Subgroups — #28379941

Order 13952.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296078	C₂	1	2	yes	maximal, complemented
2	#20296077	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296076	C₂	1	1	yes	maximal, characteristic, complemented
4	#20296075	C₂	1	1	yes	maximal, characteristic, complemented
5	#20296075	C₂	1	2	yes	maximal, complemented
6	Q₈×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
7	#20296050	C₂	1	2	yes	maximal
8	#20296050	C₂	1	2	yes	maximal
9	#20296043	C₂	1	1	yes	maximal, characteristic
10	#20296042	C₂	1	1	yes	maximal, characteristic
11	#20295578	C₂	1	1	yes	maximal, characteristic
12	#16812766	C₂²	1	1	yes	characteristic
13	#16812766	C₂²	1	1	yes	characteristic
14	#16812766	C₂²	1	2	yes	—
15	#16812766	C₂²	1	2	yes	—
16	C₄⋊#11724786	C₂²	1	1	yes	characteristic
17	C₄⋊#11724786	C₂²	1	1	yes	characteristic
18	C₄⋊#11724786	C₂²	1	2	yes	—
19	C₄⋊#11724786	C₂²	1	2	yes	—
20	C₂²⋊#11724786	C₂²	1	2	yes	—
21	C₂²⋊#11724786	C₂²	1	1	yes	characteristic
22	C₂²⋊#11724786	C₂²	1	1	yes	characteristic
23	#16812754	C₂²	1	2	yes	—
24	#16812754	C₂²	1	2	yes	—
25	C₄×#11724786	C₂²	1	2	yes	—
26	C₄×#11724786	C₂²	1	1	yes	characteristic
27	C₄×#11724786	C₂²	1	1	yes	characteristic
28	C₄×#11724786	C₂²	1	2	yes	—
29	C₄×#11724786	C₂²	1	2	yes	—
30	#16812698	C₄	1	1	yes	characteristic
31	#16812697	C₄	1	1	yes	characteristic
32	#16812689	C₄	1	2	yes	—
33	#16812678	C₄	1	1	yes	characteristic
34	#16812676	C₄	1	2	yes	—
35	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	#16812664	C₂²	1	1	yes	characteristic
37	#16812664	C₂²	1	2	yes	—
38	#16812655	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
41	C₂×#11724786	C₂³	1	2	yes	—
42	C₂×#11724786	C₂³	1	1	yes	characteristic
43	C₂×#11724786	C₂³	1	1	yes	characteristic
44	C₂×#11724786	C₂³	1	2	yes	—
45	C₂×#11724786	C₂³	1	1	yes	characteristic
46	C₂×#11724786	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
53	#13182008	C₂×C₄	1	2	yes	—
54	#13182008	C₂×C₄	1	2	yes	—
55	#13182008	C₂×C₄	1	2	yes	—
56	#13182008	C₂×C₄	1	2	yes	—
57	#13182008	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
62	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₀₉	C₂³	1	2	yes	—
64	C₄×Dic₁₀₉	C₂³	1	2	yes	—
65	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
70	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₀₉	—	2	2	no	—
78	C₂×Dic₁₀₉	—	2	1	no	—
79	C₂×Dic₁₀₉	—	2	1	no	—
80	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
86	Dic₂₁₈	—	2	2	no	—
87	Dic₂₁₈	—	2	4	no	—
88	#11724786	—	2	2	no	—
89	#11724786	—	2	1	no	—
90	#11724786	—	2	1	no	—
91	#11724786	—	2	2	no	—
92	#11724786	—	2	1	no	—
93	#11724786	—	2	1	no	—
94	C₂²×C₁₀₉	—	2	1	no	—
95	C₂²×C₁₀₉	—	2	1	no	—
96	C₂²×C₁₀₉	—	2	1	no	—
97	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₁₀₉	—	2	2	no	—
99	C₄×C₁₀₉	—	2	1	no	—
100	C₄×C₁₀₉	—	2	2	no	—
101	C₄×C₁₀₉	#131	1	2	yes	—
102	Dic₁₀₉	—	2	1	no	—
103	Dic₁₀₉	—	2	2	no	—
104	Dic₁₀₉	—	2	2	no	—
105	Dic₁₀₉	—	2	1	no	—
106	Dic₁₀₉	#131	1	2	yes	—
107	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₁₀₉	—	4	1	no	—
110	C₂×C₁₀₉	#519	1	1	yes	characteristic
111	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
112	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
113	#2986	—	109	1	no	maximal, Sylow
114	C₁₀₉	#2986	1	1	yes	characteristic, complemented, Sylow
115	#533	—	109	1	no	—
116	#446	—	109	1	no	—
