Subgroups — C₂×#22577958 #31794078

Order 17536.

1168 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577958	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984079	C₂	1	2	yes	maximal, complemented
4	C₂×#17984068	C₂	1	2	yes	maximal
5	C₂×#17984064	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂×#17984053	C₂	1	2	yes	maximal, complemented
8	C₂×#17984046	C₂	1	2	yes	maximal
9	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
12	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
15	#17984088	C₂²	1	8	yes	complemented
16	#17984079	C₂²	1	16	yes	complemented
17	#17984068	C₂²	1	16	yes	—
18	#17984064	C₂²	1	16	yes	—
19	Q₈×Dic₁₃₇	C₂²	1	16	yes	complemented
20	#17984053	C₂²	1	16	yes	complemented
21	#17984046	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
24	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
25	C₂×#14555489	C₂²	1	4	yes	—
26	C₂×#14555487	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
31	C₄²×D₁₃₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₁₃₇	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₁₃₇	C₂²	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₂₇₄	—	2	8	no	—
39	C₂×Dic₂₇₄	C₂³	1	8	yes	—
40	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
43	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₃₇	—	2	4	no	—
45	C₂×C₄×D₁₃₇	—	2	4	no	—
46	C₂×C₄×D₁₃₇	Q₈	1	8	yes	complemented
47	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
48	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
49	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
50	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
51	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
52	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
53	#14555489	C₂³	1	16	yes	—
54	#14555487	C₂³	1	16	yes	—
55	C₄×Dic₁₃₇	C₂³	1	4	yes	—
56	C₄×Dic₁₃₇	C₂³	1	4	yes	—
57	C₄×Dic₁₃₇	C₂³	1	4	yes	—
58	C₄×Dic₁₃₇	C₂³	1	16	yes	—
59	C₂×Q₈×C₁₃₇	—	2	8	no	—
60	C₂×Q₈×C₁₃₇	C₂³	1	8	yes	—
61	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
66	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
67	C₄²×C₁₃₇	C₂³	1	4	yes	—
68	C₂²×D₁₃₇	—	2	2	no	—
69	C₂²×D₁₃₇	—	2	2	no	—
70	C₂²×D₁₃₇	C₂×Q₈	1	4	yes	—
71	C₂²×D₁₃₇	C₂⁴	1	2	yes	—
72	Dic₂₇₄	—	2	16	no	—
73	C₂×Dic₁₃₇	—	2	2	no	—
74	C₂×Dic₁₃₇	—	2	2	no	—
75	C₂×Dic₁₃₇	—	2	8	no	—
76	C₂×Dic₁₃₇	C₂×Q₈	1	4	yes	—
77	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
78	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
79	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
80	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
81	C₄×D₁₃₇	—	2	8	no	—
82	C₄×D₁₃₇	—	2	8	no	—
83	C₄×D₁₃₇	C₂×Q₈	1	16	yes	complemented
84	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
85	Q₈×C₁₃₇	—	2	16	no	—
86	C₂×C₄×C₁₃₇	—	2	8	no	—
87	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
88	C₂×C₄×C₁₃₇	#41	1	4	yes	—
89	C₂×C₄×C₁₃₇	#41	1	4	yes	—
90	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
91	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
93	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
94	C₂²×C₁₃₇	C₂²×Q₈	1	2	yes	—
95	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
96	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
97	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₃₇	—	2	8	no	—
99	C₂×D₁₃₇	—	2	2	no	—
100	C₂×D₁₃₇	—	2	2	no	—
101	C₂×D₁₃₇	C₂²×Q₈	1	4	yes	—
102	C₄×C₁₃₇	—	2	8	no	—
103	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
104	C₄×C₁₃₇	C₂×#41	1	4	yes	—
105	C₄×C₁₃₇	C₂×#41	1	4	yes	—
106	Dic₁₃₇	—	2	2	no	—
107	Dic₁₃₇	—	2	2	no	—
108	Dic₁₃₇	—	2	8	no	—
109	Dic₁₃₇	C₂²×Q₈	1	4	yes	—
110	C₂×C₁₃₇	#533	1	4	yes	complemented
111	C₂×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
112	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
113	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
114	D₁₃₇	—	2	4	no	—
115	C₁₃₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
116	C₂×#533	—	137	1	no	maximal, Sylow
117	#533	—	137	16	no	—
118	C₂×C₄⋊Q₈	—	137	1	no	—
119	C₂×C₄⋊Q₈	D₁₃₇	1	1	yes	characteristic, complemented
120	C₂×#125	—	137	2	no	—
121	C₂×C₂²⋊Q₈	—	137	2	no	—
122	C₂×C₂²⋊Q₈	—	137	2	no	—
123	C₂×C₄×Q₈	—	137	2	no	—
124	C₂×C₄×Q₈	—	137	2	no	—
125	C₂×#117	—	137	2	no	—
