Subgroups — #28372820

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295583	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295333	C₂	1	2	yes	maximal, complemented
3	#20295326	C₂	1	2	yes	maximal, complemented
4	#20295321	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295316	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295311	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295306	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295062	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295022	C₂	1	2	yes	maximal
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11724784	C₂²	1	2	yes	—
13	#16812699	C₂²	1	1	yes	characteristic
14	#16812699	C₂²	1	1	yes	characteristic
15	#16812691	C₂²	1	2	yes	—
16	#16812678	—	2	1	no	—
17	#16812678	—	2	1	no	—
18	C₄×Dic₂₁₈	—	2	1	no	—
19	C₄×Dic₂₁₈	—	2	1	no	—
20	#16812658	C₂²	1	2	yes	—
21	#16812632	C₂²	1	2	yes	—
22	#16812632	C₂²	1	2	yes	—
23	#16812630	C₂²	1	2	yes	—
24	#16812630	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	2	yes	—
28	#16812623	C₂²	1	2	yes	—
29	#16812623	C₂²	1	2	yes	—
30	#16812620	C₂²	1	2	yes	—
31	#16812620	C₂²	1	2	yes	—
32	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₀₉	—	2	1	no	—
36	C₄×D₄×C₁₀₉	—	2	1	no	—
37	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	#117×C₁₀₉	—	2	1	no	—
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₉	—	2	1	no	—
42	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₁₈	—	2	1	no	—
44	C₂×Dic₂₁₈	—	2	1	no	—
45	C₂×#11724784	—	2	4	no	—
46	C₂×#11724784	C₂³	1	2	yes	—
47	C₂×#11724784	C₂³	1	2	yes	—
48	C₄⋊Dic₁₀₉	—	2	1	no	—
49	C₄⋊Dic₁₀₉	—	2	1	no	—
50	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
51	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
52	#13182008	—	2	2	no	—
53	#13182008	—	2	2	no	—
54	C₂²⋊Dic₁₀₉	—	2	2	no	—
55	C₄×Dic₁₀₉	—	2	2	no	—
56	#41×C₁₀₉	—	2	2	no	—
57	#41×C₁₀₉	Q₈	1	4	yes	complemented
58	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₀₉	—	2	1	no	—
60	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	—	2	1	no	—
62	C₂²×C₄×C₁₀₉	—	2	1	no	—
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₉	—	2	1	no	—
68	C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄²×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
78	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂³×C₁₀₉	—	2	1	no	—
82	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₀₉	—	2	1	no	—
84	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
85	D₄×C₁₀₉	—	2	4	no	—
86	D₄×C₁₀₉	—	2	1	no	—
87	D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₁₀₉	—	2	1	no	—
89	C₂×C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	1	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	4	1	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
102	C₂×C₄×C₁₀₉	#41	1	2	yes	—
103	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₁₀₉	—	2	2	no	—
105	C₂×Dic₁₀₉	—	2	2	no	—
106	Dic₂₁₈	—	4	2	no	—
107	Dic₂₁₈	—	4	2	no	—
108	#11724784	—	2	2	no	—
109	#11724784	—	2	2	no	—
110	C₂²×C₁₀₉	—	2	1	no	—
111	C₂²×C₁₀₉	—	2	1	no	—
112	C₂²×C₁₀₉	—	2	2	no	—
113	C₂²×C₁₀₉	—	4	1	no	—
114	C₂²×C₁₀₉	—	4	1	no	—
115	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
116	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	2	1	no	—
119	C₄×C₁₀₉	—	2	1	no	—
120	C₄×C₁₀₉	—	2	1	no	—
121	C₄×C₁₀₉	—	4	1	no	—
122	C₄×C₁₀₉	—	4	1	no	—
123	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₁₀₉	#135	1	2	yes	—
125	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
126	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
127	Dic₁₀₉	—	4	2	no	—
