Subgroups — C₆₇×C₄⋊#338 #30690089

Order 17152.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#338×C₆₇	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
5	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
7	#423×C₆₇	—	2	2	no	—
8	#423×C₆₇	—	2	2	no	—
9	#423×C₆₇	—	2	4	no	—
10	C₂×C₄⋊C₈×C₆₇	C₄	1	8	yes	—
11	C₄×M₁₆×C₆₇	—	2	2	no	—
12	C₄×M₁₆×C₆₇	—	2	2	no	—
13	C₂×#97×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
15	#338×C₆₇	—	2	8	no	—
16	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₆₇	—	2	4	no	—
19	C₂×M₁₆×C₆₇	—	2	4	no	—
20	C₂×M₁₆×C₆₇	—	2	8	no	—
21	C₂×M₁₆×C₆₇	—	2	4	no	—
22	C₂×M₁₆×C₆₇	—	2	4	no	—
23	C₂×M₁₆×C₆₇	—	2	8	no	—
24	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
25	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	D₄	1	2	yes	—
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₄⋊C₈×C₆₇	—	2	8	no	—
35	C₄⋊C₈×C₆₇	—	2	8	no	—
36	#97×C₆₇	—	2	4	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	—	4	4	no	—
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₆₇	#41	1	4	yes	—
52	M₁₆×C₆₇	—	2	8	no	—
53	M₁₆×C₆₇	—	2	8	no	—
54	M₁₆×C₆₇	—	4	8	no	—
55	M₁₆×C₆₇	—	4	4	no	—
56	M₁₆×C₆₇	—	4	4	no	—
57	C₂×C₈×C₆₇	—	2	4	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	8	no	—
60	C₂×C₈×C₆₇	—	2	16	no	—
61	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
62	C₄²×C₆₇	C₄²	1	2	yes	—
63	C₄²×C₆₇	C₄²	1	2	yes	—
64	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂³×C₆₇	—	4	2	no	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₆₇	#125	1	2	yes	—
73	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	—	4	2	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	#95	1	2	yes	—
79	C₂×C₄×C₆₇	#95	1	4	yes	—
80	C₂×C₄×C₆₇	#95	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₆₇	#117	1	2	yes	—
86	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
88	C₂×C₄×C₆₇	#121	1	4	yes	—
89	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₆₇	#124	1	2	yes	—
91	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₈×C₆₇	—	2	8	no	—
94	C₈×C₆₇	—	4	8	no	—
95	C₂²×C₆₇	—	2	2	no	—
96	C₂²×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	—	4	2	no	—
98	C₂²×C₆₇	—	4	2	no	—
99	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
100	C₂²×C₆₇	#382	1	1	yes	characteristic
101	C₂²×C₆₇	#384	1	1	yes	characteristic
102	C₂²×C₆₇	#389	1	4	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	4	no	—
105	C₄×C₆₇	#338	1	4	yes	complemented
106	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
107	C₄×C₆₇	#380	1	2	yes	—
108	C₄×C₆₇	#390	1	2	yes	—
109	C₄×C₆₇	#391	1	2	yes	—
110	C₄⋊#338	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#338	1	2	yes	—
116	C₂×#423	C₂×C₆₇	1	2	yes	—
117	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
118	C₂×#338	C₂×C₆₇	1	4	yes	—
119	C₆₇	C₄⋊#338	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
121	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
122	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
123	#423	—	2	2	no	—
124	#423	—	2	2	no	—
125	#423	—	2	4	no	—
126	C₂×C₄⋊C₈	C₄×C₆₇	1	8	yes	—
127	C₄×M₁₆	—	2	2	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₂×#97	C₄×C₆₇	1	2	yes	—
130	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
131	#338	—	2	8	no	—
132	C₂³×C₄	Q₈×C₆₇	1	2	yes	—
133	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	2	8	no	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	—	2	4	no	—
139	C₂×M₁₆	—	2	8	no	—
140	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
141	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
143	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
147	C₂×C₄²	D₄×C₆₇	1	2	yes	—
148	C₂×C₄²	D₄×C₆₇	1	2	yes	—
149	C₂×C₄²	D₄×C₆₇	1	2	yes	—
150	C₄⋊C₈	—	2	8	no	—
151	C₄⋊C₈	—	2	8	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂⁴	C₂×Q₈×C₆₇	1	1	yes	characteristic
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
158	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
159	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
160	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
166	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
167	C₂²×C₄	#41×C₆₇	1	4	yes	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	8	no	—
170	M₁₆	—	4	8	no	—
171	M₁₆	—	4	4	no	—
172	M₁₆	—	4	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	16	no	—
177	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	—
178	C₄²	C₄²×C₆₇	1	2	yes	—
179	C₄²	C₄²×C₆₇	1	2	yes	—
180	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
181	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
184	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
187	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
188	C₂³	#125×C₆₇	1	2	yes	—
189	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	#95×C₆₇	1	2	yes	—
195	C₂×C₄	#95×C₆₇	1	4	yes	—
196	C₂×C₄	#95×C₆₇	1	2	yes	—
197	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
200	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
201	C₂×C₄	#117×C₆₇	1	2	yes	—
202	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
203	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
204	C₂×C₄	#121×C₆₇	1	4	yes	—
205	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
206	C₂×C₄	#124×C₆₇	1	2	yes	—
207	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
208	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	4	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
216	C₂²	#382×C₆₇	1	1	yes	characteristic
217	C₂²	#384×C₆₇	1	1	yes	characteristic
218	C₂²	#389×C₆₇	1	4	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	#338×C₆₇	1	4	yes	complemented
222	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
223	C₄	#380×C₆₇	1	2	yes	—
224	C₄	#390×C₆₇	1	2	yes	—
225	C₄	#391×C₆₇	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
230	C₂	C₂×#338×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
105. C₄×C₆₇	#338	semidirect
110. C₄⋊#338	C₆₇	direct (N × H)
119. C₆₇	C₄⋊#338	direct (N × H)
221. C₄	#338×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#338×C₆₇	2
C₄⋊#338	67

Subgroups — C67×C4⋊#338 #30690089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×C₄⋊#338 #30690089