Subgroups — #31790571

Order 17536.

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×#14555510	C₂	1	1	yes	maximal, characteristic
2	#22577707	C₂	1	2	yes	maximal, complemented
3	#22577700	C₂	1	2	yes	maximal, complemented
4	#22577695	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577690	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577685	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577680	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577439	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic, complemented
10	#22577397	C₂	1	2	yes	maximal
11	#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555510	C₂²	1	1	yes	characteristic
13	C₂²×#11764040	C₂²	1	2	yes	—
14	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
16	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
17	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
19	C₄×C₂²⋊D₁₃₇	—	2	1	no	—
20	#17984047	C₂²	1	1	yes	characteristic
21	#17984045	—	2	1	no	—
22	C₄²×D₁₃₇	—	2	1	no	—
23	#17984021	C₂²	1	2	yes	—
24	#17984021	C₂²	1	2	yes	—
25	#17984019	C₂²	1	2	yes	—
26	#17984019	C₂²	1	2	yes	—
27	C₄⋊#11764040	C₂²	1	1	yes	characteristic
28	C₄⋊#11764040	C₂²	1	1	yes	characteristic
29	C₄⋊#11764040	C₂²	1	2	yes	—
30	#17984011	C₂²	1	2	yes	—
31	#17984011	C₂²	1	2	yes	—
32	#17984008	C₂²	1	2	yes	—
33	#17984008	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
36	#125×C₁₃₇	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
38	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
39	C₄×Q₈×C₁₃₇	—	2	1	no	—
40	C₄×Q₈×C₁₃₇	—	2	1	no	—
41	#117×C₁₃₇	C₂²	1	2	yes	—
42	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
43	#14555510	—	2	2	no	—
44	#14555510	Q₈	1	4	yes	complemented
45	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
46	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
48	C₂×C₄×D₁₃₇	—	2	1	no	—
49	C₂×C₄×D₁₃₇	—	2	1	no	—
50	C₂×C₄×D₁₃₇	—	2	1	no	—
51	C₂×#11764040	—	2	4	no	—
52	C₂×#11764040	C₂³	1	2	yes	—
53	C₂×#11764040	C₂³	1	2	yes	—
54	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
56	#14555489	—	2	1	no	—
57	#14555489	—	2	1	no	—
58	C₂²⋊Dic₁₃₇	—	2	1	no	—
59	#14555487	—	2	1	no	—
60	#14555487	—	2	1	no	—
61	C₄×Dic₁₃₇	—	2	1	no	—
62	C₄×Dic₁₃₇	—	2	1	no	—
63	C₄×Dic₁₃₇	—	2	1	no	—
64	C₂×Q₈×C₁₃₇	—	2	1	no	—
65	C₂×Q₈×C₁₃₇	—	2	1	no	—
66	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₃₇	—	2	2	no	—
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	—	2	1	no	—
72	C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	—	2	2	no	—
77	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
81	C₂²×D₁₃₇	—	2	1	no	—
82	Dic₂₇₄	—	2	1	no	—
83	Dic₂₇₄	C₂×Q₈	1	2	yes	—
84	D₅₄₈	—	2	1	no	—
85	D₅₄₈	C₂×Q₈	1	2	yes	—
86	C₂²⋊D₁₃₇	—	2	4	no	—
87	C₂×Dic₁₃₇	—	2	1	no	—
88	C₂×Dic₁₃₇	—	2	1	no	—
89	C₂×Dic₁₃₇	—	2	1	no	—
90	C₄×D₁₃₇	—	2	4	no	—
91	C₄×D₁₃₇	—	4	2	no	—
92	C₄×D₁₃₇	—	4	2	no	—
93	#11764040	—	2	2	no	—
94	#11764040	—	2	2	no	—
95	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	4	2	no	—
97	Q₈×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	4	1	no	—
102	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₇	#41	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
110	C₂²×C₁₃₇	—	2	1	no	—
111	C₂²×C₁₃₇	—	2	1	no	—
112	C₂²×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₃₇	—	2	2	no	—
115	C₂×D₁₃₇	—	4	1	no	—
116	C₂×D₁₃₇	—	4	1	no	—
117	C₄×C₁₃₇	—	2	1	no	—
118	C₄×C₁₃₇	—	2	1	no	—
119	C₄×C₁₃₇	—	2	1	no	—
120	C₄×C₁₃₇	—	4	2	no	—
121	C₄×C₁₃₇	—	4	2	no	—
122	C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₁₃₇	#135	1	2	yes	—
124	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
126	Dic₁₃₇	—	2	2	no	—
127	Dic₁₃₇	—	4	1	no	—
128	Dic₁₃₇	—	4	1	no	—
129	C₂×C₁₃₇	—	2	2	no	—
130	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
132	C₂×C₁₃₇	#575	1	1	yes	characteristic
133	D₁₃₇	—	4	2	no	—
134	C₁₃₇	#3076	1	1	yes	characteristic, complemented, Sylow
135	#3076	—	137	1	no	maximal, Sylow
