Subgroups — #28373183

Order 13952.

430 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295576	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295350	C₂	1	2	yes	maximal, complemented
3	#20295311	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295306	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295226	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295221	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295171	C₂	1	2	yes	maximal, complemented
8	#20295136	C₂	1	2	yes	maximal, complemented
9	#20295124	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, complemented
12	C₂×#13182029	—	2	1	no	—
13	C₂×#13182016	C₂²	1	2	yes	—
14	C₂×#13182014	C₂²	1	2	yes	—
15	#16812699	C₂²	1	1	yes	characteristic
16	#16812696	C₂²	1	1	yes	characteristic
17	#16812687	—	2	1	no	—
18	#16812673	—	2	1	no	—
19	#16812673	C₂²	1	2	yes	—
20	C₄×D₄₃₆	—	2	1	no	—
21	C₄×D₄₃₆	C₂²	1	2	yes	—
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	#16812620	C₂²	1	2	yes	—
28	#16812619	C₂²	1	2	yes	—
29	#16812619	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	#13182029	—	4	4	no	—
44	C₂×Dic₂₁₈	—	2	1	no	—
45	C₂×D₄₃₆	—	2	1	no	—
46	C₂×D₄₃₆	—	2	1	no	—
47	C₂×D₄₃₆	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	#13182016	—	2	4	no	—
51	#13182014	—	2	4	no	—
52	C₂×#11724784	C₂³	1	2	yes	—
53	C₂×#11724784	C₂³	1	2	yes	—
54	C₄⋊Dic₁₀₉	—	2	1	no	—
55	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
56	#13182006	—	2	2	no	—
57	#13182006	—	2	2	no	—
58	#41×C₁₀₉	—	2	2	no	—
59	#41×C₁₀₉	—	2	2	no	—
60	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₀₉	—	2	1	no	—
62	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₂²×C₄×C₁₀₉	—	2	1	no	—
65	C₂²×C₄×C₁₀₉	—	2	1	no	—
66	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₀₉	—	2	1	no	—
70	C₄⋊C₄×C₁₀₉	—	2	1	no	—
71	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	—	2	1	no	—
79	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
80	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂³×C₁₀₉	—	2	1	no	—
84	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₁₀₉	—	2	2	no	—
86	Q₈×C₁₀₉	—	2	1	no	—
87	Q₈×C₁₀₉	#41	1	2	yes	complemented
88	D₄×C₁₀₉	—	2	4	no	—
89	D₄×C₁₀₉	—	2	1	no	—
90	D₄×C₁₀₉	#41	1	2	yes	complemented
91	C₂×C₄×C₁₀₉	—	2	4	no	—
92	C₂×C₄×C₁₀₉	—	2	1	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	1	no	—
95	C₂×C₄×C₁₀₉	—	2	1	no	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	1	no	—
98	C₂×C₄×C₁₀₉	—	4	1	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
107	C₂²⋊D₁₀₉	—	4	4	no	—
108	C₂×Dic₁₀₉	—	2	2	no	—
109	D₄₃₆	—	2	2	no	—
110	D₄₃₆	—	2	2	no	—
111	D₄₃₆	—	4	2	no	—
112	D₄₃₆	—	4	2	no	—
113	C₄×D₁₀₉	—	4	4	no	—
114	Dic₂₁₈	—	4	2	no	—
115	#11724784	—	2	2	no	—
116	#11724784	—	2	2	no	—
117	C₂²×C₁₀₉	—	2	1	no	—
118	C₂²×C₁₀₉	—	2	2	no	—
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	—	4	1	no	—
122	C₂²×C₁₀₉	—	4	1	no	—
123	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₁₀₉	—	4	2	no	—
125	C₂×D₁₀₉	—	4	2	no	—
126	C₂×D₁₀₉	—	4	2	no	—
127	C₄×C₁₀₉	—	2	1	no	—
128	C₄×C₁₀₉	—	2	2	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	—	2	1	no	—
132	C₄×C₁₀₉	—	4	1	no	—
133	C₄×C₁₀₉	—	4	1	no	—
134	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
136	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
137	Dic₁₀₉	—	4	2	no	—
138	C₂×C₁₀₉	—	4	1	no	—
139	C₂×C₁₀₉	—	4	2	no	—
140	C₂×C₁₀₉	#525	1	1	yes	characteristic
141	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₁₀₉	#575	1	1	yes	characteristic
143	D₁₀₉	—	8	2	no	—
144	#3115	—	109	1	no	maximal, Sylow
145	C₁₀₉	#3115	1	1	yes	characteristic, complemented, Sylow
146	#532	—	109	1	no	—