126	C₂²×C₄²	—	137	1	no	—
127	C₂²×Q₈	—	137	2	no	—
128	C₂²×Q₈	C₂×D₁₃₇	1	2	yes	—
129	C₂³×C₄	—	137	1	no	—
130	C₂³×C₄	—	137	1	no	—
131	C₂³×C₄	—	137	1	no	—
132	C₄⋊Q₈	—	137	8	no	—
133	C₄⋊Q₈	C₂×D₁₃₇	1	8	yes	complemented
134	#125	—	137	16	no	—
135	C₂²⋊Q₈	—	137	16	no	—
136	C₂²⋊Q₈	—	137	16	no	—
137	C₄×Q₈	—	137	16	no	—
138	C₄×Q₈	—	137	16	no	—
139	#117	—	137	16	no	—
140	C₂×C₄⋊C₄	—	137	2	no	—
141	C₂×C₄⋊C₄	—	137	4	no	—
142	C₂×C₄⋊C₄	—	137	4	no	—
143	C₂×C₄⋊C₄	—	137	2	no	—
144	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
145	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
146	C₂×C₂²⋊C₄	—	137	4	no	—
147	C₂×C₄²	—	137	4	no	—
148	C₂×C₄²	—	137	1	no	—
149	C₂×C₄²	—	137	1	no	—
150	C₂×C₄²	—	137	1	no	—
151	C₂×C₄²	—	137	8	no	—
152	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
153	C₂⁴	—	137	1	no	—
154	C₂×Q₈	—	2	8	no	—
155	C₂×Q₈	—	274	8	no	—
156	C₂×Q₈	—	137	8	no	—
157	C₂×Q₈	C₂²×D₁₃₇	1	8	yes	—
158	C₂²×C₄	—	274	4	no	—
159	C₂²×C₄	—	274	4	no	—
160	C₂²×C₄	—	137	1	no	—
161	C₂²×C₄	—	137	1	no	—
162	C₂²×C₄	—	137	4	no	—
163	C₂²×C₄	—	137	4	no	—
164	C₂²×C₄	—	137	4	no	—
165	C₂²×C₄	—	137	1	no	—
166	C₂²×C₄	—	137	8	no	—
167	C₂²×C₄	—	137	4	no	—
168	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
169	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
172	C₄⋊C₄	—	137	16	no	—
173	C₄⋊C₄	—	137	8	no	—
174	C₄⋊C₄	—	137	16	no	—
175	C₄⋊C₄	—	137	8	no	—
176	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	—
177	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	—
178	C₂²⋊C₄	—	137	16	no	—
179	C₄²	—	137	4	no	—
180	C₄²	—	137	4	no	—
181	C₄²	—	137	4	no	—
182	C₄²	—	137	16	no	—
183	C₄²	C₂²×D₁₃₇	1	4	yes	—
184	C₂³	—	274	2	no	—
185	C₂³	—	274	2	no	—
186	C₂³	—	137	4	no	—
187	C₂³	—	137	2	no	—
188	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	274	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	274	2	no	—
193	C₂×C₄	—	274	2	no	—
194	C₂×C₄	—	274	8	no	—
195	C₂×C₄	—	274	8	no	—
196	C₂×C₄	—	274	8	no	—
197	C₂×C₄	—	137	2	no	—
198	C₂×C₄	—	137	2	no	—
199	C₂×C₄	—	137	8	no	—
200	C₂×C₄	—	137	4	no	—
201	C₂×C₄	—	137	2	no	—
202	C₂×C₄	—	137	16	no	—
203	C₂×C₄	Q₈×D₁₃₇	1	4	yes	—
204	C₂×C₄	#14555509	1	4	yes	—
205	C₂×C₄	#14555511	1	4	yes	—
206	C₂×C₄	C₂³×D₁₃₇	1	8	yes	—
207	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
208	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
209	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
210	C₂²	—	274	8	no	—
211	C₂²	—	274	2	no	—
212	C₂²	—	274	2	no	—
213	C₂²	—	137	4	no	—
214	C₂²	C₂×Q₈×D₁₃₇	1	2	yes	—
215	C₂²	C₂×#14555509	1	2	yes	—
216	C₂²	C₂×#14555511	1	2	yes	—
217	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
218	C₄	—	2	8	no	—
219	C₄	—	274	2	no	—
220	C₄	—	274	2	no	—
221	C₄	—	274	8	no	—
222	C₄	—	137	4	no	—
223	C₄	C₂×Q₈×D₁₃₇	1	4	yes	—
224	C₄	C₂×#14555509	1	4	yes	—
225	C₄	C₂×#14555511	1	4	yes	—
226	C₂	—	274	4	no	—
227	C₂	#22577958	1	4	yes	direct product factor
228	C₂	C₂²×Q₈×D₁₃₇	1	1	yes	characteristic
229	C₂	C₂²×#14555509	1	1	yes	characteristic
230	C₂	C₂²×#14555511	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×C₄⋊Q₈×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂³×D₁₃₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#533
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22577958	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
3. C₂×#17984079	C₂	semidirect
6. C₂×Q₈×Dic₁₃₇	C₂	semidirect
7. C₂×#17984053	C₂	semidirect
10. C₂×C₄⋊Q₈×C₁₃₇	C₂	semidirect
15. #17984088	C₂²	semidirect
16. #17984079	C₂²	semidirect
19. Q₈×Dic₁₃₇	C₂²	semidirect
20. #17984053	C₂²	semidirect
33. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
46. C₂×C₄×D₁₃₇	Q₈	semidirect
83. C₄×D₁₃₇	C₂×Q₈	semidirect
110. C₂×C₁₃₇	#533	semidirect
115. C₁₃₇	C₂×#533	semidirect
119. C₂×C₄⋊Q₈	D₁₃₇	semidirect
133. C₄⋊Q₈	C₂×D₁₃₇	semidirect
227. C₂	#22577958	direct (N × H)

Maximal subgroups

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

subgroup	index
#22577958	2
C₂×#17984088	2
C₂×#17984079	2
C₂×#17984068	2
C₂×#17984064	2
C₂×Q₈×Dic₁₃₇	2
C₂×#17984053	2
C₂×#17984046	2
C₂×C₄²×D₁₃₇	2
C₂×C₄⋊Q₈×C₁₃₇	2
C₂×#533	137

Subgroups — C2×#22577958 #31794078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577958 #31794078