Subgroups — #28373194

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 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	#20295367	C₂	1	2	yes	maximal, complemented
3	#20295321	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295316	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295188	C₂	1	2	yes	maximal
8	#20295131	C₂	1	2	yes	maximal, complemented
9	#20295126	C₂	1	2	yes	maximal, complemented
10	#20295083	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182019	C₂²	1	2	yes	—
13	C₂×#13182017	C₂²	1	2	yes	—
14	#16812699	C₂²	1	1	yes	characteristic
15	#16812699	C₂²	1	1	yes	characteristic
16	#16812691	—	2	1	no	—
17	#16812678	—	2	1	no	—
18	#16812678	C₂²	1	2	yes	—
19	C₄×Dic₂₁₈	—	2	1	no	—
20	C₄×Dic₂₁₈	C₂²	1	2	yes	—
21	#16812658	—	2	1	no	—
22	#16812632	C₂²	1	2	yes	—
23	#16812630	C₂²	1	2	yes	—
24	C₄⋊#11724784	C₂²	1	1	yes	characteristic
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	#16812623	C₂²	1	2	yes	—
27	#16812622	C₂²	1	2	yes	—
28	#16812622	C₂²	1	2	yes	—
29	#16812620	C₂²	1	2	yes	—
30	C₂²⋊#11724784	C₂²	1	2	yes	—
31	#16812610	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	C₂³	1	2	yes	—
45	#13182019	—	2	4	no	—
46	#13182017	—	2	4	no	—
47	C₂×#11724784	C₂³	1	2	yes	—
48	C₂×#11724784	C₂³	1	2	yes	—
49	C₄⋊Dic₁₀₉	—	2	1	no	—
50	C₄⋊Dic₁₀₉	—	2	1	no	—
51	C₄⋊Dic₁₀₉	—	2	1	no	—
52	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
53	#13182008	—	2	2	no	—
54	#13182008	—	2	2	no	—
55	C₂²⋊Dic₁₀₉	—	2	2	no	—
56	C₄×Dic₁₀₉	—	2	2	no	—
57	#41×C₁₀₉	—	2	2	no	—
58	#41×C₁₀₉	—	2	2	no	—
59	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₀₉	—	2	1	no	—
61	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	—	2	1	no	—
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₂²×C₄×C₁₀₉	—	2	1	no	—
65	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₀₉	—	2	1	no	—
69	C₄⋊C₄×C₁₀₉	—	2	1	no	—
70	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₄²×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂³×C₁₀₉	—	2	1	no	—
83	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₀₉	—	2	1	no	—
85	Q₈×C₁₀₉	#41	1	2	yes	—
86	D₄×C₁₀₉	—	2	4	no	—
87	D₄×C₁₀₉	—	2	1	no	—
88	D₄×C₁₀₉	#41	1	2	yes	—
89	C₂×C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	1	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	4	1	no	—
97	C₂×C₄×C₁₀₉	—	4	2	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
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×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₁₀₉	—	2	2	no	—
106	C₂×Dic₁₀₉	—	2	2	no	—
107	Dic₂₁₈	—	2	2	no	—
108	Dic₂₁₈	—	2	2	no	—
109	Dic₂₁₈	—	4	2	no	—
110	#11724784	—	2	2	no	—
111	#11724784	—	2	2	no	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	2	2	no	—
114	C₂²×C₁₀₉	—	2	1	no	—
115	C₂²×C₁₀₉	—	2	1	no	—
116	C₂²×C₁₀₉	—	4	1	no	—
117	C₂²×C₁₀₉	—	4	1	no	—
118	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₁₀₉	—	2	1	no	—
120	C₄×C₁₀₉	—	2	2	no	—
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	—	2	1	no	—
123	C₄×C₁₀₉	—	2	1	no	—
124	C₄×C₁₀₉	—	4	1	no	—
125	C₄×C₁₀₉	—	4	1	no	—
126	C₄×C₁₀₉	#137	1	2	yes	—
127	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
128	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
129	Dic₁₀₉	—	4	2	no	—
130	Dic₁₀₉	—	4	2	no	—
131	C₂×C₁₀₉	—	4	1	no	—
132	C₂×C₁₀₉	—	4	2	no	—
133	C₂×C₁₀₉	#525	1	1	yes	characteristic
