Subgroups — #28372814

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	C₄×#13182029	C₂	1	1	yes	maximal, characteristic
2	#20295330	C₂	1	2	yes	maximal, complemented
3	#20295323	C₂	1	2	yes	maximal, complemented
4	#20295318	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295313	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295308	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295303	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	#20295020	C₂	1	2	yes	maximal
11	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	2	yes	—
14	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
16	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
17	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
19	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
20	#16812658	C₂²	1	1	yes	characteristic
21	#16812656	—	2	1	no	—
22	C₄²×D₁₀₉	—	2	1	no	—
23	#16812632	C₂²	1	2	yes	—
24	#16812632	C₂²	1	2	yes	—
25	#16812630	C₂²	1	2	yes	—
26	#16812630	C₂²	1	2	yes	—
27	C₄⋊#11724784	C₂²	1	1	yes	characteristic
28	C₄⋊#11724784	C₂²	1	1	yes	characteristic
29	C₄⋊#11724784	C₂²	1	2	yes	—
30	#16812622	C₂²	1	2	yes	—
31	#16812622	C₂²	1	2	yes	—
32	#16812619	C₂²	1	2	yes	—
33	#16812619	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	#125×C₁₀₉	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
38	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
39	C₄×Q₈×C₁₀₉	—	2	1	no	—
40	C₄×Q₈×C₁₀₉	—	2	1	no	—
41	#117×C₁₀₉	C₂²	1	2	yes	—
42	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
43	#13182029	—	2	2	no	—
44	#13182029	Q₈	1	4	yes	complemented
45	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
46	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	1	no	—
50	C₂×C₄×D₁₀₉	—	2	1	no	—
51	C₂×#11724784	—	2	4	no	—
52	C₂×#11724784	C₂³	1	2	yes	—
53	C₂×#11724784	C₂³	1	2	yes	—
54	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
56	#13182008	—	2	1	no	—
57	#13182008	—	2	1	no	—
58	C₂²⋊Dic₁₀₉	—	2	1	no	—
59	#13182006	—	2	1	no	—
60	#13182006	—	2	1	no	—
61	C₄×Dic₁₀₉	—	2	1	no	—
62	C₄×Dic₁₀₉	—	2	1	no	—
63	C₄×Dic₁₀₉	—	2	1	no	—
64	C₂×Q₈×C₁₀₉	—	2	1	no	—
65	C₂×Q₈×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	2	no	—
70	C₄⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄⋊C₄×C₁₀₉	—	2	1	no	—
72	C₄⋊C₄×C₁₀₉	—	2	1	no	—
73	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
76	C₄²×C₁₀₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₀₉	—	2	1	no	—
83	Q₈×C₁₀₉	—	4	2	no	—
84	Q₈×C₁₀₉	—	4	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	4	1	no	—
89	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₁₀₉	#41	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₀₉	—	2	4	no	—
98	C₂×Dic₁₀₉	—	2	1	no	—
99	C₂×Dic₁₀₉	—	2	1	no	—
100	C₂×Dic₁₀₉	—	2	1	no	—
101	D₄₃₆	—	2	1	no	—
102	D₄₃₆	C₂×Q₈	1	2	yes	—
103	C₄×D₁₀₉	—	2	4	no	—
104	C₄×D₁₀₉	—	4	2	no	—
105	C₄×D₁₀₉	—	4	2	no	—
106	Dic₂₁₈	—	2	1	no	—
107	Dic₂₁₈	C₂×Q₈	1	2	yes	—
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	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₀₉	—	2	2	no	—
115	C₂×D₁₀₉	—	4	1	no	—
116	C₂×D₁₀₉	—	4	1	no	—
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	2	1	no	—
119	C₄×C₁₀₉	—	2	1	no	—
120	C₄×C₁₀₉	—	4	2	no	—
121	C₄×C₁₀₉	—	4	2	no	—
122	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₁₀₉	#135	1	2	yes	—