147	C₄×#41	D₁₀₉	1	1	yes	characteristic, complemented
148	#482	—	109	2	no	—
149	#475	—	109	1	no	—
150	#474	—	109	1	no	—
151	#460	—	109	1	no	—
152	#459	—	109	1	no	—
153	#449	—	109	2	no	—
154	#442	—	109	2	no	—
155	#440	—	109	2	no	—
156	#432	—	109	1	no	—
157	C₂×#41	—	218	1	no	—
158	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
159	C₂×SD₁₆	—	109	2	no	—
160	C₂×D₈	—	109	2	no	—
161	C₄⋊Q₈	—	109	1	no	—
162	#127	—	109	1	no	—
163	#124	—	218	1	no	—
164	#121	—	109	2	no	—
165	#121	—	218	1	no	—
166	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
167	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	109	2	no	—
171	C₄×D₄	—	218	1	no	—
172	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
173	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	#117	C₂×D₁₀₉	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
178	#107	—	109	2	no	—
179	#106	—	109	2	no	—
180	C₄⋊C₈	—	109	1	no	—
181	C₄⋊C₈	—	109	1	no	—
182	#103	—	109	2	no	—
183	#102	—	109	2	no	—
184	#102	—	109	2	no	—
185	#102	—	109	2	no	—
186	C₂²⋊C₈	—	109	2	no	—
187	#97	—	109	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	436	4	no	—
191	C₂×Q₈	—	218	1	no	—
192	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	109	2	no	—
195	C₂×D₄	—	218	2	no	—
196	C₂×D₄	—	218	1	no	—
197	C₂×D₄	—	218	1	no	—
198	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	218	2	no	—
203	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
206	SD₁₆	—	218	4	no	—
207	D₈	—	218	4	no	—
208	C₂×C₈	—	109	2	no	—
209	C₂×C₈	—	109	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	109	2	no	—
213	C₄⋊C₄	—	218	1	no	—
214	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	218	2	no	—
220	C₂²⋊C₄	—	218	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
225	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
226	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	218	2	no	—
230	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	436	2	no	—
233	Q₈	#13182029	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	436	2	no	—
237	D₄	—	436	4	no	—
238	D₄	—	436	2	no	—
239	D₄	—	218	2	no	—
240	D₄	—	218	2	no	—
241	D₄	#13182029	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	436	4	no	—
253	C₂×C₄	—	218	2	no	—
254	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
260	C₈	—	218	2	no	—
261	C₈	—	218	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	436	2	no	—
269	C₂²	—	436	2	no	—
270	C₂²	—	436	2	no	—
271	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	436	2	no	—
280	C₄	#16812724	1	2	yes	—
281	C₄	C₂×#13182029	1	1	yes	characteristic
282	C₄	C₂×#13182029	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	872	2	no	—
286	C₂	#20295537	1	1	yes	characteristic
287	C₂	C₂×#16812724	1	1	yes	characteristic
288	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
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	#3115
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. #20295576	C₂	semidirect
2. #20295350	C₂	semidirect
3. #20295311	C₂	semidirect
4. #20295306	C₂	semidirect
5. #20295226	C₂	semidirect
6. #20295221	C₂	semidirect
7. #20295171	C₂	semidirect
8. #20295136	C₂	semidirect
9. #20295124	C₂	semidirect
10. #20295083	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
87. Q₈×C₁₀₉	#41	semidirect
90. D₄×C₁₀₉	#41	semidirect
145. C₁₀₉	#3115	semidirect
147. C₄×#41	D₁₀₉	semidirect
233. Q₈	#13182029	semidirect
241. D₄	#13182029	semidirect

Maximal subgroups

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

subgroup	index
#20295576	2
#20295350	2
#20295311	2
#20295306	2
#20295226	2
#20295221	2
#20295171	2
#20295136	2
#20295124	2
#20295083	2
C₄×#41×C₁₀₉	2
#3115	109