Subgroups — #28372008

Order 13952.

380 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	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295104	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295098	C₂	1	2	yes	maximal, complemented
5	D₄×#11724784	C₂	1	2	yes	maximal, complemented
6	D₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295087	C₂	1	2	yes	maximal, complemented
8	#20295083	C₂	1	1	yes	maximal, characteristic, complemented
9	#20294982	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182013	C₂²	1	2	yes	—
13	C₂²×#11724784	C₂²	1	1	yes	characteristic
14	C₂²×#11724784	C₂²	1	1	yes	characteristic
15	C₂²×#11724784	C₂²	1	2	yes	—
16	C₄⋊#11724784	C₂²	1	1	yes	characteristic
17	C₄⋊#11724784	C₂²	1	1	yes	characteristic
18	C₄⋊#11724784	C₂²	1	2	yes	—
19	C₄⋊#11724784	C₂²	1	2	yes	—
20	C₂²⋊#11724784	C₂²	1	2	yes	—
21	C₂²⋊#11724784	C₂²	1	2	yes	—
22	C₂²⋊#11724784	C₂²	1	2	yes	—
23	#16812610	C₂²	1	2	yes	—
24	#16812610	C₂²	1	2	yes	—
25	C₄×#11724784	C₂²	1	1	yes	characteristic
26	C₄×#11724784	C₂²	1	1	yes	characteristic
27	C₄×#11724784	C₂²	1	1	yes	characteristic
28	C₄×#11724784	C₂²	1	1	yes	characteristic
29	C₄×#11724784	C₂²	1	2	yes	—
30	#126×C₁₀₉	C₄	1	2	yes	—
31	#125×C₁₀₉	C₄	1	1	yes	characteristic
32	#124×C₁₀₉	C₄	1	1	yes	characteristic
33	#123×C₁₀₉	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
35	#121×C₁₀₉	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	#117×C₁₀₉	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
41	#13182013	—	2	4	no	—
42	C₂×#11724784	—	2	2	no	—
43	C₂×#11724784	—	2	4	no	—
44	C₂×#11724784	—	2	2	no	—
45	C₂×#11724784	C₂³	1	1	yes	characteristic
46	C₂×#11724784	C₂³	1	1	yes	characteristic
47	C₂×#11724784	C₂³	1	2	yes	—
48	C₂×#11724784	C₂³	1	2	yes	—
49	C₂×#11724784	C₂³	1	1	yes	characteristic
50	C₂×#11724784	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
53	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
62	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
69	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₀₉	C₂³	1	2	yes	—
74	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	2	2	no	—
77	D₄×C₁₀₉	—	2	4	no	—
78	D₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	1	no	—
82	C₂×C₄×C₁₀₉	—	2	1	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
94	#11724784	—	2	1	no	—
95	#11724784	—	2	1	no	—
96	#11724784	—	2	2	no	—
97	#11724784	—	2	2	no	—
98	#11724784	#41	1	2	yes	—
99	#11724784	#41	1	2	yes	—
100	C₂²×C₁₀₉	—	2	1	no	—
101	C₂²×C₁₀₉	—	2	1	no	—
102	C₂²×C₁₀₉	—	2	2	no	—
103	C₂²×C₁₀₉	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	2	no	—
105	C₂²×C₁₀₉	#131	1	2	yes	—
106	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	2	1	no	—
111	C₄×C₁₀₉	—	2	1	no	—
112	C₄×C₁₀₉	—	2	2	no	—
113	C₄×C₁₀₉	—	2	2	no	—
114	C₄×C₁₀₉	#131	1	2	yes	—
115	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₀₉	—	2	2	no	—
118	C₂×C₁₀₉	—	4	2	no	—
119	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
121	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
122	#2966	—	109	1	no	maximal, Sylow
123	C₁₀₉	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	D₁₀₉	1	1	yes	characteristic
125	#446	—	109	1	no	—
126	C₈×Q₈	—	109	1	no	—
127	#436	—	109	1	no	—
128	#435	—	109	2	no	—
129	C₈×D₄	—	109	1	no	—
130	C₈×D₄	—	109	2	no	—
131	#433	—	109	2	no	—
132	#432	—	109	1	no	—
133	#405	—	109	2	no	—
134	C₂×C₄×C₈	—	109	1	no	—
135	C₂×M₁₆	—	109	2	no	—
136	C₂²×C₈	—	109	1	no	—
