Subgroups — C₂×#22569315 #31720800

Order 17152.

1706 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569315	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	8	yes	maximal, complemented
3	C₂×#16863457	C₂	1	4	yes	maximal
4	C₂×#16863453	C₂	1	2	yes	maximal
5	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863527	C₂²	1	64	yes	complemented
7	#16863457	C₂²	1	32	yes	—
8	#16863453	C₂²	1	16	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×#14523954	C₂²	1	4	yes	—
13	C₂×#14523954	C₂²	1	4	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
15	#533×C₆₇	C₄	1	8	yes	—
16	#533×C₆₇	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
18	C₂×#125×C₆₇	C₄	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂×#117×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
24	C₄⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
25	C₄⋊(C₆₇⋊C₈)	C₂³	1	32	yes	—
26	C₂²⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
27	#14523954	C₂³	1	16	yes	—
28	#14523954	C₂³	1	16	yes	—
29	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
30	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	—
34	#125×C₆₇	C₂×C₄	1	16	yes	—
35	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	—
36	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	—
37	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
38	#117×C₆₇	C₂×C₄	1	8	yes	—
39	#117×C₆₇	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
50	C₂×C₆₇⋊C₈	—	2	16	no	—
51	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	—
52	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	—	4	8	no	—
54	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	4	4	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
66	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
67	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
68	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
71	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
72	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
73	C₄²×C₆₇	C₂⁴	1	4	yes	—
74	C₄²×C₆₇	C₂⁴	1	8	yes	—
75	C₂³×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
78	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
79	Q₈×C₆₇	—	4	16	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	16	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	4	8	no	—
87	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
93	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
94	C₆₇⋊C₈	—	2	16	no	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	—	4	4	no	—
98	C₂²×C₆₇	#519	1	2	yes	—
99	C₂²×C₆₇	#568	1	4	yes	—
100	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₆₇	—	2	2	no	—
102	C₄×C₆₇	—	2	4	no	—
103	C₄×C₆₇	—	2	4	no	—
104	C₄×C₆₇	—	2	16	no	—
105	C₄×C₆₇	#519	1	2	yes	—
106	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
107	C₂×#2987	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	4	2	no	—
109	C₂×C₆₇	#2987	1	4	yes	complemented
110	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#568	1	2	yes	—
112	C₂×#533	D₆₇	1	1	yes	characteristic
113	#2987	—	67	16	no	—
114	C₂×#446	—	67	8	no	—
115	C₂×#433	—	67	4	no	—
116	C₂×#432	—	67	2	no	—
117	C₆₇	C₂×#2987	1	1	yes	characteristic, complemented, Sylow
118	#533	Dic₆₇	1	8	yes	—
119	#533	C₂×D₆₇	1	8	yes	—
120	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
121	C₂×#125	Dic₆₇	1	2	yes	—
122	C₂×C₂²⋊Q₈	Dic₆₇	1	4	yes	—
123	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
124	C₂×#117	C₂×D₆₇	1	2	yes	—
125	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
126	#446	—	67	64	no	—
127	#433	—	67	32	no	—
128	#432	—	67	16	no	—
129	C₂×C₄⋊C₈	—	67	4	no	—
130	C₂×C₄⋊C₈	—	67	8	no	—
131	C₂×C₂²⋊C₈	—	67	4	no	—
132	C₂×#97	—	67	4	no	—
133	C₂×#97	—	67	4	no	—
134	C₂×C₄×C₈	—	67	4	no	—
135	C₂²×Q₈	C₂×Dic₆₇	1	4	yes	—
136	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
137	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
138	C₂²×C₈	—	67	8	no	—
139	C₄⋊Q₈	C₂×Dic₆₇	1	16	yes	—
140	#125	C₂×Dic₆₇	1	16	yes	—
141	C₂²⋊Q₈	C₂×Dic₆₇	1	32	yes	—
142	C₄×Q₈	C₂×Dic₆₇	1	16	yes	—
143	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
144	#117	C₂×Dic₆₇	1	8	yes	—
145	#117	C₂²×D₆₇	1	8	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
147	C₂×C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
148	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
149	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
150	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
153	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
154	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
156	C₄⋊C₈	—	67	16	no	—
157	C₄⋊C₈	—	67	32	no	—
158	C₂²⋊C₈	—	67	16	no	—
159	#97	—	67	16	no	—
160	#97	—	67	16	no	—
161	C₄×C₈	—	67	16	no	—
162	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
163	C₂×Q₈	—	4	8	no	—
164	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
172	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
173	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
175	C₂×C₈	—	67	16	no	—
176	C₂×C₈	—	134	16	no	—
177	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
178	C₄⋊C₄	C₂²×Dic₆₇	1	32	yes	—
179	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
180	C₂²⋊C₄	C₂²×Dic₆₇	1	16	yes	—
181	C₄²	C₂²×Dic₆₇	1	4	yes	—
182	C₄²	C₂²×Dic₆₇	1	4	yes	—
183	C₄²	C₂²×Dic₆₇	1	4	yes	—
184	C₄²	C₂²×Dic₆₇	1	8	yes	—
185	C₄²	C₂³×D₆₇	1	8	yes	—
186	C₄²	C₂³×D₆₇	1	4	yes	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₂³×Dic₆₇	1	2	yes	—
190	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
191	Q₈	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
200	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
201	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
202	C₂×C₄	C₂³×Dic₆₇	1	16	yes	—
203	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
204	C₂×C₄	#14524091	1	4	yes	—
205	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
206	C₈	—	134	16	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	#16863874	1	2	yes	—
211	C₂²	#16864163	1	4	yes	—
212	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	4	no	—
215	C₄	—	2	16	no	—
216	C₄	—	2	2	no	—
217	C₄	#16863874	1	2	yes	—
218	C₄	C₂×#14524091	1	2	yes	—
219	C₂	—	4	2	no	—
220	C₂	#22569315	1	4	yes	direct product factor
221	C₂	C₂×#16863874	1	1	yes	characteristic
222	C₂	C₂×#16864163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	C₂×#2987
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #22569315	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
6. #16863527	C₂²	semidirect
109. C₂×C₆₇	#2987	semidirect
117. C₆₇	C₂×#2987	semidirect
220. C₂	#22569315	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569315	2
C₂×#16863527	2
C₂×#16863457	2
C₂×#16863453	2
C₂×#533×C₆₇	2
C₂×#2987	67

Subgroups — C2×#22569315 #31720800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569315 #31720800