Subgroups — #28376627

Order 13952.

1112 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	#20295761	C₂	1	3	yes	maximal, complemented
2	#20295755	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295749	C₂	1	3	yes	maximal, complemented
4	#20295724	C₂	1	6	yes	maximal, complemented
5	#20295698	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295687	C₂	1	2	yes	maximal, complemented
7	#20295576	C₂	1	3	yes	maximal, complemented
8	#20295563	C₂	1	6	yes	maximal, complemented
9	C₄×#13182030	C₂	1	2	yes	maximal, complemented
10	C₄×#13182029	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182030	C₂²	1	2	yes	—
13	C₂×#13182029	C₂²	1	3	yes	—
14	#16812699	C₂²	1	3	yes	—
15	#16812696	C₂²	1	3	yes	—
16	#16812694	C₂²	1	12	yes	—
17	#16812690	C₂²	1	6	yes	—
18	#16812688	C₂²	1	6	yes	—
19	#16812687	C₂²	1	6	yes	—
20	#16812684	C₂²	1	12	yes	—
21	#16812675	C₂²	1	6	yes	—
22	#16812674	C₂²	1	6	yes	—
23	#16812673	C₂²	1	6	yes	—
24	#16812673	C₂²	1	6	yes	—
25	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
26	C₄×Dic₂₁₈	C₂²	1	3	yes	—
27	C₄×Dic₂₁₈	C₂²	1	3	yes	—
28	C₄×D₄₃₆	C₂²	1	3	yes	—
29	C₄×D₄₃₆	C₂²	1	3	yes	—
30	C₄×D₄₃₆	C₂²	1	6	yes	—
31	C₄×D₄₃₆	C₂²	1	6	yes	—
32	#16812661	C₂²	1	6	yes	—
33	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	—
34	C₄×C₂²⋊D₁₀₉	C₂²	1	6	yes	—
35	#16812658	C₂²	1	1	yes	characteristic
36	#16812657	C₂²	1	6	yes	—
37	#16812656	C₂²	1	6	yes	—
38	C₄²×D₁₀₉	C₂²	1	6	yes	—
39	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
42	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
44	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	—
45	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
46	#13182030	—	2	8	no	—
47	#13182029	—	2	12	no	—
48	C₂×Dic₂₁₈	C₂³	1	3	yes	—
49	C₂×D₄₃₆	C₂³	1	6	yes	—
50	C₂×D₄₃₆	C₂³	1	3	yes	—
51	C₂×D₄₃₆	C₂³	1	6	yes	—
52	C₂×C₂²⋊D₁₀₉	C₂³	1	6	yes	—
53	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
54	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
55	C₂×C₄×D₁₀₉	C₂³	1	6	yes	—
56	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
57	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
60	#13182008	C₂³	1	6	yes	—
61	#13182008	C₂³	1	6	yes	—
62	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
63	#13182006	C₂³	1	6	yes	—
64	#13182006	C₂³	1	6	yes	—
65	#13182006	C₂³	1	12	yes	—
66	#13182006	C₂³	1	12	yes	—
67	C₄×Dic₁₀₉	C₂³	1	2	yes	—
68	C₄×Dic₁₀₉	C₂³	1	6	yes	—
69	C₄×Dic₁₀₉	C₂³	1	6	yes	—
70	C₂×Q₈×C₁₀₉	—	2	4	no	—
71	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
72	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
74	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
75	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
77	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
78	C₄⋊C₄×C₁₀₉	C₂³	1	6	yes	—
79	C₄²×C₁₀₉	C₂³	1	3	yes	—
80	C₄²×C₁₀₉	C₂³	1	3	yes	—
81	C₄²×C₁₀₉	C₂³	1	3	yes	—
82	C₄²×C₁₀₉	C₂³	1	3	yes	—
83	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₀₉	C₂⁴	1	6	yes	—
