Subgroups — #28379820

Order 13952.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#20296032	C₂	1	2	yes	maximal
8	#20296029	C₂	1	1	yes	maximal, characteristic
9	#20295990	C₂	1	2	yes	maximal
10	C₂×C₄×#11724786	C₂	1	1	yes	maximal, characteristic
11	#20295578	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182038	C₂²	1	2	yes	—
13	C₂²×#11724786	C₂²	1	1	yes	characteristic
14	C₂²×#11724786	C₂²	1	1	yes	characteristic
15	#16812766	C₂²	1	2	yes	—
16	#16812766	C₂²	1	2	yes	—
17	#16812766	C₂²	1	2	yes	—
18	C₄⋊#11724786	C₂²	1	1	yes	characteristic
19	C₄⋊#11724786	C₂²	1	1	yes	characteristic
20	C₄⋊#11724786	C₂²	1	2	yes	—
21	C₄⋊#11724786	C₂²	1	2	yes	—
22	#16812754	C₂²	1	2	yes	—
23	#16812754	C₂²	1	2	yes	—
24	C₄×#11724786	C₂²	1	1	yes	characteristic
25	C₄×#11724786	C₂²	1	1	yes	characteristic
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	#13182038	—	2	4	no	—
42	C₂×#11724786	—	2	2	no	—
43	C₂×#11724786	Q₈	1	4	yes	complemented
44	C₂×#11724786	C₂³	1	1	yes	characteristic
45	C₂×#11724786	C₂³	1	1	yes	characteristic
46	C₂×#11724786	C₂³	1	2	yes	—
47	C₂×#11724786	C₂³	1	2	yes	—
48	C₂×#11724786	C₂³	1	1	yes	characteristic
49	C₂×#11724786	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
51	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
56	#13182008	C₂×C₄	1	2	yes	—
57	#13182008	C₂×C₄	1	2	yes	—
58	#13182008	C₂×C₄	1	2	yes	—
59	#13182008	C₂×C₄	1	2	yes	—
60	#13182008	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₀₉	C₂³	1	2	yes	—
67	C₄×Dic₁₀₉	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
73	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₁₀₉	—	2	2	no	—
75	C₂×C₄×C₁₀₉	—	2	2	no	—
76	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₀₉	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
87	C₂×Dic₁₀₉	#41	1	2	yes	—
88	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
89	Dic₂₁₈	—	2	2	no	—
90	Dic₂₁₈	—	2	4	no	—
91	#11724786	—	2	1	no	—
92	#11724786	—	2	1	no	—
93	#11724786	—	2	2	no	—
94	#11724786	—	2	2	no	—
95	#11724786	C₂×Q₈	1	2	yes	—
96	#11724786	C₂×Q₈	1	2	yes	—
97	C₂²×C₁₀₉	—	2	1	no	—
98	C₂²×C₁₀₉	—	2	1	no	—
99	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
100	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₀₉	—	2	2	no	—
102	C₄×C₁₀₉	—	2	1	no	—
103	C₄×C₁₀₉	—	2	2	no	—
104	C₄×C₁₀₉	#131	1	2	yes	—
105	Dic₁₀₉	—	2	2	no	—
106	Dic₁₀₉	—	2	2	no	—
107	Dic₁₀₉	—	2	1	no	—
108	Dic₁₀₉	C₄×Q₈	1	2	yes	—
109	Dic₁₀₉	#131	1	2	yes	—
110	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
111	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
112	C₂×C₁₀₉	—	2	2	no	—
113	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
115	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
116	#2960	—	109	1	no	maximal, Sylow
117	C₁₀₉	#2960	1	1	yes	characteristic, complemented, Sylow
118	#533	—	109	1	no	—
119	#446	—	109	1	no	—
120	#446	—	109	1	no	—
121	#446	—	109	2	no	—
122	C₈×Q₈	—	109	1	no	—
123	C₈×Q₈	—	109	1	no	—
124	C₈×Q₈	—	109	2	no	—
125	#424	—	109	2	no	—
126	#423	—	109	1	no	—
