Subgroups — C₂×#22578414 #31797441

Order 17536.

536 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984167	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
3	#22578414	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984157	C₂	1	2	yes	maximal, complemented
5	C₂×#17984156	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10587739	—	2	1	no	—
8	#17984167	—	2	2	no	—
9	#17984167	C₂²	1	4	yes	—
10	C₂×C₄⋊#10587739	—	2	1	no	—
11	C₂×C₄⋊#10587739	—	2	2	no	—
12	C₂×C₄⋊#10587739	C₂²	1	2	yes	—
13	C₂×C₄⋊#10587739	C₂²	1	4	yes	—
14	C₂×C₄⋊#10587739	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10587739	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊#10587739	—	2	1	no	—
17	C₂×C₄×#10587739	—	2	1	no	—
18	#17984157	C₂²	1	8	yes	complemented
19	#17984156	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#14555494	C₄	1	1	yes	characteristic, complemented
22	C₂×#14555493	C₄	1	2	yes	complemented
23	M₁₆×D₁₃₇	C₄	1	4	yes	complemented
24	M₁₆×D₁₃₇	C₂²	1	4	yes	—
25	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
26	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
27	C₂²×#10587739	—	2	1	no	—
28	C₂²×#10587739	—	2	1	no	—
29	C₂²×#10587739	—	2	4	no	—
30	C₂²×#10587739	—	2	2	no	—
31	C₂²×#10587739	—	4	4	no	—
32	C₄⋊#10587739	—	2	4	no	—
33	C₄⋊#10587739	—	2	2	no	—
34	C₄⋊#10587739	—	2	1	no	—
35	C₄⋊#10587739	—	2	1	no	—
36	C₄⋊#10587739	C₂³	1	2	yes	—
37	C₄⋊#10587739	C₂³	1	4	yes	—
38	C₄⋊#10587739	C₂³	1	2	yes	—
39	C₂²⋊#10587739	—	2	4	no	—
40	C₄×#10587739	—	2	4	no	—
41	C₂³×D₁₃₇	D₄	1	1	yes	characteristic
42	C₂²×Dic₁₃₇	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃₇	D₄	1	2	yes	—
47	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
48	C₂×C₄×D₁₃₇	Q₈	1	1	yes	characteristic
49	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
51	#14555494	C₂×C₄	1	4	yes	complemented
52	#14555493	C₂×C₄	1	8	yes	complemented
53	C₂×#11764040	C₂×C₄	1	2	yes	complemented
54	C₈×D₁₃₇	C₂×C₄	1	4	yes	complemented
55	C₈×D₁₃₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
58	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
59	C₂×#10587739	—	2	2	no	—
60	C₂×#10587739	—	2	2	no	—
61	C₂×#10587739	—	2	4	no	—
62	C₂×#10587739	—	4	8	no	—
63	C₂×#10587739	—	4	2	no	—
64	C₂×#10587739	—	4	4	no	—
65	C₂×#10587739	—	4	2	no	—
66	C₂²×D₁₃₇	—	2	2	no	—
67	C₂²×D₁₃₇	—	2	2	no	—
68	C₂²×D₁₃₇	C₄⋊C₄	1	2	yes	—
69	C₂²×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×D₁₃₇	C₂×D₄	1	2	yes	—
71	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
73	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂×Dic₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
76	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
77	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
78	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
79	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