128	Dic₁₀₉	—	4	2	no	—
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	—	4	2	no	—
131	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
133	C₂×C₁₀₉	#575	1	1	yes	characteristic
134	#3076	—	109	1	no	maximal, Sylow
135	C₁₀₉	#3076	1	1	yes	characteristic, complemented, Sylow
136	#533	—	109	1	no	—
137	C₄×#41	D₁₀₉	1	1	yes	characteristic
138	#479	—	109	2	no	—
139	#478	—	109	2	no	—
140	#477	—	109	1	no	—
141	#476	—	109	1	no	—
142	#475	—	109	1	no	—
143	#474	—	109	1	no	—
144	#427	—	109	2	no	—
145	C₂×C₄⋊C₈	—	109	1	no	—
146	#416	—	109	2	no	—
147	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
148	C₂²×C₈	—	109	2	no	—
149	C₄⋊Q₈	—	109	1	no	—
150	C₄⋊Q₈	—	109	1	no	—
151	#125	—	109	2	no	—
152	C₂²⋊Q₈	—	218	1	no	—
153	C₂²⋊Q₈	—	218	1	no	—
154	C₄×Q₈	—	218	1	no	—
155	C₄×Q₈	—	218	1	no	—
156	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
161	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	—	109	2	no	—
164	#117	C₂×D₁₀₉	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
167	#107	—	109	2	no	—
168	#107	—	109	2	no	—
169	#106	—	109	2	no	—
170	#106	—	109	2	no	—
171	C₄⋊C₈	—	109	1	no	—
172	C₄⋊C₈	—	109	1	no	—
173	C₄⋊C₈	—	109	2	no	—
174	#103	—	109	2	no	—
175	#103	—	109	2	no	—
176	#102	—	109	2	no	—
177	#102	—	109	2	no	—
178	#41	—	2	2	no	—
179	#41	Dic₂₁₈	1	4	yes	complemented
180	C₂×Q₈	—	218	1	no	—
181	C₂×Q₈	—	218	1	no	—
182	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₂×C₈	—	109	2	no	—
192	C₂×C₈	—	109	2	no	—
193	C₂×C₈	—	218	4	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	109	2	no	—
197	C₄⋊C₄	—	109	2	no	—
198	C₄⋊C₄	—	218	2	no	—
199	C₄⋊C₄	—	218	2	no	—
200	C₄⋊C₄	—	218	1	no	—
201	C₄⋊C₄	—	218	1	no	—
202	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	218	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	218	2	no	—
212	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
213	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
214	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
215	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	436	2	no	—
220	Q₈	—	436	2	no	—
221	Q₈	C₂×Dic₂₁₈	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Dic₂₁₈	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	218	2	no	—
234	C₂×C₄	—	218	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
241	C₂×C₄	#13182029	1	2	yes	—
242	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
243	C₈	—	218	2	no	—
244	C₈	—	218	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#16812678	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	436	2	no	—
259	C₄	—	436	2	no	—
260	C₄	#16812678	1	2	yes	—
261	C₄	#16812719	1	2	yes	—
262	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
263	C₄	C₂×#13182029	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#16812678	1	1	yes	characteristic
267	C₂	C₂×#16812719	1	1	yes	characteristic
268	C₂	#20295824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×#41×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	#3076
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20295583	C₂	semidirect
2. #20295333	C₂	semidirect
3. #20295326	C₂	semidirect
4. #20295321	C₂	semidirect
5. #20295316	C₂	semidirect
6. #20295311	C₂	semidirect
7. #20295306	C₂	semidirect
8. #20295062	C₂	semidirect
9. C₂×C₄⋊#11724784	C₂	semidirect
57. #41×C₁₀₉	Q₈	semidirect
135. C₁₀₉	#3076	semidirect
179. #41	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295583	2
#20295333	2
#20295326	2
#20295321	2
#20295316	2
#20295311	2
#20295306	2
#20295062	2
C₂×C₄⋊#11724784	2
#20295022	2
C₄×#41×C₁₀₉	2
#3076	109