Subgroups — C₂×#22560346 #30856056

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22560346	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863160	C₂	1	4	yes	maximal
5	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863420	—	2	2	no	—
9	#16863420	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
11	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
15	#16863160	C₂²	1	16	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#423×C₆₇	—	2	2	no	—
19	#423×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	8	no	—
23	C₂×#11761612	C₂³	1	4	yes	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	4	no	—
26	C₂²×C₆₇⋊C₈	—	2	4	no	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
30	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
31	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
32	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	8	no	—
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₆₇	—	2	8	no	—
47	#11761612	—	2	8	no	—
48	#11761612	—	4	4	no	—
49	C₂×C₆₇⋊C₈	—	2	8	no	—
50	C₂×C₆₇⋊C₈	—	2	8	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	4	4	no	—
55	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
56	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	8	no	—
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	4	yes	—
68	C₂²×C₄×C₆₇	#41	1	4	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	M₁₆×C₆₇	—	2	8	no	—
71	M₁₆×C₆₇	—	4	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₂×C₈×C₆₇	—	4	4	no	—
74	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
76	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂×D₄	1	4	yes	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₆₇	#125	1	4	yes	—
84	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	8	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	#108	1	8	yes	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₆₇	#117	1	4	yes	—
94	C₂×C₄×C₆₇	#117	1	4	yes	—
95	C₂×C₄×C₆₇	#124	1	2	yes	—
96	C₂×C₄×C₆₇	#124	1	2	yes	—
97	C₂×C₄×C₆₇	#124	1	2	yes	—
98	C₂×C₄×C₆₇	#124	1	2	yes	—
99	C₂×C₄×C₆₇	#125	1	4	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
104	C₈×C₆₇	—	4	4	no	—
105	C₆₇⋊C₈	—	2	8	no	—
106	C₆₇⋊C₈	—	4	4	no	—
107	C₂²×C₆₇	—	2	8	no	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	#382	1	2	yes	—
111	C₂²×C₆₇	#382	1	2	yes	—
112	C₂²×C₆₇	C₂×#108	1	4	yes	—
113	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#125	1	2	yes	—
115	C₄×C₆₇	—	2	4	no	—
116	C₄×C₆₇	#382	1	2	yes	—
117	C₄×C₆₇	#382	1	2	yes	—
118	C₄×C₆₇	C₂×#108	1	8	yes	—
119	C₄×C₆₇	C₂×#117	1	2	yes	—
120	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
121	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
122	C₂×#1837	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	2	2	no	—
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	#1837	1	4	yes	complemented
126	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
127	C₂×C₆₇	C₂²×#108	1	2	yes	—
128	C₂×#423	—	67	1	no	—
129	C₂×#423	D₆₇	1	1	yes	characteristic
130	C₂²×C₄×C₈	—	67	1	no	—
131	#1837	—	67	8	no	—
132	C₂×#341	—	67	4	no	—
133	C₆₇	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	67	2	no	—
135	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
136	C₂³×C₈	—	67	2	no	—
137	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
138	#423	—	2	2	no	—
139	#423	—	67	4	no	—
140	#423	—	134	2	no	—
141	#423	C₂×D₆₇	1	4	yes	—
142	C₂×C₄⋊C₈	—	2	2	no	—
143	C₂×C₄⋊C₈	—	134	2	no	—