81	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
82	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
84	C₄×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
85	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
86	#11764040	C₂²×C₄	1	4	yes	complemented
87	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
92	C₈×C₁₃₇	C₂²×C₄	1	4	yes	complemented
93	C₂²×C₁₃₇	—	2	2	no	—
94	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×D₁₃₇	—	2	2	no	—
97	C₂×D₁₃₇	—	2	2	no	—
98	C₂×D₁₃₇	—	2	2	no	—
99	C₂×D₁₃₇	—	2	4	no	—
100	C₂×D₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×D₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×D₁₃₇	#137	1	2	yes	—
104	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	#10587739	—	4	2	no	—
106	#10587739	—	4	4	no	—
107	#10587739	—	4	2	no	—
108	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
111	Dic₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	Dic₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
113	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₁₃₇	—	2	2	no	—
115	C₂×C₁₃₇	#428	1	2	yes	complemented
116	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	D₁₃₇	—	2	2	no	—
118	D₁₃₇	#428	1	2	yes	complemented
119	D₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	D₁₃₇	C₂×#137	1	1	yes	characteristic
121	C₁₃₇	C₂×#428	1	1	yes	characteristic, complemented, Sylow
122	C₂×#428	—	137	1	no	maximal, Sylow
123	C₂²×M₁₆	—	137	1	no	—
124	C₂×#117	—	137	1	no	—
125	C₂²×C₄⋊C₄	—	137	1	no	—
126	#428	—	137	8	no	—
127	C₂×#107	—	137	2	no	—
128	C₂×#106	—	137	2	no	—
129	C₂³×C₄	—	274	1	no	—
130	C₂³×C₄	—	137	1	no	—
131	C₂×M₁₆	—	137	1	no	—
132	C₂×M₁₆	—	137	4	no	—
133	C₂×M₁₆	—	137	4	no	—
134	C₂×M₁₆	—	137	2	no	—
135	C₂×M₁₆	#10587739	1	1	yes	characteristic, complemented
136	C₂²×C₈	—	137	2	no	—
137	#117	—	274	2	no	—
138	#117	—	137	4	no	—
139	C₂×C₄⋊C₄	—	274	1	no	—
140	C₂×C₄⋊C₄	—	274	2	no	—
141	C₂×C₄⋊C₄	—	137	2	no	—
142	C₂×C₄⋊C₄	—	137	4	no	—
143	C₂×C₄⋊C₄	—	137	1	no	—
144	C₂×C₄⋊C₄	—	137	1	no	—
145	C₂×C₂²⋊C₄	—	274	1	no	—
146	C₂×C₄²	—	274	1	no	—
147	#107	—	137	8	no	—
148	#106	—	137	8	no	—
149	C₂⁴	—	137	1	no	—
150	C₂²×C₄	—	274	1	no	—
151	C₂²×C₄	—	274	1	no	—
152	C₂²×C₄	—	274	4	no	—
153	C₂²×C₄	—	274	2	no	—
154	C₂²×C₄	—	548	4	no	—
155	C₂²×C₄	—	137	1	no	—
156	C₂²×C₄	—	137	1	no	—
157	C₂²×C₄	—	137	2	no	—
158	C₂²×C₄	—	137	2	no	—
159	C₂²×C₄	—	137	1	no	—
160	C₂²×C₄	—	137	2	no	—
161	C₂²×C₄	—	137	1	no	—
162	C₂²×C₄	—	137	1	no	—
163	C₂²×C₄	—	137	2	no	—
164	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
165	M₁₆	—	137	4	no	—
166	M₁₆	—	137	8	no	—
167	M₁₆	C₂×#10587739	1	4	yes	complemented
168	C₂×C₈	—	137	2	no	—
169	C₂×C₈	—	137	4	no	—
170	C₂×C₈	—	137	4	no	—
171	C₂×C₈	C₂×#10587739	1	2	yes	complemented
172	C₄⋊C₄	—	274	4	no	—
173	C₄⋊C₄	—	274	2	no	—
174	C₄⋊C₄	—	274	1	no	—
175	C₄⋊C₄	—	274	1	no	—
176	C₄⋊C₄	—	137	2	no	—
177	C₄⋊C₄	—	137	4	no	—
178	C₄⋊C₄	—	137	2	no	—
