Subgroups — C₂×#22559957 #30853296

Order 17152.

1310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559957	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863150	C₂	1	4	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₆₇	C₄	1	4	yes	—
8	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
9	C₂×#14523954	C₄	1	8	yes	complemented
10	C₂×#14523954	C₂²	1	4	yes	—
11	C₄²×Dic₆₇	C₄	1	4	yes	—
12	C₄²×Dic₆₇	C₂²	1	4	yes	—
13	#16863150	C₂²	1	16	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₆₇	C₄	1	8	yes	complemented
17	C₂×#97×C₆₇	C₂²	1	4	yes	—
18	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
19	C₂²×C₆₇⋊C₈	—	2	8	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
22	C₂×C₄×Dic₆₇	—	2	16	no	—
23	C₂×C₄×Dic₆₇	—	2	8	no	—
24	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₆₇	C₂×C₄	1	16	yes	—
26	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
27	#14523954	C₂×C₄	1	16	yes	complemented
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
37	#97×C₆₇	C₂×C₄	1	16	yes	complemented
38	C₂²×Dic₆₇	—	2	8	no	—
39	C₂²×Dic₆₇	—	2	8	no	—
40	C₂²×Dic₆₇	M₁₆	1	16	yes	—
41	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
42	C₂×C₆₇⋊C₈	—	2	16	no	—
43	C₂×C₆₇⋊C₈	—	2	8	no	—
44	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
45	C₄×Dic₆₇	—	2	8	no	—
46	C₄×Dic₆₇	—	2	8	no	—
47	C₄×Dic₆₇	—	2	32	no	—
48	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	#41	1	8	yes	—
56	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₆₇	—	2	16	no	—
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
60	C₄²×C₆₇	C₄²	1	8	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
64	C₂³×C₆₇	#117	1	4	yes	—
65	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
66	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	8	no	—
68	C₂×C₄×C₆₇	—	2	16	no	—
69	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
70	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₆₇	#117	1	8	yes	—
72	C₂×C₄×C₆₇	#117	1	4	yes	—
73	C₂×C₄×C₆₇	#117	1	8	yes	—
74	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
77	C₂×Dic₆₇	—	2	8	no	—
78	C₂×Dic₆₇	—	2	8	no	—
79	C₂×Dic₆₇	—	2	32	no	—
80	C₂×Dic₆₇	C₂×M₁₆	1	16	yes	—
81	C₈×C₆₇	—	2	8	no	—
82	C₆₇⋊C₈	—	2	8	no	—
83	C₂²×C₆₇	#376	1	4	yes	—
84	C₂²×C₆₇	C₄×M₁₆	1	8	yes	—
85	C₂²×C₆₇	#432	1	16	yes	—
86	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×#117	1	2	yes	—
88	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
89	C₄×C₆₇	—	2	8	no	—
90	C₄×C₆₇	#376	1	4	yes	—
91	C₄×C₆₇	C₂×#117	1	2	yes	—
92	C₄×C₆₇	C₂×#117	1	2	yes	—
93	Dic₆₇	—	2	16	no	—
94	C₂×#1745	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	#1745	1	8	yes	complemented
96	C₂×C₆₇	C₂×#376	1	1	yes	characteristic
97	C₂×C₆₇	C₂×C₄×M₁₆	1	2	yes	—
98	C₂×C₆₇	C₂×#432	1	4	yes	—
99	C₂²×#97	—	67	1	no	—
100	C₂²×#97	D₆₇	1	1	yes	characteristic
101	C₂×C₄³	—	67	1	no	—
102	#1745	—	67	8	no	—
103	C₂×#336	—	67	4	no	—
104	C₆₇	C₂×#1745	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	67	2	no	—
106	C₂³×C₈	C₂×D₆₇	1	2	yes	—
107	C₂²×C₄²	—	67	4	no	—
108	C₂²×C₄²	—	67	2	no	—
109	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
110	C₂×#97	—	67	8	no	—
111	C₂×#97	—	67	4	no	—
112	C₂×#97	Dic₆₇	1	8	yes	complemented
