Subgroups — C₂×#22560349 #30856060

Order 17152.

830 conjugacy classes of subgroups.

The subgroup lattice has 238 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×#14523954	C₂	1	1	yes	maximal, characteristic
3	#22560349	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×#14523954	C₄	1	4	yes	—
12	C₂×#14523954	C₄	1	4	yes	—
13	C₂×#14523954	C₂²	1	4	yes	—
14	#16863160	C₂²	1	16	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	—	2	2	no	—
18	#423×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	8	no	—
22	C₂×#11761612	C₂³	1	4	yes	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	4	no	—
25	C₂²×C₆₇⋊C₈	—	2	4	no	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
30	#14523954	C₂×C₄	1	16	yes	—
31	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	8	no	—
35	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
43	C₄⋊C₈×C₆₇	—	2	8	no	—
44	#11761612	—	2	8	no	—
45	#11761612	—	4	4	no	—
46	C₂×C₆₇⋊C₈	—	2	8	no	—
47	C₂×C₆₇⋊C₈	—	2	8	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	2	4	no	—
50	C₂×C₆₇⋊C₈	—	2	4	no	—
51	C₂×C₆₇⋊C₈	—	4	4	no	—
52	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
53	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	4	4	no	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	#41	1	4	yes	—
64	C₂²×C₄×C₆₇	#41	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₆₇	—	2	8	no	—
67	M₁₆×C₆₇	—	4	4	no	—
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
71	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
73	C₄²×C₆₇	C₂×D₄	1	4	yes	—
74	C₂³×C₆₇	—	2	4	no	—
75	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₆₇	#125	1	4	yes	—
78	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	8	no	—
80	C₂×C₄×C₆₇	—	4	4	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#117	1	4	yes	—
86	C₂×C₄×C₆₇	#117	1	4	yes	—
87	C₂×C₄×C₆₇	#124	1	2	yes	—
88	C₂×C₄×C₆₇	#124	1	2	yes	—
89	C₂×C₄×C₆₇	#124	1	2	yes	—
90	C₂×C₄×C₆₇	#124	1	2	yes	—
91	C₂×C₄×C₆₇	#125	1	4	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
96	C₈×C₆₇	—	4	4	no	—
97	C₆₇⋊C₈	—	2	8	no	—
98	C₆₇⋊C₈	—	4	4	no	—
99	C₂²×C₆₇	—	2	8	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	#382	1	2	yes	—
103	C₂²×C₆₇	#382	1	2	yes	—
104	C₂²×C₆₇	#430	1	4	yes	—
105	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₆₇	C₂×#125	1	2	yes	—
107	C₄×C₆₇	—	2	8	no	—
108	C₄×C₆₇	#382	1	2	yes	—
109	C₄×C₆₇	#382	1	2	yes	—
110	C₄×C₆₇	C₂×#117	1	2	yes	—
111	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
112	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
113	C₂×#1838	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	2	no	—
115	C₂×C₆₇	—	2	2	no	—
116	C₂×C₆₇	#1838	1	4	yes	complemented
117	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#430	1	2	yes	—
119	C₂×#423	—	67	1	no	—
120	C₂×#423	D₆₇	1	1	yes	characteristic
121	C₂²×#97	—	67	1	no	—
122	#1838	—	67	8	no	—
123	C₂×#341	—	67	4	no	—
124	C₆₇	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	67	2	no	—
126	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
127	C₂³×C₈	—	67	2	no	—
128	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
129	#423	—	2	2	no	—
130	#423	—	67	4	no	—
131	#423	—	134	2	no	—
132	#423	C₂×D₆₇	1	4	yes	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₄⋊C₈	—	134	2	no	—
135	C₂×#97	—	67	4	no	—
136	C₂×#97	—	67	4	no	—
137	C₂×#97	—	67	4	no	—
138	#341	—	67	16	no	—
139	C₂³×C₄	Dic₁₃₄	1	2	yes	—
140	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	8	no	—
143	C₂×M₁₆	—	67	4	no	—
144	C₂×M₁₆	—	134	8	no	—
145	C₂×M₁₆	—	134	2	no	—
146	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	67	4	no	—
149	C₂²×C₈	—	67	4	no	—
150	C₂²×C₈	—	67	4	no	—
151	C₂²×C₈	—	134	2	no	—
152	C₂²×C₈	—	134	4	no	—
153	C₂²×C₈	—	134	4	no	—
154	C₂×C₄²	Dic₁₃₄	1	2	yes	—
155	C₂×C₄²	C₄×D₆₇	1	2	yes	—
156	C₂×C₄²	C₄×D₆₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
159	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
160	C₄⋊C₈	—	2	8	no	—
161	C₄⋊C₈	—	134	8	no	—
162	#97	—	67	16	no	—
163	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
166	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
172	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
173	C₂²×C₄	#11761627	1	4	yes	—
174	C₂²×C₄	#11761629	1	4	yes	—
175	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	4	no	—
178	M₁₆	—	268	4	no	—
179	M₁₆	—	134	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	268	4	no	—
183	C₂×C₈	—	67	8	no	—
184	C₂×C₈	—	134	4	no	—
185	C₂×C₈	—	134	8	no	—
186	C₂×C₈	—	134	8	no	—
187	C₂×C₈	—	134	4	no	—
188	C₂×C₈	—	134	4	no	—
189	C₄²	Dic₆₇⋊C₄	1	4	yes	—
190	C₄²	Dic₆₇⋊C₄	1	4	yes	—
191	C₄²	C₂×C₄×D₆₇	1	4	yes	—
192	C₄²	C₂×C₂²⋊D₆₇	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
195	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
196	C₂³	#14524034	1	4	yes	—
197	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	C₂×Dic₆₇⋊C₄	1	8	yes	—
202	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
204	C₂×C₄	#14524001	1	4	yes	—
205	C₂×C₄	#14524001	1	4	yes	—
206	C₂×C₄	#14524030	1	2	yes	—
207	C₂×C₄	#14524030	1	2	yes	—
208	C₂×C₄	#14524032	1	2	yes	—
209	C₂×C₄	#14524032	1	2	yes	—
210	C₂×C₄	#14524034	1	4	yes	—
211	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×#11761627	1	2	yes	—
214	C₂×C₄	C₂×#11761629	1	2	yes	—
215	C₈	—	4	4	no	—
216	C₈	—	268	4	no	—
217	C₈	—	134	8	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	#16863261	1	2	yes	—
222	C₂²	#16863261	1	2	yes	—
223	C₂²	#16863444	1	4	yes	—
224	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
225	C₂²	C₂×#14524034	1	2	yes	—
226	C₄	—	2	8	no	—
227	C₄	#16863261	1	2	yes	—
228	C₄	#16863261	1	2	yes	—
229	C₄	C₂×#14524001	1	2	yes	—
230	C₄	C₂×#14524030	1	1	yes	characteristic
231	C₄	C₂×#14524032	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	2	2	no	—
234	C₂	#22560349	1	4	yes	direct product factor
235	C₂	C₂×#16863261	1	1	yes	characteristic
236	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×#1838
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. #22560349	C₂	direct (N × H)
116. C₂×C₆₇	#1838	semidirect
124. C₆₇	C₂×#1838	semidirect
234. C₂	#22560349	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×#14523954	2
#22560349	2
C₂×#16863160	2
C₂×#423×C₆₇	2
C₂×#1838	67

Subgroups — C2×#22560349 #30856060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560349 #30856060