127	#405	—	109	2	no	—
128	C₂×C₄×C₈	—	109	1	no	—
129	C₂×M₁₆	—	109	2	no	—
130	C₂²×C₈	—	109	1	no	—
131	C₂²×C₈	—	109	1	no	—
132	C₄⋊Q₈	—	109	1	no	—
133	C₄⋊Q₈	—	109	1	no	—
134	#125	—	109	2	no	—
135	C₂²⋊Q₈	—	109	1	no	—
136	C₂²⋊Q₈	—	109	2	no	—
137	C₂²⋊Q₈	#89137	1	1	yes	characteristic
138	C₄×Q₈	—	109	1	no	—
139	C₄×Q₈	—	109	1	no	—
140	C₄×Q₈	—	109	2	no	—
141	#117	—	109	2	no	—
142	C₂×C₄²	—	109	1	no	—
143	C₄⋊C₈	—	109	1	no	—
144	C₄⋊C₈	—	109	1	no	—
145	C₄⋊C₈	—	109	2	no	—
146	C₄⋊C₈	—	109	2	no	—
147	C₄⋊C₈	—	109	2	no	—
148	C₄⋊C₈	—	109	2	no	—
149	C₄⋊C₈	—	109	2	no	—
150	#97	—	109	2	no	—
151	#97	—	109	2	no	—
152	C₄×C₈	—	109	1	no	—
153	C₄×C₈	—	109	1	no	—
154	C₄×C₈	—	109	1	no	—
155	C₄×C₈	—	109	1	no	—
156	C₄×C₈	—	109	2	no	—
157	C₄×C₈	—	109	2	no	—
158	C₂×Q₈	—	109	2	no	—
159	C₂×Q₈	—	109	1	no	—
160	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
161	C₂²×C₄	—	109	1	no	—
162	C₂²×C₄	—	109	1	no	—
163	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
164	M₁₆	—	218	4	no	—
165	C₂×C₈	—	109	4	no	—
166	C₂×C₈	—	109	1	no	—
167	C₂×C₈	—	109	1	no	—
168	C₂×C₈	—	109	1	no	—
169	C₂×C₈	—	109	1	no	—
170	C₂×C₈	—	109	2	no	—
171	C₂×C₈	—	109	2	no	—
172	C₂×C₈	—	218	2	no	—
173	C₄⋊C₄	—	109	2	no	—
174	C₄⋊C₄	—	109	2	no	—
175	C₄⋊C₄	—	109	2	no	—
176	C₄⋊C₄	—	109	2	no	—
177	C₄⋊C₄	—	109	2	no	—
178	C₄⋊C₄	—	109	2	no	—
179	C₄⋊C₄	—	109	1	no	—
180	C₄⋊C₄	C₂×#89137	1	2	yes	—
181	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
182	C₂²⋊C₄	—	109	2	no	—
183	C₂²⋊C₄	C₂×#89137	1	2	yes	—
184	C₄²	—	109	1	no	—
185	C₄²	—	109	1	no	—
186	C₄²	—	109	1	no	—
187	C₄²	—	109	1	no	—
188	C₄²	—	109	2	no	—
189	C₄²	—	109	2	no	—
190	C₂³	C₂²×#89137	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	218	4	no	—
193	Q₈	—	218	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	109	2	no	—
196	C₂×C₄	—	109	2	no	—
197	C₂×C₄	—	109	1	no	—
198	C₂×C₄	—	109	1	no	—
199	C₂×C₄	—	109	1	no	—
200	C₂×C₄	—	109	1	no	—
201	C₂×C₄	—	109	2	no	—
202	C₂×C₄	—	109	2	no	—
203	C₂×C₄	—	218	2	no	—
204	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
205	C₂×C₄	C₂²×#89137	1	2	yes	—
206	C₂×C₄	C₂²×#89137	1	2	yes	—
207	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
208	C₈	—	109	2	no	—
209	C₈	—	109	2	no	—
210	C₈	—	218	1	no	—
211	C₈	—	218	1	no	—
212	C₈	—	218	2	no	—
213	C₈	—	218	2	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×#89137	1	2	yes	—
217	C₂²	C₂³×#89137	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	109	1	no	—
222	C₄	—	109	1	no	—
223	C₄	—	109	2	no	—
224	C₄	—	109	2	no	—
225	C₄	—	218	2	no	—
226	C₄	—	218	1	no	—
227	C₄	—	218	2	no	—
228	C₄	#16812786	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈×#89137	1	1	yes	characteristic
231	C₂	C₂×#16812786	1	1	yes	characteristic
232	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	#2960
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
43. C₂×#11724786	Q₈	semidirect
117. C₁₀₉	#2960	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
#20296032	2
#20296029	2
#20295990	2
C₂×C₄×#11724786	2
#20295578	2
#2960	109