137	C₂²×C₈	—	109	1	no	—
138	C₂²×C₈	—	109	2	no	—
139	#126	Dic₁₀₉	1	2	yes	—
140	#125	Dic₁₀₉	1	1	yes	characteristic
141	#124	Dic₁₀₉	1	1	yes	characteristic
142	#123	Dic₁₀₉	1	2	yes	—
143	C₂²⋊Q₈	Dic₁₀₉	1	1	yes	characteristic
144	#121	Dic₁₀₉	1	1	yes	characteristic
145	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
146	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
147	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
148	#117	C₂×D₁₀₉	1	2	yes	—
149	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄⋊C₈	—	109	1	no	—
151	C₄⋊C₈	—	109	1	no	—
152	C₄⋊C₈	—	109	2	no	—
153	C₄⋊C₈	—	109	2	no	—
154	C₂²⋊C₈	—	109	2	no	—
155	C₂²⋊C₈	—	109	2	no	—
156	C₂²⋊C₈	—	109	2	no	—
157	#97	—	109	2	no	—
158	#97	—	109	2	no	—
159	C₄×C₈	—	109	1	no	—
160	C₄×C₈	—	109	1	no	—
161	C₄×C₈	—	109	1	no	—
162	C₄×C₈	—	109	1	no	—
163	C₄×C₈	—	109	2	no	—
164	C₂×Q₈	C₂×Dic₁₀₉	1	1	yes	characteristic
165	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
166	C₂×D₄	C₂×Dic₁₀₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
171	M₁₆	—	218	4	no	—
172	C₂×C₈	—	109	1	no	—
173	C₂×C₈	—	109	1	no	—
174	C₂×C₈	—	109	1	no	—
175	C₂×C₈	—	109	1	no	—
176	C₂×C₈	—	109	2	no	—
177	C₂×C₈	—	109	2	no	—
178	C₂×C₈	—	218	2	no	—
179	C₂×C₈	—	218	4	no	—
180	C₂×C₈	—	218	2	no	—
181	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
182	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
185	C₄⋊C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
186	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
190	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
191	C₂²⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
192	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
194	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
195	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₄²	C₂²×D₁₀₉	1	2	yes	—
197	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
198	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
213	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
214	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
215	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
217	C₈	—	109	2	no	—
218	C₈	—	109	2	no	—
219	C₈	—	218	1	no	—
220	C₈	—	218	1	no	—
221	C₈	—	218	2	no	—
222	C₈	—	218	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	#16812706	1	2	yes	—
229	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	#16812706	1	2	yes	—
238	C₄	#41×D₁₀₉	1	1	yes	characteristic
239	C₄	#41×D₁₀₉	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	#41×Dic₁₀₉	1	1	yes	characteristic
243	C₂	C₂×#16812706	1	1	yes	characteristic
244	C₂	C₂×#16812706	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#529×C₁₀₉	C₂
prime radical	#121×C₁₀₉	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#529	D₁₀₉
hypocenter	C₁₀₉	#2966
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #20295157	C₂	semidirect
2. Q₈×#11724784	C₂	semidirect
3. #20295104	C₂	semidirect
4. #20295098	C₂	semidirect
5. D₄×#11724784	C₂	semidirect
6. D₄×#11724784	C₂	semidirect
7. #20295087	C₂	semidirect
8. #20295083	C₂	semidirect
9. #20294982	C₂	semidirect
10. C₂×C₄×#11724784	C₂	semidirect
123. C₁₀₉	#2966	semidirect

Maximal subgroups

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

subgroup	index
#20295157	2
Q₈×#11724784	2
#20295104	2
#20295098	2
D₄×#11724784	2
#20295087	2
#20295083	2
#20294982	2
C₂×C₄×#11724784	2
#529×C₁₀₉	2
#2966	109