124	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
126	Dic₁₀₉	—	2	2	no	—
127	Dic₁₀₉	—	4	1	no	—
128	Dic₁₀₉	—	4	1	no	—
129	C₂×C₁₀₉	—	2	2	no	—
130	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
132	C₂×C₁₀₉	#575	1	1	yes	characteristic
133	D₁₀₉	—	4	2	no	—
134	#3076	—	109	1	no	maximal, Sylow
135	C₁₀₉	#3076	1	1	yes	characteristic, complemented, Sylow
136	#533	D₁₀₉	1	1	yes	characteristic, complemented
137	C₄×#41	—	109	1	no	—
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	—	109	1	no	—
148	C₂²×C₈	—	109	2	no	—
149	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
151	#125	C₂×D₁₀₉	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	—	109	1	no	—
157	C₄×Q₈	—	109	1	no	—
158	C₄×D₄	—	109	1	no	—
159	C₄×D₄	—	109	1	no	—
160	C₄×D₄	—	218	1	no	—
161	C₄×D₄	—	218	1	no	—
162	#117	—	109	1	no	—
163	#117	—	218	1	no	—
164	#117	C₂×D₁₀₉	1	2	yes	—
165	C₂×C₄²	—	218	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	—	109	4	no	—
179	#41	—	218	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	109	1	no	—
183	C₂×D₄	—	109	1	no	—
184	C₂×D₄	—	218	1	no	—
185	C₂²×C₄	—	218	1	no	—
186	C₂²×C₄	—	218	1	no	—
187	C₂²×C₄	—	218	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	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	109	1	no	—
199	C₄⋊C₄	—	109	1	no	—
200	C₄⋊C₄	—	218	1	no	—
201	C₄⋊C₄	—	218	1	no	—
202	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
203	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	218	1	no	—
206	C₂²⋊C₄	—	218	1	no	—
207	C₂²⋊C₄	—	218	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	218	1	no	—
210	C₄²	—	218	1	no	—
211	C₄²	—	218	1	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	—	218	1	no	—
217	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	109	2	no	—
221	Q₈	—	218	1	no	—
222	D₄	—	109	2	no	—
223	D₄	—	218	4	no	—
224	D₄	—	218	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	436	2	no	—
230	C₂×C₄	—	436	2	no	—
231	C₂×C₄	—	218	1	no	—
232	C₂×C₄	—	218	1	no	—
233	C₂×C₄	—	218	1	no	—
234	C₂×C₄	—	218	4	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	Q₈×D₁₀₉	1	2	yes	—
241	C₂×C₄	#13182030	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	—	436	1	no	—
248	C₂²	—	436	1	no	—
249	C₂²	—	218	2	no	—
250	C₂²	#16812675	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	436	1	no	—
258	C₄	—	436	1	no	—
259	C₄	—	218	2	no	—
260	C₄	#16812675	1	2	yes	—
261	C₄	#16812719	1	2	yes	—
262	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
263	C₄	C₂×#13182030	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	436	2	no	—
266	C₂	C₂×#16812675	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	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#13182029	C₂²
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3076
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #20295330	C₂	semidirect
3. #20295323	C₂	semidirect
4. #20295318	C₂	semidirect
5. #20295313	C₂	semidirect
6. #20295308	C₂	semidirect
7. #20295303	C₂	semidirect
8. #20295062	C₂	semidirect
9. C₂×C₄⋊#11724784	C₂	semidirect
11. #533×C₁₀₉	C₂	semidirect
44. #13182029	Q₈	semidirect
135. C₁₀₉	#3076	semidirect
136. #533	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182029	2
#20295330	2
#20295323	2
#20295318	2
#20295313	2
#20295308	2
#20295303	2
#20295062	2
C₂×C₄⋊#11724784	2
#20295020	2
#533×C₁₀₉	2
#3076	109