85	Q₈×C₁₀₉	—	2	6	no	—
86	Q₈×C₁₀₉	#41	1	4	yes	complemented
87	C₂×C₄×C₁₀₉	—	2	6	no	—
88	C₂×C₄×C₁₀₉	—	4	3	no	—
89	C₂×C₄×C₁₀₉	#41	1	4	yes	complemented
90	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
91	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
92	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
93	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
95	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₁₀₉	—	2	12	no	—
97	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
98	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
99	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
100	D₄₃₆	—	2	6	no	—
101	D₄₃₆	—	2	12	no	—
102	D₄₃₆	—	4	6	no	—
103	C₄×D₁₀₉	—	2	12	no	—
104	C₄×D₁₀₉	—	2	12	no	—
105	C₄×D₁₀₉	—	4	6	no	—
106	Dic₂₁₈	—	2	6	no	—
107	C₂²×C₁₀₉	—	2	1	no	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
110	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
111	C₂×D₁₀₉	—	2	6	no	—
112	C₂×D₁₀₉	—	2	6	no	—
113	C₂×D₁₀₉	—	2	6	no	—
114	C₄×C₁₀₉	—	2	3	no	—
115	C₄×C₁₀₉	—	2	3	no	—
116	C₄×C₁₀₉	—	2	3	no	—
117	C₄×C₁₀₉	C₂×#41	1	6	yes	—
118	C₄×C₁₀₉	C₂×#41	1	2	yes	—
119	C₄×C₁₀₉	C₂×#41	1	2	yes	—
120	Dic₁₀₉	—	2	6	no	—
121	Dic₁₀₉	—	2	2	no	—
122	Dic₁₀₉	—	2	2	no	—
123	C₂×C₁₀₉	—	2	2	no	—
124	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
125	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
126	C₂×C₁₀₉	#585	1	1	yes	characteristic
127	D₁₀₉	—	4	6	no	—
128	#3480	—	109	1	no	maximal, Sylow
129	C₁₀₉	#3480	1	1	yes	characteristic, complemented, Sylow
130	#559	—	109	3	no	—
131	#557	—	109	1	no	—
132	#556	—	109	3	no	—
133	#553	—	109	6	no	—
134	#550	—	109	1	no	—
135	#548	—	109	2	no	—
136	#532	—	109	3	no	—
137	#529	—	109	6	no	—
138	C₄×#41	—	109	2	no	—
139	C₄×#41	—	109	3	no	—
140	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
141	C₂×#41	—	109	2	no	—
142	C₂×#41	—	109	3	no	—
143	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
144	C₄⋊Q₈	—	109	3	no	—
145	#127	—	109	3	no	—
146	#126	—	109	12	no	—
147	#125	—	109	6	no	—
148	#124	—	109	6	no	—
149	#124	—	109	6	no	—
150	#123	—	109	12	no	—
151	C₂²⋊Q₈	—	109	6	no	—
152	#121	—	109	6	no	—
153	#121	—	109	6	no	—
154	#121	—	109	6	no	—
155	C₄×Q₈	—	109	2	no	—
156	C₄×Q₈	—	109	3	no	—
157	C₄×Q₈	—	109	3	no	—
158	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₁₀₉	1	3	yes	—
160	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₄×Q₈	C₂×D₁₀₉	1	3	yes	—
162	C₄×D₄	—	109	6	no	—
163	C₄×D₄	—	109	6	no	—
164	C₄×D₄	—	109	6	no	—
165	C₄×D₄	—	109	6	no	—
166	C₄×D₄	—	109	3	no	—
167	C₄×D₄	—	109	3	no	—
168	C₄×D₄	—	109	6	no	—
169	#117	—	109	1	no	—
170	#117	—	109	6	no	—
171	#117	—	109	6	no	—
172	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	—
173	C₂×C₄²	—	109	6	no	—
174	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
175	#41	—	218	8	no	—
176	#41	—	218	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	109	3	no	—
179	C₂×Q₈	C₂²×D₁₀₉	1	3	yes	—
180	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