113	C₂×#97	C₂×D₆₇	1	4	yes	—
114	C₄³	—	67	4	no	—
115	C₄³	—	67	4	no	—
116	#336	—	67	16	no	—
117	C₂³×C₄	—	67	4	no	—
118	C₂³×C₄	C₄×D₆₇	1	2	yes	—
119	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
120	C₂²×C₈	—	2	8	no	—
121	C₂²×C₈	—	67	8	no	—
122	C₂²×C₈	—	67	4	no	—
123	C₂²×C₈	—	134	8	no	—
124	C₂²×C₈	C₂×Dic₆₇	1	8	yes	—
125	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
126	C₂×C₄²	—	67	16	no	—
127	C₂×C₄²	—	67	4	no	—
128	C₂×C₄²	—	67	4	no	—
129	C₂×C₄²	—	134	16	no	—
130	C₂×C₄²	—	134	8	no	—
131	C₂×C₄²	C₄×D₆₇	1	2	yes	—
132	C₂×C₄²	C₄×D₆₇	1	4	yes	—
133	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
134	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
135	#97	—	67	16	no	—
136	#97	C₂×Dic₆₇	1	16	yes	complemented
137	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	67	16	no	—
140	C₂²×C₄	—	67	8	no	—
141	C₂²×C₄	—	134	8	no	—
142	C₂²×C₄	—	134	8	no	—
143	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
144	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
148	C₂²×C₄	#11761628	1	8	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	C₂×C₈	—	2	16	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	67	8	no	—
153	C₂×C₈	—	134	16	no	—
154	C₂×C₈	—	134	8	no	—
155	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
156	C₄²	—	134	32	no	—
157	C₄²	—	134	8	no	—
158	C₄²	—	134	8	no	—
159	C₄²	C₄×Dic₆₇	1	8	yes	—
160	C₄²	C₂×C₄×D₆₇	1	4	yes	—
161	C₄²	C₂²×Dic₆₇	1	4	yes	—
162	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
163	C₂³	#14524000	1	4	yes	—
164	C₂³	M₁₆×D₆₇	1	8	yes	—
165	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	67	16	no	—
169	C₂×C₄	—	134	8	no	—
170	C₂×C₄	—	134	8	no	—
171	C₂×C₄	—	134	32	no	—
172	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
173	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
174	C₂×C₄	#14524000	1	4	yes	—
175	C₂×C₄	#14524000	1	8	yes	—
176	C₂×C₄	#14524002	1	8	yes	—
177	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	C₂×#11761628	1	4	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	134	8	no	—
182	C₂²	#16863242	1	4	yes	—
183	C₂²	M₁₆×Dic₆₇	1	8	yes	—
184	C₂²	#16863451	1	16	yes	—
185	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
186	C₂²	C₂×#14524000	1	2	yes	—
187	C₂²	C₂×M₁₆×D₆₇	1	4	yes	—
188	C₄	—	2	8	no	—
189	C₄	—	134	16	no	—
190	C₄	#16863242	1	4	yes	—
191	C₄	C₂×#14524000	1	2	yes	—
192	C₄	C₂×#14524002	1	2	yes	—
193	C₂	#22559957	1	8	yes	direct product factor
194	C₂	C₂×#16863242	1	1	yes	characteristic
195	C₂	C₂×M₁₆×Dic₆₇	1	2	yes	—
196	C₂	C₂×#16863451	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂×#1745
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #22559957	C₂	direct (N × H)
9. C₂×#14523954	C₄	semidirect
16. C₂×#97×C₆₇	C₄	semidirect
27. #14523954	C₂×C₄	semidirect
37. #97×C₆₇	C₂×C₄	semidirect
95. C₂×C₆₇	#1745	semidirect
104. C₆₇	C₂×#1745	semidirect
112. C₂×#97	Dic₆₇	semidirect
136. #97	C₂×Dic₆₇	semidirect
193. C₂	#22559957	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×#14523954	2
C₂×C₄²×Dic₆₇	2
#22559957	2
C₂×#16863150	2
C₂²×#97×C₆₇	2
C₂×#1745	67

Subgroups — C2×#22559957 #30853296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559957 #30853296