Subgroups — C₂×#22558259 #30767072

Order 17152.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	#22558259	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	1	no	—
6	C₂²×Dic₂₆₈	C₄	1	2	yes	—
7	C₂×#14524043	—	2	1	no	—
8	#16863776	—	2	1	no	—
9	#16863776	—	2	1	no	—
10	#16863776	C₄	1	2	yes	—
11	#16863776	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
13	#16863149	—	2	1	no	—
14	#16863149	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₆₇	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	—	2	1	no	—
18	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	4	4	no	—
24	C₂×Dic₂₆₈	—	4	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	4	1	no	—
28	#14524043	—	2	2	no	—
29	#14524043	—	2	2	no	—
30	#14524043	—	4	1	no	—
31	#14524043	—	4	1	no	—
32	C₂×C₄⋊Dic₆₇	—	4	1	no	—
33	C₂×C₄⋊Dic₆₇	—	4	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	—	4	2	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	4	no	—
47	C₂×Dic₁₃₄	—	4	4	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	Dic₂₆₈	—	4	4	no	—
51	Dic₂₆₈	—	4	4	no	—
52	Dic₂₆₈	—	4	4	no	—
53	Dic₂₆₈	—	4	2	no	—
54	Dic₂₆₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	8	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	D₈	1	2	yes	—
68	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	D₈	1	1	yes	characteristic
73	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₆₇	—	4	2	no	—
76	C₂³×C₆₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	#100	1	2	yes	—
79	C₂×C₄×C₆₇	#102	1	2	yes	—
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
84	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×Dic₆₇	—	4	4	no	—
89	C₂×Dic₆₇	—	8	4	no	—
90	Dic₁₃₄	—	4	4	no	—
91	Dic₁₃₄	—	4	4	no	—
92	Dic₁₃₄	—	8	4	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	#102	1	2	yes	—
96	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₆₇⋊C₈	—	8	2	no	—
99	C₂²×C₆₇	#331	1	2	yes	—
100	C₂²×C₆₇	#358	1	2	yes	—
101	C₂²×C₆₇	#362	1	2	yes	—
102	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	#331	1	2	yes	—
106	C₄×C₆₇	#358	1	2	yes	—
107	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
108	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
109	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
110	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
111	Dic₆₇	—	8	4	no	—
112	C₂×#1362	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	#1362	1	4	yes	complemented
114	C₂×C₆₇	C₂×#331	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
116	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
117	C₂×#494	—	67	1	no	—
118	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
119	C₂×#335	—	67	1	no	—
120	#1362	—	67	4	no	—
121	C₆₇	C₂×#1362	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₁₆	—	67	2	no	—
123	C₂²×Q₁₆	—	134	1	no	—
124	C₂×C₄⋊Q₈	—	134	1	no	—
125	C₂²×C₁₆	—	2	1	no	—
126	#494	—	67	2	no	—
127	#494	—	67	2	no	—
128	#494	—	134	1	no	—
129	#494	—	134	1	no	—
130	C₂×C₄⋊C₈	—	134	1	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
134	#335	—	67	2	no	—
135	#335	—	134	1	no	—
136	C₂²×Q₈	—	134	2	no	—
137	C₂×Q₁₆	—	268	4	no	—
138	C₂×Q₁₆	—	268	2	no	—
139	C₂×Q₁₆	—	268	2	no	—
140	C₂×Q₁₆	—	67	4	no	—
141	C₂×Q₁₆	—	134	2	no	—
142	C₂×Q₁₆	—	134	2	no	—
143	C₂×Q₁₆	—	134	2	no	—
144	C₂²×C₈	—	268	1	no	—
145	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	268	1	no	—
148	C₄⋊Q₈	—	268	1	no	—
149	C₄⋊Q₈	—	134	2	no	—
150	C₄⋊Q₈	—	134	2	no	—
151	C₂×C₄⋊C₄	—	268	1	no	—
152	C₂×C₄⋊C₄	—	268	1	no	—
153	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₄⋊C₈	—	268	1	no	—
157	C₄⋊C₈	—	134	2	no	—
158	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
159	C₄×C₈	D₂₆₈	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
162	C₂×Q₈	—	268	2	no	—
163	C₂×Q₈	—	268	4	no	—
164	C₂×Q₈	—	268	4	no	—
165	C₂×Q₈	—	134	4	no	—
166	C₂²×C₄	—	268	2	no	—
167	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
169	C₂²×C₄	#11761617	1	1	yes	characteristic
170	Q₁₆	—	268	4	no	—
171	Q₁₆	—	268	4	no	—
172	Q₁₆	—	268	4	no	—
173	Q₁₆	—	268	2	no	—
174	Q₁₆	—	268	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	268	2	no	—
177	C₂×C₈	—	536	2	no	—
178	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
179	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
180	C₂×C₈	#11761613	1	2	yes	—
181	C₂×C₈	#11761615	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	C₄⋊C₄	—	268	2	no	—
185	C₄⋊C₄	—	268	2	no	—
186	C₄⋊C₄	—	268	2	no	—
187	C₄⋊C₄	—	268	2	no	—
188	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
189	C₄²	D₅₃₆	1	1	yes	characteristic
190	C₄²	#11761617	1	1	yes	characteristic
191	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
192	C₁₆	—	4	2	no	—
193	C₂³	#14523965	1	1	yes	characteristic
194	Q₈	—	268	4	no	—
195	Q₈	—	268	4	no	—
196	Q₈	—	536	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	268	4	no	—
199	C₂×C₄	—	536	4	no	—
200	C₂×C₄	#14523959	1	2	yes	—
201	C₂×C₄	#14523963	1	2	yes	—
202	C₂×C₄	#14523965	1	1	yes	characteristic
203	C₂×C₄	#14523965	1	1	yes	characteristic
204	C₂×C₄	#14523965	1	1	yes	characteristic
205	C₂×C₄	#14523988	1	2	yes	—
206	C₂×C₄	Dic₅₃₆	1	2	yes	—
207	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
209	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	536	2	no	—
213	C₈	#14523963	1	2	yes	—
214	C₈	C₂×#11761613	1	1	yes	characteristic
215	C₈	C₂×#11761615	1	1	yes	characteristic
216	C₂²	#16863139	1	2	yes	—
217	C₂²	#16863201	1	2	yes	—
218	C₂²	#16863213	1	2	yes	—
219	C₂²	C₂×#14523965	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	536	4	no	—
223	C₄	#16863139	1	2	yes	—
224	C₄	#16863201	1	2	yes	—
225	C₄	C₂×#14523959	1	1	yes	characteristic
226	C₄	C₂×#14523963	1	1	yes	characteristic
227	C₄	C₂×#14523988	1	1	yes	characteristic
228	C₄	C₂×Dic₅₃₆	1	1	yes	characteristic
229	C₂	#22558259	1	4	yes	direct product factor
230	C₂	C₂×#16863139	1	1	yes	characteristic
231	C₂	C₂×#16863201	1	1	yes	characteristic
232	C₂	C₂×#16863213	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1362
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558259	C₂	direct (N × H)
113. C₂×C₆₇	#1362	semidirect
121. C₆₇	C₂×#1362	semidirect
229. C₂	#22558259	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×#16863776	2
C₂×#16863149	2
#22558259	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#1362	67

Subgroups — C2×#22558259 #30767072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558259 #30767072