Subgroups — C₂×#22560383 #30856290

Order 17152.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523978	C₂	1	4	yes	maximal
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
4	#22560383	C₂	1	8	yes	maximal, direct product factor
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	C₂×#14523978	—	2	16	no	—
9	C₂×#14523978	C₂²	1	16	yes	—
10	#16863420	—	2	2	no	—
11	#16863420	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
13	C₂×#14523954	C₄	1	4	yes	—
14	C₂×#14523954	C₄	1	4	yes	—
15	C₂×#14523954	C₂²	1	4	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	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	D₄	1	8	yes	—
28	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
30	#14523978	—	2	32	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
32	#14523954	C₂×C₄	1	16	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	2	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₄⋊C₈×C₆₇	—	2	8	no	—
46	#11761612	—	2	8	no	—
47	#11761612	—	4	4	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	4	4	no	—
50	C₂×C₆₇⋊C₈	C₄⋊C₄	1	16	yes	—
51	C₂×C₆₇⋊C₈	C₄⋊C₄	1	16	yes	—
52	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
53	C₂×C₆₇⋊C₈	C₂×D₄	1	8	yes	—
54	C₂×C₆₇⋊C₈	C₂×Q₈	1	8	yes	—
55	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	4	4	no	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₆₇	—	2	8	no	—
69	M₁₆×C₆₇	—	4	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	4	4	no	—
72	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
75	C₄²×C₆₇	C₂×D₄	1	4	yes	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
80	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	8	no	—
82	C₂×C₄×C₆₇	—	4	4	no	—
83	C₂×C₄×C₆₇	—	4	4	no	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#127	1	2	yes	—
90	C₂×C₄×C₆₇	#127	1	2	yes	—
91	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
98	C₈×C₆₇	—	4	4	no	—
99	C₆₇⋊C₈	—	4	4	no	—
100	C₆₇⋊C₈	C₂×C₄⋊C₄	1	16	yes	—
101	C₂²×C₆₇	—	2	8	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	#384	1	2	yes	—
105	C₂²×C₆₇	#384	1	2	yes	—
106	C₂²×C₆₇	#430	1	4	yes	—
107	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
109	C₄×C₆₇	—	2	8	no	—
110	C₄×C₆₇	#384	1	2	yes	—
111	C₄×C₆₇	#384	1	2	yes	—
112	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
113	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
114	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×#1849	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	#1849	1	4	yes	complemented
119	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#430	1	2	yes	—
121	C₂²×#108	—	67	4	no	—
122	C₂×#423	—	67	1	no	—
123	C₂×#423	D₆₇	1	1	yes	characteristic
124	C₂²×#97	—	67	1	no	—
125	#1849	—	67	8	no	—
126	C₆₇	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
127	C₂²×M₁₆	—	67	2	no	—
128	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
129	C₂³×C₈	—	67	2	no	—
130	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
131	C₂×#108	—	67	16	no	—
132	C₂×#108	—	134	16	no	—
133	#423	—	2	2	no	—
134	#423	—	67	4	no	—
135	#423	—	134	2	no	—
136	#423	C₂×D₆₇	1	4	yes	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	—	134	2	no	—
139	C₂×#97	—	67	4	no	—
140	C₂×#97	—	67	4	no	—
141	C₂×#97	—	67	4	no	—
142	C₂³×C₄	Dic₁₃₄	1	2	yes	—
143	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	67	4	no	—
147	C₂×M₁₆	—	134	8	no	—
148	C₂×M₁₆	—	134	2	no	—
149	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	67	4	no	—
152	C₂²×C₈	—	67	4	no	—
153	C₂²×C₈	—	67	8	no	—
154	C₂²×C₈	—	67	8	no	—
155	C₂²×C₈	—	67	4	no	—
156	C₂²×C₈	—	134	2	no	—
157	C₂×C₄²	Dic₁₃₄	1	2	yes	—
158	C₂×C₄²	C₄×D₆₇	1	2	yes	—
159	C₂×C₄²	C₄×D₆₇	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
162	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
163	#108	—	134	32	no	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄⋊C₈	—	134	8	no	—
166	#97	—	67	16	no	—
167	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
170	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
171	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
173	C₂²×C₄	D₄×D₆₇	1	4	yes	—
174	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
176	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
177	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
178	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
179	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
180	M₁₆	—	2	8	no	—
181	M₁₆	—	4	4	no	—
182	M₁₆	—	268	4	no	—
183	M₁₆	—	134	8	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	268	4	no	—
187	C₂×C₈	—	67	16	no	—
188	C₂×C₈	—	67	16	no	—
189	C₂×C₈	—	67	8	no	—
190	C₂×C₈	—	67	8	no	—
191	C₂×C₈	—	67	8	no	—
192	C₂×C₈	—	134	4	no	—
193	C₄²	Dic₆₇⋊C₄	1	4	yes	—
194	C₄²	Dic₆₇⋊C₄	1	4	yes	—
195	C₄²	C₂×C₄×D₆₇	1	4	yes	—
196	C₄²	C₂×C₂²⋊D₆₇	1	4	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
199	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
200	C₂³	#14524042	1	4	yes	—
201	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
206	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
207	C₂×C₄	C₂×Dic₆₇⋊C₄	1	8	yes	—
208	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
210	C₂×C₄	#14524039	1	2	yes	—
211	C₂×C₄	#14524039	1	2	yes	—
212	C₂×C₄	#14524041	1	2	yes	—
213	C₂×C₄	#14524041	1	2	yes	—
214	C₂×C₄	#14524042	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
217	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
219	C₈	—	4	4	no	—
220	C₈	—	268	4	no	—
221	C₈	—	67	16	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	8	no	—
225	C₂²	#16863265	1	2	yes	—
226	C₂²	#16863265	1	2	yes	—
227	C₂²	#16863444	1	4	yes	—
228	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
229	C₂²	C₂×#14524042	1	2	yes	—
230	C₄	—	2	8	no	—
231	C₄	#16863265	1	2	yes	—
232	C₄	#16863265	1	2	yes	—
233	C₄	C₂×C₄⋊C₄×D₆₇	1	2	yes	—
234	C₄	C₂×#14524039	1	1	yes	characteristic
235	C₄	C₂×#14524041	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	2	2	no	—
238	C₂	#22560383	1	4	yes	direct product factor
239	C₂	C₂×#16863265	1	1	yes	characteristic
240	C₂	C₂×#16863444	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×#1849
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
4. #22560383	C₂	direct (N × H)
118. C₂×C₆₇	#1849	semidirect
126. C₆₇	C₂×#1849	semidirect
238. C₂	#22560383	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523978	2
C₂×#16863420	2
C₂²×#14523954	2
#22560383	2
C₂×#423×C₆₇	2
C₂×#1849	67

Subgroups — C2×#22560383 #30856290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560383 #30856290