Subgroups — #31791315

Order 17536.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578031	C₂	1	1	yes	maximal, characteristic
2	#22577813	C₂	1	2	yes	maximal, complemented
3	#22577809	C₂	1	2	yes	maximal, complemented
4	#22577802	C₂	1	2	yes	maximal, complemented
5	#22577799	C₂	1	2	yes	maximal, complemented
6	#22577446	C₂	1	1	yes	maximal, characteristic
7	#22577443	C₂	1	1	yes	maximal, characteristic
8	#22577439	C₂	1	1	yes	maximal, characteristic
9	C₂×#17984019	C₂	1	1	yes	maximal, characteristic
10	#22577359	C₂	1	1	yes	maximal, characteristic
11	#544×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555494	C₂²	1	1	yes	characteristic
13	C₂²×#11764040	C₂²	1	1	yes	characteristic
14	#17984088	C₂²	1	2	yes	—
15	#17984087	—	2	1	no	—
16	#17984080	C₂²	1	2	yes	—
17	#17984079	—	2	1	no	—
18	#17984065	—	2	2	no	—
19	#17984047	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
21	#17984021	C₂²	1	1	yes	characteristic
22	#17984021	C₂²	1	1	yes	characteristic
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	#17984019	C₂²	1	1	yes	characteristic
28	#17984019	C₂²	1	1	yes	characteristic
29	#17984019	C₂²	1	2	yes	—
30	#17984019	C₂²	1	2	yes	—
31	#17983999	C₂²	1	2	yes	—
32	C₄×#11764040	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
34	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
35	#125×C₁₃₇	—	2	1	no	—
36	#125×C₁₃₇	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
38	#117×C₁₃₇	C₂²	1	1	yes	characteristic
39	#117×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₇₄	—	2	2	no	—
42	C₂²×Dic₁₃₇	—	2	1	no	—
43	#14555494	Q₈	1	4	yes	complemented
44	C₂×#11764040	Q₈	1	4	yes	complemented
45	C₂×#11764040	C₂³	1	1	yes	characteristic
46	C₂×#11764040	C₂³	1	1	yes	characteristic
47	C₂×#11764040	C₂³	1	2	yes	—
48	C₄⋊Dic₁₃₇	—	2	1	no	—
49	C₄⋊Dic₁₃₇	—	2	2	no	—
50	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
51	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
53	#14555489	—	2	2	no	—
54	#14555489	—	2	2	no	—
55	C₂²⋊Dic₁₃₇	—	2	1	no	—
56	C₄×Dic₁₃₇	—	2	1	no	—
57	C₂×Q₈×C₁₃₇	—	2	2	no	—
58	C₂²×C₄×C₁₃₇	—	2	1	no	—
59	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₇	—	2	1	no	—
61	C₄⋊C₄×C₁₃₇	—	2	2	no	—
62	C₄⋊C₄×C₁₃₇	—	2	2	no	—
63	C₄⋊C₄×C₁₃₇	—	2	2	no	—
64	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
71	C₄²×C₁₃₇	—	2	1	no	—
72	C₄²×C₁₃₇	C₂³	1	2	yes	—
73	Dic₂₇₄	—	8	2	no	—
74	C₂×Dic₁₃₇	—	2	1	no	—
75	C₂×Dic₁₃₇	—	2	1	no	—
76	C₂×Dic₁₃₇	—	2	2	no	—
77	C₂×Dic₁₃₇	—	8	1	no	—
78	#11764040	C₂×Q₈	1	2	yes	—
79	#11764040	C₂×Q₈	1	2	yes	—
80	#11764040	C₂×Q₈	1	4	yes	—
81	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₁₃₇	—	8	2	no	—
83	C₂×C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	8	1	no	—
87	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
93	C₂²×C₁₃₇	—	2	1	no	—
94	C₂²×C₁₃₇	—	2	1	no	—
95	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₃₇	—	2	2	no	—
98	C₄×C₁₃₇	—	2	2	no	—
99	C₄×C₁₃₇	—	4	1	no	—
100	C₄×C₁₃₇	—	4	2	no	—
101	C₄×C₁₃₇	—	4	1	no	—
102	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
103	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
104	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
105	Dic₁₃₇	—	4	1	no	—
106	Dic₁₃₇	—	4	2	no	—
107	Dic₁₃₇	—	4	1	no	—
108	C₂×C₁₃₇	—	2	2	no	—
109	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₁₃₇	#577	1	1	yes	characteristic
111	C₂×C₁₃₇	#577	1	1	yes	characteristic
112	C₁₃₇	#3163	1	1	yes	characteristic, complemented, Sylow