179	C₂²⋊C₄	—	274	4	no	—
180	C₄²	—	274	4	no	—
181	C₂³	—	274	2	no	—
182	C₂³	—	274	2	no	—
183	C₂³	—	137	2	no	—
184	C₂³	—	137	2	no	—
185	C₂³	—	137	1	no	—
186	C₂³	—	137	1	no	—
187	C₂³	C₄⋊#10587739	1	1	yes	characteristic
188	C₂×C₄	—	274	2	no	—
189	C₂×C₄	—	274	2	no	—
190	C₂×C₄	—	274	4	no	—
191	C₂×C₄	—	548	8	no	—
192	C₂×C₄	—	548	4	no	—
193	C₂×C₄	—	548	2	no	—
194	C₂×C₄	—	548	2	no	—
195	C₂×C₄	—	137	2	no	—
196	C₂×C₄	—	137	2	no	—
197	C₂×C₄	—	137	1	no	—
198	C₂×C₄	—	137	1	no	—
199	C₂×C₄	—	137	2	no	—
200	C₂×C₄	—	137	1	no	—
201	C₂×C₄	—	137	2	no	—
202	C₂×C₄	—	137	2	no	—
203	C₂×C₄	—	137	2	no	—
204	C₂×C₄	—	137	2	no	—
205	C₂×C₄	—	137	1	no	—
206	C₂×C₄	—	137	1	no	—
207	C₂×C₄	—	137	1	no	—
208	C₂×C₄	—	137	2	no	—
209	C₂×C₄	C₄⋊#10587739	1	2	yes	—
210	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
211	C₂×C₄	C₂²×#10587739	1	2	yes	—
212	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
213	C₈	—	137	4	no	—
214	C₈	C₂²×#10587739	1	4	yes	complemented
215	C₂²	—	2	2	no	—
216	C₂²	—	274	2	no	—
217	C₂²	—	274	2	no	—
218	C₂²	—	274	4	no	—
219	C₂²	—	274	2	no	—
220	C₂²	—	137	2	no	—
221	C₂²	—	137	1	no	—
222	C₂²	—	137	1	no	—
223	C₂²	—	137	2	no	—
224	C₂²	—	137	2	no	—
225	C₂²	C₂×C₄⋊#10587739	1	2	yes	—
226	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
227	C₄	—	548	4	no	—
228	C₄	—	548	2	no	—
229	C₄	—	548	2	no	—
230	C₄	—	137	1	no	—
231	C₄	—	137	2	no	—
232	C₄	—	137	1	no	—
233	C₄	C₂×C₄⋊#10587739	1	2	yes	—
234	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
235	C₄	C₂³×#10587739	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	274	2	no	—
238	C₂	—	137	1	no	—
239	C₂	—	137	1	no	—
240	C₂	—	137	2	no	—
241	C₂	#22578414	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃₇	D₄
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	C₂×#428
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22578414	C₂	direct (N × H)
4. C₂×#17984157	C₂	semidirect
5. C₂×#17984156	C₂	semidirect
18. #17984157	C₂²	semidirect
19. #17984156	C₂²	semidirect
21. C₂×#14555494	C₄	semidirect
22. C₂×#14555493	C₄	semidirect
23. M₁₆×D₁₃₇	C₄	semidirect
26. C₂×M₁₆×C₁₃₇	C₄	semidirect
51. #14555494	C₂×C₄	semidirect
52. #14555493	C₂×C₄	semidirect
53. C₂×#11764040	C₂×C₄	semidirect
54. C₈×D₁₃₇	C₂×C₄	semidirect
57. M₁₆×C₁₃₇	C₂×C₄	semidirect
58. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
86. #11764040	C₂²×C₄	semidirect
92. C₈×C₁₃₇	C₂²×C₄	semidirect
115. C₂×C₁₃₇	#428	semidirect
118. D₁₃₇	#428	semidirect
121. C₁₃₇	C₂×#428	semidirect
135. C₂×M₁₆	#10587739	semidirect
167. M₁₆	C₂×#10587739	semidirect
171. C₂×C₈	C₂×#10587739	semidirect
214. C₈	C₂²×#10587739	semidirect
241. C₂	#22578414	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984167	2
C₂²×C₄⋊#10587739	2
#22578414	2
C₂×#17984157	2
C₂×#17984156	2
C₂×M₁₆×D₁₃₇	2
C₂×#428	137

Subgroups — C2×#22578414 #31797441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578414 #31797441