Subgroups — #28372487

Order 13952.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812719	C₂	1	2	yes	maximal, complemented
2	#20295576	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295248	C₂	1	2	yes	maximal, complemented
4	#20295241	C₂	1	2	yes	maximal, complemented
5	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295226	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295221	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295020	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	2	yes	—
13	#16812719	—	2	8	no	—
14	C₂×#13182019	C₂²	1	2	yes	—
15	C₂×#13182017	C₂²	1	2	yes	—
16	C₂×#13182016	C₂²	1	2	yes	—
17	C₂×#13182014	C₂²	1	2	yes	—
18	C₂²×#11724784	C₂²	1	2	yes	—
19	#16812699	C₂²	1	1	yes	characteristic
20	#16812696	C₂²	1	1	yes	characteristic
21	#16812687	C₂²	1	2	yes	—
22	#16812673	—	2	1	no	—
23	#16812673	—	2	1	no	—
24	C₄×D₄₃₆	—	2	1	no	—
25	C₄×D₄₃₆	—	2	1	no	—
26	C₄⋊#11724784	C₂²	1	1	yes	characteristic
27	C₄⋊#11724784	C₂²	1	1	yes	characteristic
28	C₄⋊#11724784	C₂²	1	2	yes	—
29	#16812622	C₂²	1	2	yes	—
30	#16812622	C₂²	1	2	yes	—
31	#16812619	C₂²	1	2	yes	—
32	#16812619	C₂²	1	2	yes	—
33	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₀₉	—	2	1	no	—
37	C₄×D₄×C₁₀₉	—	2	1	no	—
38	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	—	2	1	no	—
41	#117×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₀₉	—	2	1	no	—
43	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
44	#13182029	—	2	4	no	—
45	#13182029	—	2	4	no	—
46	C₂×Dic₂₁₈	C₂³	1	2	yes	—
47	C₂×D₄₃₆	—	2	1	no	—
48	C₂×D₄₃₆	—	2	1	no	—
49	C₂×D₄₃₆	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
51	C₂×C₄×D₁₀₉	—	2	2	no	—
52	#13182019	—	2	4	no	—
53	#13182017	—	2	4	no	—
54	#13182016	—	2	4	no	—
55	#13182014	—	2	4	no	—
56	C₂×#11724784	—	2	4	no	—
57	C₂×#11724784	C₂³	1	2	yes	—
58	C₂×#11724784	C₂³	1	2	yes	—
59	C₄⋊Dic₁₀₉	—	2	1	no	—
60	C₄⋊Dic₁₀₉	—	2	1	no	—
61	#13182006	—	2	2	no	—
62	#13182006	—	2	2	no	—
63	#41×C₁₀₉	—	2	2	no	—
64	#41×C₁₀₉	D₄	1	4	yes	complemented
65	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₁₀₉	—	2	1	no	—
67	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₉	—	2	1	no	—
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	—	2	1	no	—
71	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₉	—	2	1	no	—
75	C₄⋊C₄×C₁₀₉	—	2	1	no	—
76	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
81	C₄²×C₁₀₉	—	2	1	no	—
82	C₄²×C₁₀₉	—	2	1	no	—
83	C₄²×C₁₀₉	—	2	1	no	—
84	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
85	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
86	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂³×C₁₀₉	—	2	1	no	—
89	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₀₉	—	2	2	no	—
91	Q₈×C₁₀₉	—	2	1	no	—
92	Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
93	D₄×C₁₀₉	—	2	4	no	—
94	D₄×C₁₀₉	—	2	1	no	—
95	D₄×C₁₀₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	—	2	1	no	—
97	C₂×C₄×C₁₀₉	—	2	4	no	—
98	C₂×C₄×C₁₀₉	—	2	2	no	—
99	C₂×C₄×C₁₀₉	—	2	1	no	—
100	C₂×C₄×C₁₀₉	—	2	1	no	—
101	C₂×C₄×C₁₀₉	—	4	1	no	—
102	C₂×C₄×C₁₀₉	—	4	2	no	—
103	C₂×C₄×C₁₀₉	—	4	2	no	—
104	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₀₉	#41	1	2	yes	—
111	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₀₉	—	4	4	no	—
113	C₂×Dic₁₀₉	—	2	2	no	—
114	D₄₃₆	—	2	2	no	—
115	D₄₃₆	—	2	2	no	—
116	D₄₃₆	—	4	2	no	—
117	D₄₃₆	—	4	2	no	—
118	C₄×D₁₀₉	—	4	4	no	—
119	Dic₂₁₈	—	2	2	no	—
120	Dic₂₁₈	—	2	2	no	—
121	#11724784	—	2	2	no	—
122	#11724784	—	2	2	no	—
123	C₂²×C₁₀₉	—	2	1	no	—
124	C₂²×C₁₀₉	—	2	1	no	—
125	C₂²×C₁₀₉	—	2	2	no	—
126	C₂²×C₁₀₉	—	4	1	no	—
127	C₂²×C₁₀₉	—	4	1	no	—
128	C₂²×C₁₀₉	#121	1	2	yes	—
129	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₁₀₉	—	4	2	no	—
131	C₂×D₁₀₉	—	4	2	no	—
132	C₂×D₁₀₉	—	4	2	no	—
133	C₄×C₁₀₉	—	2	2	no	—
134	C₄×C₁₀₉	—	2	1	no	—
135	C₄×C₁₀₉	—	2	1	no	—
136	C₄×C₁₀₉	—	2	1	no	—
137	C₄×C₁₀₉	—	4	1	no	—
138	C₄×C₁₀₉	—	4	1	no	—
139	C₄×C₁₀₉	#121	1	2	yes	—
140	C₄×C₁₀₉	#135	1	2	yes	complemented
141	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
143	Dic₁₀₉	—	4	2	no	—
144	C₂×C₁₀₉	—	2	2	no	—
145	C₂×C₁₀₉	—	4	2	no	—
146	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
147	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