181	C₂×D₄	—	109	6	no	—
182	C₂×D₄	—	109	6	no	—
183	C₂×D₄	—	109	6	no	—
184	C₂×D₄	—	109	3	no	—
185	C₂²×C₄	—	109	6	no	—
186	C₂²×C₄	—	109	6	no	—
187	C₂²×C₄	—	109	6	no	—
188	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
189	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
190	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₄⋊C₄	—	109	6	no	—
192	C₄⋊C₄	—	109	6	no	—
193	C₄⋊C₄	—	109	1	no	—
194	C₄⋊C₄	—	109	1	no	—
195	C₄⋊C₄	—	109	6	no	—
196	C₄⋊C₄	—	109	6	no	—
197	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
198	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
199	C₄⋊C₄	C₂²×D₁₀₉	1	6	yes	—
200	C₂²⋊C₄	—	109	2	no	—
201	C₂²⋊C₄	—	109	6	no	—
202	C₂²⋊C₄	—	109	12	no	—
203	C₂²⋊C₄	—	109	6	no	—
204	C₂²⋊C₄	—	109	12	no	—
205	C₄²	—	109	6	no	—
206	C₄²	—	109	6	no	—
207	C₄²	—	109	2	no	—
208	C₄²	C₂²×D₁₀₉	1	3	yes	—
209	C₄²	C₂²×D₁₀₉	1	3	yes	—
210	C₄²	C₂²×D₁₀₉	1	3	yes	—
211	C₄²	C₂²×D₁₀₉	1	3	yes	—
212	C₂³	—	109	6	no	—
213	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	218	6	no	—
216	Q₈	#13182029	1	4	yes	complemented
217	D₄	—	436	6	no	—
218	D₄	—	218	12	no	—
219	D₄	—	218	12	no	—
220	D₄	—	218	6	no	—
221	C₂×C₄	—	2	6	no	—
222	C₂×C₄	—	4	3	no	—
223	C₂×C₄	—	109	2	no	—
224	C₂×C₄	—	109	2	no	—
225	C₂×C₄	—	109	6	no	—
226	C₂×C₄	—	436	6	no	—
227	C₂×C₄	—	218	12	no	—
228	C₂×C₄	—	218	12	no	—
229	C₂×C₄	#13182030	1	4	yes	complemented
230	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
231	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
232	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
233	C₂×C₄	C₂³×D₁₀₉	1	3	yes	—
234	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	218	6	no	—
239	C₂²	—	218	6	no	—
240	C₂²	—	218	6	no	—
241	C₂²	C₂×#13182030	1	2	yes	—
242	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	218	2	no	—
247	C₄	—	218	6	no	—
248	C₄	—	218	2	no	—
249	C₄	C₂×#13182029	1	6	yes	—
250	C₄	C₂×#13182030	1	2	yes	—
251	C₄	C₂×#13182030	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	436	6	no	—
254	C₂	C₂²×#13182029	1	1	yes	characteristic
255	C₂	C₂²×#13182030	1	1	yes	characteristic
256	C₂	#20295862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	#3480
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20295761	C₂	semidirect
2. #20295755	C₂	semidirect
3. #20295749	C₂	semidirect
4. #20295724	C₂	semidirect
5. #20295698	C₂	semidirect
6. #20295687	C₂	semidirect
7. #20295576	C₂	semidirect
8. #20295563	C₂	semidirect
9. C₄×#13182030	C₂	semidirect
10. C₄×#13182029	C₂	semidirect
11. C₂×C₄×Q₈×C₁₀₉	C₂	semidirect
86. Q₈×C₁₀₉	#41	semidirect
89. C₂×C₄×C₁₀₉	#41	semidirect
129. C₁₀₉	#3480	semidirect
140. C₂×C₄×Q₈	D₁₀₉	semidirect
216. Q₈	#13182029	semidirect
229. C₂×C₄	#13182030	semidirect

Maximal subgroups

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

subgroup	index
#20295761	2
#20295755	2
#20295749	2
#20295724	2
#20295698	2
#20295687	2
#20295576	2
#20295563	2
C₄×#13182030	2
C₄×#13182029	2
C₂×C₄×Q₈×C₁₀₉	2
#3480	109