144	C₂×C₄×C₈	—	67	2	no	—
145	C₂×C₄×C₈	—	67	4	no	—
146	C₂×C₄×C₈	—	67	2	no	—
147	C₂×C₄×C₈	—	67	4	no	—
148	#341	—	67	16	no	—
149	C₂³×C₄	Dic₁₃₄	1	2	yes	—
150	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	8	no	—
153	C₂×M₁₆	—	67	4	no	—
154	C₂×M₁₆	—	134	8	no	—
155	C₂×M₁₆	—	134	2	no	—
156	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	67	4	no	—
159	C₂²×C₈	—	67	4	no	—
160	C₂²×C₈	—	67	4	no	—
161	C₂²×C₈	—	134	2	no	—
162	C₂²×C₈	—	134	4	no	—
163	C₂²×C₈	—	134	4	no	—
164	C₂×C₄²	Dic₁₃₄	1	2	yes	—
165	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₄×D₆₇	1	2	yes	—
167	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
170	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
171	C₄⋊C₈	—	2	8	no	—
172	C₄⋊C₈	—	134	8	no	—
173	C₄×C₈	—	67	8	no	—
174	C₄×C₈	—	67	8	no	—
175	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
178	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
179	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
185	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
186	C₂²×C₄	#11761627	1	4	yes	—
187	C₂²×C₄	#11761629	1	4	yes	—
188	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
189	M₁₆	—	2	8	no	—
190	M₁₆	—	4	4	no	—
191	M₁₆	—	268	4	no	—
192	M₁₆	—	134	8	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	268	4	no	—
196	C₂×C₈	—	67	8	no	—
197	C₂×C₈	—	134	4	no	—
198	C₂×C₈	—	134	8	no	—
199	C₂×C₈	—	134	8	no	—
200	C₂×C₈	—	134	4	no	—
201	C₂×C₈	—	134	4	no	—
202	C₄²	Dic₆₇⋊C₄	1	2	yes	—
203	C₄²	Dic₆₇⋊C₄	1	4	yes	—
204	C₄²	Dic₆₇⋊C₄	1	2	yes	—
205	C₄²	C₂×C₄×D₆₇	1	2	yes	—
206	C₄²	C₂×C₄×D₆₇	1	2	yes	—
207	C₄²	C₂×C₂²⋊D₆₇	1	4	yes	—
208	C₂³	—	2	4	no	—
209	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
210	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
211	C₂³	#14524034	1	4	yes	—
212	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	#14523978	1	8	yes	—
217	C₂×C₄	C₂×Dic₆₇⋊C₄	1	4	yes	—
218	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×Dic₆₇⋊C₄	1	4	yes	—
221	C₂×C₄	#14524001	1	4	yes	—
222	C₂×C₄	#14524001	1	4	yes	—
223	C₂×C₄	#14524030	1	2	yes	—
224	C₂×C₄	#14524030	1	2	yes	—
225	C₂×C₄	#14524032	1	2	yes	—
226	C₂×C₄	#14524032	1	2	yes	—
227	C₂×C₄	#14524034	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₂×C₄	C₂×#11761627	1	2	yes	—
231	C₂×C₄	C₂×#11761629	1	2	yes	—
232	C₈	—	4	4	no	—
233	C₈	—	268	4	no	—
234	C₈	—	134	8	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	#16863261	1	2	yes	—
239	C₂²	#16863261	1	2	yes	—
240	C₂²	C₂×#14523978	1	4	yes	—
241	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×#14524034	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	#16863261	1	2	yes	—
245	C₄	#16863261	1	2	yes	—
246	C₄	C₂×#14523978	1	8	yes	—
247	C₄	C₂×#14524001	1	2	yes	—
248	C₄	C₂×#14524030	1	1	yes	characteristic
249	C₄	C₂×#14524032	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	2	2	no	—
252	C₂	#22560346	1	4	yes	direct product factor
253	C₂	C₂×#16863261	1	1	yes	characteristic
254	C₂	C₂²×#14523978	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	C₂×#1837
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #22560346	C₂	direct (N × H)
125. C₂×C₆₇	#1837	semidirect
133. C₆₇	C₂×#1837	semidirect
252. C₂	#22560346	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×#16863420	2
C₂²×C₄×C₆₇⋊C₈	2
#22560346	2
C₂×#16863160	2
C₂×#423×C₆₇	2
C₂×#1837	67

Subgroups — C2×#22560346 #30856056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560346 #30856056