134	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
135	C₂×C₁₀₉	#575	1	1	yes	characteristic
136	#3116	—	109	1	no	maximal, Sylow
137	C₁₀₉	#3116	1	1	yes	characteristic, complemented, Sylow
138	#533	—	109	1	no	—
139	C₄×#41	D₁₀₉	1	1	yes	characteristic
140	#485	—	109	2	no	—
141	#477	—	109	1	no	—
142	#476	—	109	1	no	—
143	#462	—	109	1	no	—
144	#461	—	109	1	no	—
145	#452	—	109	2	no	—
146	#441	—	109	2	no	—
147	#440	—	109	2	no	—
148	#432	—	109	1	no	—
149	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
150	C₂×Q₁₆	—	109	2	no	—
151	C₂×SD₁₆	—	109	2	no	—
152	C₄⋊Q₈	—	109	1	no	—
153	C₄⋊Q₈	—	109	1	no	—
154	#125	—	218	1	no	—
155	C₂²⋊Q₈	—	109	2	no	—
156	C₂²⋊Q₈	—	218	1	no	—
157	C₄×Q₈	—	109	2	no	—
158	C₄×Q₈	—	218	1	no	—
159	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
164	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
165	#117	—	2	1	no	—
166	#117	—	218	1	no	—
167	#117	C₂×D₁₀₉	1	1	yes	characteristic
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
170	#107	—	109	2	no	—
171	#106	—	109	2	no	—
172	C₄⋊C₈	—	109	1	no	—
173	C₄⋊C₈	—	109	1	no	—
174	#103	—	109	2	no	—
175	#103	—	109	2	no	—
176	#103	—	109	2	no	—
177	#102	—	109	2	no	—
178	C₂²⋊C₈	—	109	2	no	—
179	#97	—	109	2	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	109	2	no	—
183	C₂×Q₈	—	218	1	no	—
184	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
193	Q₁₆	—	218	4	no	—
194	SD₁₆	—	218	4	no	—
195	C₂×C₈	—	109	2	no	—
196	C₂×C₈	—	109	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	109	2	no	—
200	C₄⋊C₄	—	218	2	no	—
201	C₄⋊C₄	—	218	2	no	—
202	C₄⋊C₄	—	218	1	no	—
203	C₄⋊C₄	—	218	1	no	—
204	C₄⋊C₄	—	218	1	no	—
205	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	218	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	218	2	no	—
215	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
216	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
217	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
218	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	436	2	no	—
223	Q₈	—	218	2	no	—
224	Q₈	—	218	2	no	—
225	Q₈	#13182029	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#13182029	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	218	2	no	—
240	C₂×C₄	—	218	2	no	—
241	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
247	C₈	—	218	2	no	—
248	C₈	—	218	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	436	2	no	—
264	C₄	—	436	2	no	—
265	C₄	#16812729	1	2	yes	—
266	C₄	C₂×#13182029	1	1	yes	characteristic
267	C₄	C₂×#13182029	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	#20295537	1	1	yes	characteristic
271	C₂	C₂×#16812729	1	1	yes	characteristic
272	C₂	#20295824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×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	#3116
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ 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. #20295367	C₂	semidirect
3. #20295321	C₂	semidirect
4. #20295316	C₂	semidirect
5. #20295236	C₂	semidirect
6. #20295231	C₂	semidirect
8. #20295131	C₂	semidirect
9. #20295126	C₂	semidirect
10. #20295083	C₂	semidirect
137. C₁₀₉	#3116	semidirect

Maximal subgroups

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

subgroup	index
#20295583	2
#20295367	2
#20295321	2
#20295316	2
#20295236	2
#20295231	2
#20295188	2
#20295131	2
#20295126	2
#20295083	2
C₄×#41×C₁₀₉	2
#3116	109