Subgroups — C₂×#22558160 #30765016

Order 17152.

410 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863178	C₂	1	2	yes	maximal
2	#22558160	C₂	1	4	yes	maximal, direct product factor
3	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	2	no	—
5	#16863178	—	2	2	no	—
6	#16863178	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
10	#423×C₆₇	—	2	2	no	—
11	#423×C₆₇	C₄	1	2	yes	—
12	#423×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
14	#14523983	—	2	4	no	—
15	#14523983	—	4	2	no	—
16	C₂×C₆₇⋊C₁₆	—	4	2	no	—
17	C₂³×C₄×C₆₇	—	2	1	no	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	—	2	1	no	—
20	C₂×M₁₆×C₆₇	—	4	1	no	—
21	C₂×M₁₆×C₆₇	—	4	1	no	—
22	C₂×M₁₆×C₆₇	—	4	1	no	—
23	C₂×M₁₆×C₆₇	—	4	1	no	—
24	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
27	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	—	2	1	no	—
29	C₂²×C₈×C₆₇	D₄	1	2	yes	—
30	C₂×C₄²×C₆₇	—	2	1	no	—
31	C₂×C₄²×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	—	2	1	no	—
33	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₈	1	2	yes	—
35	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₈×C₆₇	—	2	4	no	—
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	C₆₇⋊C₁₆	—	4	4	no	—
40	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	—	2	1	no	—
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	4	4	no	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₆₇	—	4	1	no	—
57	M₁₆×C₆₇	—	4	2	no	—
58	M₁₆×C₆₇	—	4	1	no	—
59	M₁₆×C₆₇	—	4	2	no	—
60	M₁₆×C₆₇	—	4	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	4	1	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
68	C₄²×C₆₇	—	2	1	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	1	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	C₂×C₈	1	2	yes	—
73	C₄²×C₆₇	C₂×C₈	1	2	yes	—
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	1	no	—
76	C₂³×C₆₇	—	4	2	no	—
77	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	1	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#99	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	—	4	2	no	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	1	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	4	1	no	—
111	C₂²×C₆₇	—	4	1	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	#324	1	1	yes	characteristic
115	C₂²×C₆₇	#324	1	1	yes	characteristic
116	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	2	1	no	—
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	4	2	no	—
121	C₄×C₆₇	#324	1	1	yes	characteristic
122	C₄×C₆₇	#324	1	1	yes	characteristic
123	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
124	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
125	C₂×#1327	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	2	1	no	—
128	C₂×C₆₇	—	4	1	no	—
129	C₂×C₆₇	—	4	1	no	—
130	C₂×C₆₇	#1327	1	2	yes	complemented
131	C₂×C₆₇	C₂×#324	1	1	yes	characteristic
132	C₂×#423	D₆₇	1	1	yes	characteristic
133	C₂×#349	—	67	2	no	—
134	#1327	—	67	4	no	—
135	C₆₇	C₂×#1327	1	1	yes	characteristic, complemented, Sylow
136	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
138	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
139	C₂×M₃₂	—	134	2	no	—
140	#423	—	2	2	no	—
141	#423	Dic₆₇	1	2	yes	—
142	#423	C₂×D₆₇	1	2	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	#349	—	67	4	no	—
145	#349	—	134	2	no	—
146	C₂³×C₄	—	2	1	no	—
147	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	4	1	no	—
150	C₂×M₁₆	—	4	1	no	—
151	C₂×M₁₆	—	4	1	no	—
152	C₂×M₁₆	—	4	1	no	—
153	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
154	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
155	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
156	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
164	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
166	M₃₂	—	268	2	no	—
167	M₃₂	—	134	4	no	—
168	C₂×C₁₆	—	268	2	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₂⁴	#11761612	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
185	C₂²×C₄	#11761612	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
187	M₁₆	—	4	1	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
198	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
204	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
205	C₁₆	—	268	4	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
210	C₂³	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
211	C₂³	C₂×#11761612	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
230	C₂×C₄	#14523958	1	1	yes	characteristic
231	C₂×C₄	#14523958	1	1	yes	characteristic
232	C₂×C₄	#14523962	1	1	yes	characteristic
233	C₂×C₄	#14523962	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	4	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	#16863122	1	1	yes	characteristic
247	C₂²	#16863122	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	#16863122	1	1	yes	characteristic
254	C₄	#16863122	1	1	yes	characteristic
255	C₄	C₂×#14523958	1	1	yes	characteristic
256	C₄	C₂×#14523962	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	#22558160	1	2	yes	direct product factor
262	C₂	C₂×#16863122	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523958
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₆₇
Fitting	C₂×#423×C₆₇	C₂
socle	C₂²×C₆₇	#324
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×M₁₆×C₆₇	C₂³
hypercenter	C₂×#423	D₆₇
hypocenter	C₆₇	C₂×#1327
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
2. #22558160	C₂	direct (N × H)
130. C₂×C₆₇	#1327	semidirect
135. C₆₇	C₂×#1327	semidirect
261. C₂	#22558160	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×#16863178	2
#22558160	2
C₂×#423×C₆₇	2
C₂×#1327	67

Subgroups — C2×#22558160 #30765016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558160 #30765016