113	#3163	—	137	1	no	maximal, Sylow
114	#544	—	137	1	no	—
115	#544	D₁₃₇	1	1	yes	characteristic
116	#501	—	137	2	no	—
117	#501	—	137	2	no	—
118	#499	—	137	2	no	—
119	#498	—	137	2	no	—
120	#428	—	137	1	no	—
121	#428	—	137	1	no	—
122	#427	—	137	1	no	—
123	C₂×#106	—	137	1	no	—
124	#405	—	137	1	no	—
125	C₂×M₁₆	—	137	1	no	—
126	C₂²×C₈	—	137	1	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	—	274	1	no	—
129	C₄⋊Q₈	—	137	2	no	—
130	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	—	274	1	no	—
133	#125	—	137	2	no	—
134	#125	C₂×D₁₃₇	1	2	yes	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	—	274	2	no	—
137	#117	—	137	1	no	—
138	#117	C₂×D₁₃₇	1	1	yes	characteristic
139	#117	C₂×D₁₃₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	137	1	no	—
141	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic
142	#107	—	137	1	no	—
143	#107	—	137	1	no	—
144	#107	—	137	2	no	—
145	#107	—	137	2	no	—
146	#106	—	137	2	no	—
147	#106	—	137	2	no	—
148	#106	—	137	1	no	—
149	#106	—	137	1	no	—
150	#106	—	137	2	no	—
151	#106	—	137	2	no	—
152	#97	—	137	2	no	—
153	C₄×C₈	—	137	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	274	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	274	1	no	—
158	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
159	M₁₆	—	137	4	no	—
160	C₂×C₈	—	137	4	no	—
161	C₂×C₈	—	137	1	no	—
162	C₂×C₈	—	137	2	no	—
163	C₂×C₈	—	137	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	274	2	no	—
169	C₄⋊C₄	—	274	2	no	—
170	C₄⋊C₄	—	274	1	no	—
171	C₄⋊C₄	—	274	2	no	—
172	C₄⋊C₄	—	137	2	no	—
173	C₄⋊C₄	—	137	1	no	—
174	C₄⋊C₄	—	137	1	no	—
175	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
177	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
178	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
179	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	274	1	no	—
182	C₂²⋊C₄	C₂²⋊D₁₃₇	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	274	1	no	—
185	C₄²	C₂²×D₁₃₇	1	2	yes	—
186	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	1096	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	274	1	no	—
194	C₂×C₄	—	274	1	no	—
195	C₂×C₄	—	274	2	no	—
196	C₂×C₄	—	1096	1	no	—
197	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
200	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
201	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
202	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
203	C₈	—	137	2	no	—
204	C₈	—	137	2	no	—
205	C₈	—	137	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#17984086	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	548	2	no	—
216	C₄	—	548	1	no	—
217	C₄	—	548	1	no	—
218	C₄	#17984086	1	2	yes	—
219	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#17984086	1	1	yes	characteristic
223	C₂	#22578211	1	1	yes	characteristic
224	C₂	#22578214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#544×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	#544	D₁₃₇
hypocenter	C₁₃₇	#3163
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #22577813	C₂	semidirect
3. #22577809	C₂	semidirect
4. #22577802	C₂	semidirect
5. #22577799	C₂	semidirect
43. #14555494	Q₈	semidirect
44. C₂×#11764040	Q₈	semidirect
112. C₁₃₇	#3163	semidirect

Maximal subgroups

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

subgroup	index
#22578031	2
#22577813	2
#22577809	2
#22577802	2
#22577799	2
#22577446	2
#22577443	2
#22577439	2
C₂×#17984019	2
#22577359	2
#544×C₁₃₇	2
#3163	137