#16812691	C₂²	1	6	yes	—
18	#16812690	C₂²	1	6	yes	—
19	#16812679	C₂²	1	6	yes	—
20	#16812677	C₂²	1	12	yes	—
21	#16812677	C₂²	1	12	yes	—
22	#16812675	C₂²	1	6	yes	—
23	Q₈×Dic₁₀₉	C₂²	1	3	yes	—
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
26	C₄×Dic₂₁₈	C₂²	1	3	yes	—
27	C₄×Dic₂₁₈	C₂²	1	6	yes	—
28	#16812664	C₂²	1	1	yes	characteristic
29	#16812664	C₂²	1	6	yes	—
30	#16812664	C₂²	1	6	yes	—
31	C₄×D₄₃₆	C₂²	1	3	yes	—
32	C₄×D₄₃₆	C₂²	1	6	yes	—
33	#16812661	C₂²	1	6	yes	—
34	#16812661	C₂²	1	3	yes	—
35	#16812657	C₂²	1	6	yes	—
36	#16812657	C₂²	1	3	yes	—
37	C₄⋊C₄×D₁₀₉	C₂²	1	6	yes	—
38	C₄⋊C₄×D₁₀₉	C₂²	1	3	yes	—
39	C₄²×D₁₀₉	C₂²	1	3	yes	—
40	C₄²×D₁₀₉	C₂²	1	6	yes	—
41	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
42	#125×C₁₀₉	C₂²	1	6	yes	—
43	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
44	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
45	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
46	#13182030	Q₈	1	8	yes	complemented
47	C₂×Dic₂₁₈	C₂³	1	6	yes	—
48	C₂×Dic₂₁₈	C₂³	1	6	yes	—
49	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
50	Q₈×D₁₀₉	—	2	12	no	—
51	C₂×D₄₃₆	C₂³	1	3	yes	—
52	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
53	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
54	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
55	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
56	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
57	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
58	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
59	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
60	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
61	#13182008	C₂³	1	2	yes	—
62	#13182008	C₂³	1	6	yes	—
63	#13182008	C₂³	1	6	yes	—
64	#13182008	C₂³	1	12	yes	—
65	#13182006	C₂³	1	12	yes	—
66	#13182006	C₂³	1	6	yes	—
67	C₄×Dic₁₀₉	C₂³	1	2	yes	—
68	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
69	C₄×Dic₁₀₉	C₂³	1	3	yes	—
70	C₄×Dic₁₀₉	C₂³	1	6	yes	—
71	C₄×Dic₁₀₉	C₂³	1	3	yes	—
72	C₄×Dic₁₀₉	C₂³	1	6	yes	—
73	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
74	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
77	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
78	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
79	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
80	C₄²×C₁₀₉	C₂³	1	6	yes	—
81	C₄²×C₁₀₉	C₂³	1	3	yes	—
82	C₂²×D₁₀₉	C₂⁴	1	3	yes	—
83	Q₈×C₁₀₉	—	2	6	no	—
84	Q₈×C₁₀₉	C₂×Q₈	1	4	yes	complemented
85	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
87	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
88	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
89	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
92	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
93	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
95	D₄₃₆	C₂×Q₈	1	12	yes	complemented
96	C₄×D₁₀₉	—	2	6	no	—
97	C₄×D₁₀₉	—	2	12	no	—
98	C₄×D₁₀₉	—	4	6	no	—
99	C₄×D₁₀₉	—	4	3	no	—
100	C₄×D₁₀₉	C₂×Q₈	1	12	yes	complemented
101	Dic₂₁₈	—	2	12	no	—
102	Dic₂₁₈	—	4	6	no	—
103	Dic₂₁₈	#41	1	4	yes	complemented
104	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
105	C₂×D₁₀₉	—	2	3	no	—
106	C₂×D₁₀₉	—	2	3	no	—
107	C₂×D₁₀₉	C₂²×Q₈	1	6	yes	—
108	C₄×C₁₀₉	—	2	6	no	—
109	C₄×C₁₀₉	—	2	3	no	—
110	C₄×C₁₀₉	—	2	2	no	—
111	C₄×C₁₀₉	C₂²×Q₈	1	6	yes	—
112	C₄×C₁₀₉	C₂×#41	1	2	yes	—
113	Dic₁₀₉	—	2	3	no	—
114	Dic₁₀₉	—	2	6	no	—
115	Dic₁₀₉	—	2	1	no	—
116	Dic₁₀₉	C₂²×Q₈	1	2	yes	—
117	Dic₁₀₉	C₂×#41	1	4	yes	—
118	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
119	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
120	C₂×C₁₀₉	#585	1	1	yes	characteristic
121	D₁₀₉	—	2	6	no	—
122	Q₈×#41	—	109	1	no	maximal, Sylow
123	C₁₀₉	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
124	Q₈²	—	109	1	no	—
125	#557	—	109	2	no	—
126	#557	D₁₀₉	1	1	yes	characteristic, complemented
127	#554	—	109	6	no	—
128	#554	—	109	3	no	—
129	D₄×Q₈	—	109	3	no	—
130	#533	—	109	3	no	—
131	#533	—	109	6	no	—
