Subgroups — C₂×#22558278 #30767398

Order 17152.

286 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558278	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	1	no	—
15	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₆₇⋊C₈	—	2	1	no	—
17	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	4	2	no	—
21	C₄×C₆₇⋊C₈	—	2	1	no	—
22	C₄×C₆₇⋊C₈	C₈	1	2	yes	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	C₂×C₁₆×C₆₇	—	4	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₆₇⋊C₈	—	2	4	no	—
35	C₂×C₆₇⋊C₈	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	C₁₆	1	4	yes	complemented
38	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
39	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
40	C₆₇⋊C₁₆	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	D₈	1	2	yes	—
48	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
54	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₆₇	—	4	2	no	—
56	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#107	1	2	yes	—
62	C₂×C₄×C₆₇	#108	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₆₇	—	2	1	no	—
69	C₈×C₆₇	—	2	1	no	—
70	C₈×C₆₇	#107	1	2	yes	—
71	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
72	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₆₇⋊C₈	—	2	4	no	—
74	C₆₇⋊C₈	—	2	2	no	—
75	C₆₇⋊C₈	C₂×C₁₆	1	4	yes	complemented
76	C₂²×C₆₇	#335	1	2	yes	—
77	C₂²×C₆₇	C₄⋊C₁₆	1	2	yes	—
78	C₂²×C₆₇	#364	1	2	yes	—
79	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	#335	1	2	yes	—
83	C₄×C₆₇	C₄⋊C₁₆	1	2	yes	—
84	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
85	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
86	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
87	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
88	C₂×#1370	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	#1370	1	4	yes	complemented
90	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
91	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
92	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
93	C₂×C₄⋊C₁₆	—	67	1	no	—
94	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
95	C₂×C₈²	—	67	1	no	—
96	#1370	—	67	4	no	—
97	C₆₇	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	2	1	no	—
99	C₂²×C₁₆	—	134	1	no	—
100	C₂×C₄×C₈	—	67	1	no	—
101	C₂×C₄×C₈	—	67	1	no	—
102	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
103	C₄⋊C₁₆	—	2	1	no	—
104	C₄⋊C₁₆	—	67	2	no	—
105	C₄⋊C₁₆	—	134	1	no	—
106	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
107	C₈²	—	67	2	no	—
108	C₈²	—	67	2	no	—
109	C₂²×C₈	—	67	1	no	—
110	C₂²×C₈	—	67	1	no	—
111	C₂²×C₈	—	134	1	no	—
112	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	2	no	—
117	C₂×C₁₆	—	268	2	no	—
118	C₂×C₁₆	—	134	2	no	—
119	C₄×C₈	—	67	2	no	—
120	C₄×C₈	—	67	2	no	—
121	C₄×C₈	—	67	2	no	—
122	C₄×C₈	—	134	1	no	—
123	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
124	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	#11761611	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	67	4	no	—
132	C₂×C₈	—	67	2	no	—
133	C₂×C₈	—	67	2	no	—
134	C₂×C₈	—	134	4	no	—
135	C₂×C₈	—	134	2	no	—
136	C₂×C₈	—	134	2	no	—
137	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#11761613	1	2	yes	—
140	C₂×C₈	#11761618	1	2	yes	—
141	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
143	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
144	C₄²	C₈×D₆₇	1	1	yes	characteristic
145	C₄²	#11761611	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
147	C₁₆	—	4	2	no	—
148	C₁₆	—	268	2	no	—
149	C₂³	Dic₆₇⋊C₈	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
152	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
153	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
154	C₂×C₄	#14523976	1	2	yes	—
155	C₂×C₄	#14523978	1	2	yes	—
156	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
157	C₂×C₄	#14523982	1	2	yes	—
158	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	67	4	no	—
164	C₈	—	134	4	no	—
165	C₈	—	134	2	no	—
166	C₈	#14523976	1	2	yes	—
167	C₈	C₂×#11761613	1	1	yes	characteristic
168	C₈	C₂×#11761618	1	1	yes	characteristic
169	C₂²	#16863148	1	2	yes	—
170	C₂²	Dic₆₇⋊C₁₆	1	2	yes	—
171	C₂²	#16863216	1	2	yes	—
172	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16863148	1	2	yes	—
176	C₄	Dic₆₇⋊C₁₆	1	2	yes	—
177	C₄	C₂×#14523976	1	1	yes	characteristic
178	C₄	C₂×#14523978	1	1	yes	characteristic
179	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
180	C₄	C₂×#14523982	1	1	yes	characteristic
181	C₂	#22558278	1	4	yes	direct product factor
182	C₂	C₂×#16863148	1	1	yes	characteristic
183	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic
184	C₂	C₂×#16863216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	C₂×#1370
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558278	C₂	direct (N × H)
37. C₂×C₆₇⋊C₈	C₁₆	semidirect
75. C₆₇⋊C₈	C₂×C₁₆	semidirect
89. C₂×C₆₇	#1370	semidirect
97. C₆₇	C₂×#1370	semidirect
181. C₂	#22558278	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×C₄⋊(C₆₇⋊C₁₆)	2
C₂×C₈×C₆₇⋊C₈	2
#22558278	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#1370	67

Subgroups — C2×#22558278 #30767398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558278 #30767398