117	#446	—	109	1	no	—
118	#446	—	109	2	no	—
119	C₈×Q₈	—	109	1	no	—
120	C₈×Q₈	—	109	1	no	—
121	C₈×Q₈	—	109	2	no	—
122	#433	—	109	2	no	—
123	#433	—	109	2	no	—
124	#431	—	109	1	no	—
125	#431	—	109	1	no	—
126	C₄⋊Q₈	—	109	1	no	—
127	C₄⋊Q₈	—	109	1	no	—
128	#125	—	109	2	no	—
129	C₂²⋊Q₈	—	109	1	no	—
130	C₂²⋊Q₈	—	109	2	no	—
131	C₂²⋊Q₈	#89137	1	1	yes	characteristic
132	C₄×Q₈	—	109	1	no	—
133	C₄×Q₈	—	109	1	no	—
134	C₄×Q₈	—	109	2	no	—
135	#117	—	109	2	no	—
136	C₂×C₄²	—	109	1	no	—
137	C₄⋊C₈	—	109	1	no	—
138	C₄⋊C₈	—	109	1	no	—
139	C₄⋊C₈	—	109	1	no	—
140	C₄⋊C₈	—	109	1	no	—
141	C₄⋊C₈	—	109	2	no	—
142	C₄⋊C₈	—	109	2	no	—
143	C₄⋊C₈	—	109	2	no	—
144	C₄⋊C₈	—	109	2	no	—
145	C₂²⋊C₈	—	109	2	no	—
146	C₂²⋊C₈	—	109	1	no	—
147	C₂²⋊C₈	—	109	1	no	—
148	#97	—	109	2	no	—
149	#97	—	109	2	no	—
150	C₄×C₈	—	109	2	no	—
151	C₄×C₈	—	109	1	no	—
152	C₄×C₈	—	109	1	no	—
153	C₄×C₈	—	109	2	no	—
154	C₄×C₈	—	109	2	no	—
155	C₂×Q₈	—	109	2	no	—
156	C₂×Q₈	—	109	1	no	—
157	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
158	C₂²×C₄	—	109	1	no	—
159	C₂²×C₄	—	109	1	no	—
160	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
161	C₂×C₈	—	109	2	no	—
162	C₂×C₈	—	109	1	no	—
163	C₂×C₈	—	109	1	no	—
164	C₂×C₈	—	109	2	no	—
165	C₂×C₈	—	109	1	no	—
166	C₂×C₈	—	109	1	no	—
167	C₄⋊C₄	—	109	2	no	—
168	C₄⋊C₄	—	109	2	no	—
169	C₄⋊C₄	—	109	2	no	—
170	C₄⋊C₄	—	109	2	no	—
171	C₄⋊C₄	—	109	2	no	—
172	C₄⋊C₄	—	109	2	no	—
173	C₄⋊C₄	—	109	1	no	—
174	C₄⋊C₄	C₂×#89137	1	2	yes	—
175	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
176	C₂²⋊C₄	—	109	2	no	—
177	C₂²⋊C₄	C₂×#89137	1	2	yes	—
178	C₄²	—	109	1	no	—
179	C₄²	—	109	1	no	—
180	C₄²	—	109	1	no	—
181	C₄²	—	109	1	no	—
182	C₄²	—	109	2	no	—
183	C₄²	—	109	2	no	—
184	C₂³	C₂²×#89137	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	218	4	no	—
187	Q₈	—	218	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	109	1	no	—
190	C₂×C₄	—	109	1	no	—
191	C₂×C₄	—	109	1	no	—
192	C₂×C₄	—	109	1	no	—
193	C₂×C₄	—	109	2	no	—
194	C₂×C₄	—	109	2	no	—
195	C₂×C₄	—	218	1	no	—
196	C₂×C₄	—	218	1	no	—
197	C₂×C₄	—	218	2	no	—
198	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
199	C₂×C₄	C₂²×#89137	1	2	yes	—
200	C₂×C₄	C₂²×#89137	1	2	yes	—
201	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
202	C₈	—	218	2	no	—
203	C₈	—	218	1	no	—
204	C₈	—	218	1	no	—
205	C₈	—	218	2	no	—
206	C₈	—	218	1	no	—
207	C₈	—	218	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂³×#89137	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	109	1	no	—
216	C₄	—	109	1	no	—
217	C₄	—	109	2	no	—
218	C₄	—	218	1	no	—
219	C₄	—	218	2	no	—
220	C₄	—	218	1	no	—
221	C₄	—	218	2	no	—
222	C₄	#16812786	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	#20296094	1	1	yes	characteristic
225	C₂	C₂×#16812786	1	1	yes	characteristic
226	C₂	D₁₀₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₂²⋊Q₈	#89137
hypocenter	C₁₀₉	#2986
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20296078	C₂	semidirect
2. #20296077	C₂	semidirect
3. #20296076	C₂	semidirect
4. #20296075	C₂	semidirect
5. #20296075	C₂	semidirect
6. Q₈×#11724786	C₂	semidirect
114. C₁₀₉	#2986	semidirect

Maximal subgroups

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

subgroup	index
#20296078	2
#20296077	2
#20296076	2
#20296075	2
Q₈×#11724786	2
#20296050	2
#20296043	2
#20296042	2
#20295578	2
#2986	109