132	C₄×#41	—	109	2	no	—
133	C₄×#41	—	109	1	no	—
134	C₂×C₄×Q₈	—	109	3	no	—
135	C₂×#41	—	109	1	no	—
136	C₂²×Q₈	—	109	3	no	—
137	C₄⋊Q₈	—	109	6	no	—
138	C₄⋊Q₈	—	109	6	no	—
139	C₄⋊Q₈	—	109	3	no	—
140	C₄⋊Q₈	C₂×D₁₀₉	1	3	yes	—
141	#125	—	109	6	no	—
142	#125	—	109	6	no	—
143	#125	C₂×D₁₀₉	1	6	yes	—
144	C₂²⋊Q₈	—	109	6	no	—
145	C₂²⋊Q₈	—	109	12	no	—
146	C₂²⋊Q₈	—	109	12	no	—
147	C₂²⋊Q₈	—	109	6	no	—
148	C₄×Q₈	—	109	3	no	—
149	C₄×Q₈	—	109	1	no	—
150	C₄×Q₈	—	109	2	no	—
151	C₄×Q₈	—	109	1	no	—
152	C₄×Q₈	—	109	6	no	—
153	C₄×Q₈	—	109	6	no	—
154	C₄×Q₈	—	109	3	no	—
155	C₄×Q₈	—	109	6	no	—
156	C₄×Q₈	C₂×D₁₀₉	1	3	yes	—
157	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
158	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
159	C₄×D₄	—	109	6	no	—
160	C₄×D₄	—	109	3	no	—
161	C₄×D₄	—	109	3	no	—
162	C₄×D₄	—	109	6	no	—
163	#117	—	109	6	no	—
164	#117	—	109	3	no	—
165	C₂×C₄⋊C₄	—	109	6	no	—
166	C₂×C₄⋊C₄	—	109	3	no	—
167	C₂×C₄²	—	109	3	no	—
168	C₂×C₄²	—	109	6	no	—
169	#41	—	109	8	no	—
170	C₂×Q₈	—	109	1	no	—
171	C₂×Q₈	—	109	6	no	—
172	C₂×Q₈	—	109	6	no	—
173	C₂×Q₈	—	218	12	no	—
174	C₂×Q₈	C₂²×D₁₀₉	1	3	yes	—
175	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
176	C₂×D₄	—	109	3	no	—
177	C₂²×C₄	—	109	3	no	—
178	C₂²×C₄	—	109	3	no	—
179	C₂²×C₄	—	109	6	no	—
180	C₂²×C₄	—	109	6	no	—
181	C₂²×C₄	—	109	3	no	—
182	C₄⋊C₄	—	109	2	no	—
183	C₄⋊C₄	—	109	6	no	—
184	C₄⋊C₄	—	109	6	no	—
185	C₄⋊C₄	—	109	12	no	—
186	C₄⋊C₄	—	109	6	no	—
187	C₄⋊C₄	—	109	3	no	—
188	C₄⋊C₄	—	109	6	no	—
189	C₄⋊C₄	—	109	3	no	—
190	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
192	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
193	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
194	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
195	C₂²⋊C₄	—	109	12	no	—
196	C₂²⋊C₄	—	109	6	no	—
197	C₄²	—	109	2	no	—
198	C₄²	—	109	1	no	—
199	C₄²	—	109	6	no	—
200	C₄²	—	109	3	no	—
201	C₄²	—	109	3	no	—
202	C₄²	—	109	6	no	—
203	C₄²	C₂²×D₁₀₉	1	6	yes	—
204	C₄²	C₂²×D₁₀₉	1	3	yes	—
205	C₂³	—	109	3	no	—
206	Q₈	—	2	6	no	—
207	Q₈	—	109	4	no	—
208	Q₈	—	436	6	no	—
209	Q₈	—	218	12	no	—
210	Q₈	Q₈×D₁₀₉	1	4	yes	complemented
211	D₄	—	109	12	no	—
212	C₂×C₄	—	109	1	no	—
213	C₂×C₄	—	109	2	no	—
214	C₂×C₄	—	109	12	no	—
215	C₂×C₄	—	109	1	no	—
216	C₂×C₄	—	109	3	no	—
217	C₂×C₄	—	109	6	no	—
218	C₂×C₄	—	436	6	no	—
219	C₂×C₄	—	436	3	no	—
220	C₂×C₄	—	218	12	no	—
221	C₂×C₄	—	218	6	no	—
222	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
223	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
224	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
225	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
226	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
227	C₂²	—	109	6	no	—
228	C₂²	—	218	3	no	—
229	C₂²	—	218	3	no	—
230	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	3	no	—
234	C₄	—	109	2	no	—
235	C₄	—	109	4	no	—
236	C₄	—	218	3	no	—
237	C₄	—	218	1	no	—
238	C₄	—	218	6	no	—
239	C₄	C₂×Q₈×D₁₀₉	1	6	yes	—
240	C₄	#41×D₁₀₉	1	2	yes	—
241	C₂	—	218	6	no	—
242	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
243	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
244	C₂	#20295863	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#557×C₁₀₉	C₂
prime radical	C₂×#13182030	C₂²
hypercenter	#557	D₁₀₉
hypocenter	C₁₀₉	Q₈×#41
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₁₈	C₂	semidirect
2. #20295752	C₂	semidirect
3. #20295736	C₂	semidirect
4. #20295732	C₂	semidirect
5. #20295691	C₂	semidirect
6. #20295582	C₂	semidirect
7. #20295581	C₂	semidirect
10. C₄×Q₈×D₁₀₉	C₂	semidirect
11. #557×C₁₀₉	C₂	semidirect
46. #13182030	Q₈	semidirect
84. Q₈×C₁₀₉	C₂×Q₈	semidirect
95. D₄₃₆	C₂×Q₈	semidirect
100. C₄×D₁₀₉	C₂×Q₈	semidirect
103. Dic₂₁₈	#41	semidirect
123. C₁₀₉	Q₈×#41	semidirect
126. #557	D₁₀₉	semidirect
210. Q₈	Q₈×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₁₈	2
#20295752	2
#20295736	2
#20295732	2
#20295691	2
#20295582	2
#20295581	2
C₄×#13182030	2
C₄×Q₈×D₁₀₉	2
#557×C₁₀₉	2
Q₈×#41	109