Subgroups — #31797591

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578455	C₂	1	4	yes	maximal, complemented
2	#22578454	C₂	1	2	yes	maximal, complemented
3	#22578452	C₂	1	2	yes	maximal, complemented
4	#22578408	C₂	1	2	yes	maximal
5	#22578407	C₂	1	1	yes	maximal, characteristic
6	C₄×#14555520	C₂	1	1	yes	maximal, characteristic
7	C₄×#14555520	C₂	1	2	yes	maximal
8	C₄×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14555520	C₂²	1	2	yes	—
10	C₂×#14555520	C₂²	1	1	yes	characteristic
11	C₂×#14555520	C₂²	1	1	yes	characteristic
12	#17984155	C₂²	1	4	yes	—
13	#17984155	C₂²	1	4	yes	—
14	C₄⋊#11764047	C₂²	1	2	yes	—
15	C₄⋊#11764047	C₂²	1	2	yes	—
16	#17984143	C₂²	1	4	yes	—
17	#17984143	C₂²	1	2	yes	—
18	C₄×#11764047	C₂²	1	2	yes	—
19	C₄×#11764047	C₂²	1	4	yes	—
20	C₂×Q₈×D₁₃₇	C₄	1	1	yes	characteristic
21	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₇₄	C₄	1	2	yes	—
23	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
24	#17984053	C₂²	1	1	yes	characteristic
25	#17984053	C₂²	1	2	yes	—
26	C₄⋊C₄×D₁₃₇	C₄	1	2	yes	—
27	C₄⋊C₄×D₁₃₇	C₄	1	1	yes	characteristic
28	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
29	C₄²×D₁₃₇	C₂²	1	2	yes	—
30	C₄×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
31	#14555520	—	2	4	no	—
32	#14555520	—	2	2	no	—
33	#14555520	Q₈	1	4	yes	complemented
34	C₂×#11764047	C₂³	1	4	yes	—
35	C₂×#11764047	C₂³	1	2	yes	—
36	C₂×#11764047	C₂³	1	2	yes	—
37	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
39	Q₈×D₁₃₇	—	2	4	no	—
40	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
45	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
47	#14555489	C₂×C₄	1	2	yes	—
48	#14555489	C₂×C₄	1	2	yes	—
49	#14555489	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₃₇	C₂³	1	2	yes	—
53	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
54	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
55	C₄×Dic₁₃₇	C₂³	1	2	yes	—
56	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
60	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
61	#11764047	—	2	4	no	—
62	#11764047	—	2	2	no	—
63	#11764047	C₂×Q₈	1	4	yes	complemented
64	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	Dic₂₇₄	—	2	4	no	—
66	Dic₂₇₄	M₁₆	1	4	yes	complemented
67	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
72	C₄×D₁₃₇	—	2	2	no	—
73	C₄×D₁₃₇	—	2	4	no	—
74	C₄×D₁₃₇	—	4	1	no	—
75	C₄×D₁₃₇	—	4	2	no	—
76	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
77	C₄×D₁₃₇	#41	1	2	yes	—
78	Q₈×C₁₃₇	—	2	2	no	—
79	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×D₁₃₇	—	2	1	no	—
86	C₂×D₁₃₇	—	2	1	no	—
87	C₂×D₁₃₇	C₄×Q₈	1	2	yes	—
88	C₄×C₁₃₇	—	2	2	no	—
89	C₄×C₁₃₇	—	2	2	no	—
90	C₄×C₁₃₇	—	2	1	no	—
91	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
92	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
93	Dic₁₃₇	—	2	1	no	—
94	Dic₁₃₇	—	2	1	no	—
95	Dic₁₃₇	—	2	2	no	—
96	Dic₁₃₇	C₂×M₁₆	1	4	yes	—
97	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
98	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
99	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
100	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
101	C₂×C₁₃₇	#568	1	1	yes	characteristic
102	D₁₃₇	—	2	2	no	—
103	C₁₃₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