148	C₂×C₁₀₉	#575	1	1	yes	characteristic
149	D₁₀₉	—	8	2	no	—
150	#3035	—	109	1	no	maximal, Sylow
151	C₁₀₉	#3035	1	1	yes	characteristic, complemented, Sylow
152	C₂×#135	—	109	2	no	—
153	#532	—	109	1	no	—
154	C₄×#41	D₁₀₉	1	1	yes	characteristic, complemented
155	#464	—	109	2	no	—
156	#463	—	109	2	no	—
157	#462	—	109	1	no	—
158	#461	—	109	1	no	—
159	#460	—	109	1	no	—
160	#459	—	109	1	no	—
161	C₂×C₄⋊C₈	—	109	1	no	—
162	#416	—	109	2	no	—
163	C₂×#41	—	109	2	no	—
164	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
165	#135	—	218	8	no	—
166	C₂×Q₁₆	—	109	2	no	—
167	C₂×SD₁₆	—	109	2	no	—
168	C₂×SD₁₆	—	109	2	no	—
169	C₂×D₈	—	109	2	no	—
170	C₂²×C₈	—	109	2	no	—
171	C₄⋊Q₈	—	109	1	no	—
172	#127	—	109	1	no	—
173	#124	—	109	2	no	—
174	#121	—	218	1	no	—
175	#121	—	218	1	no	—
176	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
177	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	218	1	no	—
181	C₄×D₄	—	218	1	no	—
182	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
183	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	#117	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
188	C₄⋊C₈	—	109	1	no	—
189	C₄⋊C₈	—	109	1	no	—
190	C₄⋊C₈	—	109	2	no	—
191	#103	—	109	2	no	—
192	#103	—	109	2	no	—
193	#102	—	109	2	no	—
194	#102	—	109	2	no	—
195	#41	—	2	2	no	—
196	#41	—	218	4	no	—
197	#41	—	218	4	no	—
198	#41	D₄₃₆	1	4	yes	complemented
199	C₂×Q₈	—	109	2	no	—
200	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	109	2	no	—
203	C₂×D₄	—	218	2	no	—
204	C₂×D₄	—	218	1	no	—
205	C₂×D₄	—	218	1	no	—
206	C₂×D₄	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	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
214	Q₁₆	—	218	4	no	—
215	SD₁₆	—	218	4	no	—
216	SD₁₆	—	218	4	no	—
217	D₈	—	218	4	no	—
218	C₂×C₈	—	109	2	no	—
219	C₂×C₈	—	109	2	no	—
220	C₂×C₈	—	218	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	218	1	no	—
224	C₄⋊C₄	—	218	1	no	—
225	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	218	2	no	—
231	C₂²⋊C₄	—	218	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
236	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
237	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	218	2	no	—
241	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	218	2	no	—
244	Q₈	—	218	2	no	—
245	Q₈	C₂×D₄₃₆	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	1	no	—
248	D₄	—	436	2	no	—
249	D₄	—	436	4	no	—
250	D₄	—	436	2	no	—
251	D₄	—	218	2	no	—
252	D₄	—	218	2	no	—
253	D₄	C₂×D₄₃₆	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	436	4	no	—
263	C₂×C₄	—	218	2	no	—
264	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
269	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
270	C₂×C₄	#13182029	1	2	yes	—
271	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
272	C₈	—	218	2	no	—
273	C₈	—	218	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	436	2	no	—
280	C₂²	—	436	2	no	—
281	C₂²	—	436	2	no	—
282	C₂²	#16812673	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	436	2	no	—
291	C₄	#16812673	1	2	yes	—
292	C₄	#16812719	1	2	yes	complemented
293	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
294	C₄	C₂×#13182029	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	872	2	no	—
298	C₂	C₂×#16812673	1	1	yes	characteristic
299	C₂	C₂×#16812719	1	1	yes	characteristic
300	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
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	#3035
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. C₂×#16812719	C₂	semidirect
2. #20295576	C₂	semidirect
3. #20295248	C₂	semidirect
4. #20295241	C₂	semidirect
5. #20295236	C₂	semidirect
6. #20295231	C₂	semidirect
7. #20295226	C₂	semidirect
8. #20295221	C₂	semidirect
9. C₂×C₄⋊#11724784	C₂	semidirect
10. #20295020	C₂	semidirect
11. C₄×#41×C₁₀₉	C₂	semidirect
64. #41×C₁₀₉	D₄	semidirect
140. C₄×C₁₀₉	#135	semidirect
151. C₁₀₉	#3035	semidirect
154. C₄×#41	D₁₀₉	semidirect
198. #41	D₄₃₆	semidirect
292. C₄	#16812719	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812719	2
#20295576	2
#20295248	2
#20295241	2
#20295236	2
#20295231	2
#20295226	2
#20295221	2
C₂×C₄⋊#11724784	2
#20295020	2
C₄×#41×C₁₀₉	2
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