136	#533	D₁₃₇	1	1	yes	characteristic, complemented
137	C₄×#41	—	137	1	no	—
138	#479	—	137	2	no	—
139	#478	—	137	2	no	—
140	#477	—	137	1	no	—
141	#476	—	137	1	no	—
142	#475	—	137	1	no	—
143	#474	—	137	1	no	—
144	#427	—	137	2	no	—
145	C₂×C₄⋊C₈	—	137	1	no	—
146	#416	—	137	2	no	—
147	C₂×#41	—	137	1	no	—
148	C₂²×C₈	—	137	2	no	—
149	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
151	#125	C₂×D₁₃₇	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	—	137	1	no	—
157	C₄×Q₈	—	137	1	no	—
158	C₄×D₄	—	274	1	no	—
159	C₄×D₄	—	274	1	no	—
160	C₄×D₄	—	137	1	no	—
161	C₄×D₄	—	137	1	no	—
162	#117	—	274	1	no	—
163	#117	—	137	1	no	—
164	#117	C₂×D₁₃₇	1	2	yes	—
165	C₂×C₄²	—	274	1	no	—
166	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
167	#107	—	137	2	no	—
168	#107	—	137	2	no	—
169	#106	—	137	2	no	—
170	#106	—	137	2	no	—
171	C₄⋊C₈	—	137	1	no	—
172	C₄⋊C₈	—	137	1	no	—
173	C₄⋊C₈	—	137	2	no	—
174	#103	—	137	2	no	—
175	#103	—	137	2	no	—
176	#102	—	137	2	no	—
177	#102	—	137	2	no	—
178	#41	—	274	2	no	—
179	#41	—	137	4	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	137	1	no	—
183	C₂×D₄	—	274	1	no	—
184	C₂×D₄	—	137	1	no	—
185	C₂²×C₄	—	274	1	no	—
186	C₂²×C₄	—	274	1	no	—
187	C₂²×C₄	—	274	1	no	—
188	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
191	C₂×C₈	—	274	4	no	—
192	C₂×C₈	—	137	2	no	—
193	C₂×C₈	—	137	2	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	—	274	1	no	—
199	C₄⋊C₄	—	274	1	no	—
200	C₄⋊C₄	—	137	1	no	—
201	C₄⋊C₄	—	137	1	no	—
202	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
203	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	274	1	no	—
206	C₂²⋊C₄	—	274	1	no	—
207	C₂²⋊C₄	—	274	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	274	1	no	—
210	C₄²	—	274	1	no	—
211	C₄²	—	274	1	no	—
212	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
213	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
214	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
215	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
216	C₂³	—	274	1	no	—
217	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	274	1	no	—
221	Q₈	—	137	2	no	—
222	D₄	—	274	4	no	—
223	D₄	—	274	1	no	—
224	D₄	—	137	2	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	—	274	1	no	—
230	C₂×C₄	—	274	4	no	—
231	C₂×C₄	—	274	1	no	—
232	C₂×C₄	—	274	1	no	—
233	C₂×C₄	—	548	2	no	—
234	C₂×C₄	—	548	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
241	C₂×C₄	#14555511	1	2	yes	—
242	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
243	C₈	—	274	2	no	—
244	C₈	—	274	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	274	2	no	—
248	C₂²	—	548	1	no	—
249	C₂²	—	548	1	no	—
250	C₂²	#17984064	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₃₇	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	—	274	2	no	—
258	C₄	—	548	1	no	—
259	C₄	—	548	1	no	—
260	C₄	#17984064	1	2	yes	—
261	C₄	#17984108	1	2	yes	—
262	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
263	C₄	C₂×#14555511	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	548	2	no	—
266	C₂	C₂×#17984064	1	1	yes	characteristic
267	C₂	C₂×#17984108	1	1	yes	characteristic
268	C₂	#22578201	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#533×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×#14555510	C₂²
hypercenter	#533	D₁₃₇
hypocenter	C₁₃₇	#3076
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #22577707	C₂	semidirect
3. #22577700	C₂	semidirect
4. #22577695	C₂	semidirect
5. #22577690	C₂	semidirect
6. #22577685	C₂	semidirect
7. #22577680	C₂	semidirect
8. #22577439	C₂	semidirect
9. C₂×C₄⋊#11764040	C₂	semidirect
11. #533×C₁₃₇	C₂	semidirect
44. #14555510	Q₈	semidirect
134. C₁₃₇	#3076	semidirect
136. #533	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555510	2
#22577707	2
#22577700	2
#22577695	2
#22577690	2
#22577685	2
#22577680	2
#22577439	2
C₂×C₄⋊#11764040	2
#22577397	2
#533×C₁₃₇	2
#3076	137