104	Q₈×M₁₆	—	137	1	no	maximal, Sylow
105	C₂×C₄×Q₈	—	137	1	no	—
106	#446	—	137	2	no	—
107	#446	—	137	4	no	—
108	C₈×Q₈	—	137	2	no	—
109	#423	—	137	1	no	—
110	#423	—	137	2	no	—
111	C₄×M₁₆	—	137	1	no	—
112	C₄×M₁₆	—	137	2	no	—
113	C₂²×Q₈	—	137	1	no	—
114	C₂×M₁₆	—	137	2	no	—
115	C₂×M₁₆	—	137	1	no	—
116	C₂×M₁₆	—	137	1	no	—
117	C₄×Q₈	—	137	1	no	—
118	C₄×Q₈	—	137	1	no	—
119	C₄×Q₈	—	137	2	no	—
120	C₄×Q₈	—	137	2	no	—
121	C₄×Q₈	—	137	1	no	—
122	C₄×Q₈	#10587739	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	137	1	no	—
124	C₂×C₄⋊C₄	—	137	2	no	—
125	C₂×C₄²	—	137	1	no	—
126	C₂×C₄²	—	137	2	no	—
127	C₄⋊C₈	—	137	2	no	—
128	C₄⋊C₈	—	137	2	no	—
129	C₄⋊C₈	—	137	4	no	—
130	C₄⋊C₈	—	137	4	no	—
131	#97	—	137	4	no	—
132	#97	—	137	2	no	—
133	C₄×C₈	—	137	2	no	—
134	C₄×C₈	—	137	4	no	—
135	C₂×Q₈	—	274	4	no	—
136	C₂×Q₈	—	137	2	no	—
137	C₂×Q₈	—	137	1	no	—
138	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
139	C₂²×C₄	—	137	2	no	—
140	C₂²×C₄	—	137	1	no	—
141	C₂²×C₄	—	137	1	no	—
142	C₂²×C₄	—	137	2	no	—
143	C₂²×C₄	—	137	1	no	—
144	M₁₆	—	274	4	no	—
145	M₁₆	—	274	2	no	—
146	M₁₆	—	137	4	no	—
147	C₂×C₈	—	137	4	no	—
148	C₂×C₈	—	137	2	no	—
149	C₂×C₈	—	137	2	no	—
150	C₄⋊C₄	—	137	2	no	—
151	C₄⋊C₄	—	137	2	no	—
152	C₄⋊C₄	—	137	2	no	—
153	C₄⋊C₄	—	137	2	no	—
154	C₄⋊C₄	—	137	1	no	—
155	C₄⋊C₄	C₂×#10587739	1	2	yes	—
156	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
157	C₄²	—	137	1	no	—
158	C₄²	—	137	2	no	—
159	C₄²	—	137	1	no	—
160	C₄²	—	137	2	no	—
161	C₄²	—	137	2	no	—
162	C₄²	—	137	1	no	—
163	C₄²	C₂×#10587739	1	2	yes	—
164	C₄²	C₂×#10587739	1	1	yes	characteristic
165	C₂³	—	137	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	274	4	no	—
168	Q₈	—	137	4	no	—
169	C₂×C₄	—	274	4	no	—
170	C₂×C₄	—	274	2	no	—
171	C₂×C₄	—	548	2	no	—
172	C₂×C₄	—	548	1	no	—
173	C₂×C₄	—	137	1	no	—
174	C₂×C₄	—	137	2	no	—
175	C₂×C₄	—	137	1	no	—
176	C₂×C₄	—	137	1	no	—
177	C₂×C₄	—	137	2	no	—
178	C₂×C₄	—	137	2	no	—
179	C₂×C₄	—	137	2	no	—
180	C₂×C₄	C₂²×#10587739	1	2	yes	—
181	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
182	C₂×C₄	C₂²×#10587739	1	2	yes	—
183	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
184	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
185	C₈	—	274	4	no	—
186	C₈	—	274	2	no	—
187	C₈	—	137	4	no	—
188	C₂²	—	274	1	no	—
189	C₂²	—	274	1	no	—
190	C₂²	—	137	2	no	—
191	C₂²	C₂³×#10587739	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	274	1	no	—
196	C₄	—	274	1	no	—
197	C₄	—	274	2	no	—
198	C₄	—	137	1	no	—
199	C₄	—	137	1	no	—
200	C₄	—	137	4	no	—
201	C₄	Q₈×#10587739	1	2	yes	—
202	C₄	D₁₃₇⋊M₁₆	1	2	yes	—
203	C₂	—	274	2	no	—
204	C₂	C₂×Q₈×#10587739	1	1	yes	characteristic
205	C₂	C₂×D₁₃₇⋊M₁₆	1	1	yes	characteristic
206	C₂	#22578482	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₁₃₇	C₄
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₄×Q₈	#10587739
hypocenter	C₁₃₇	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578455	C₂	semidirect
2. #22578454	C₂	semidirect
3. #22578452	C₂	semidirect
33. #14555520	Q₈	semidirect
63. #11764047	C₂×Q₈	semidirect
66. Dic₂₇₄	M₁₆	semidirect
103. C₁₃₇	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22578455	2
#22578454	2
#22578452	2
#22578408	2
#22578407	2
C₄×#14555520	2
C₄×Q₈×D₁₃₇	2
Q₈×M₁₆	137