Subgroups — C₆₇×#33169 #30712420

Order 17152.

1046 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×#464×C₆₇	C₂	1	3	yes	maximal
2	C₂×#462×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2658	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1983	C₂	1	3	yes	maximal
6	C₆₇×#1981	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₆₇	C₂²	1	3	yes	—
9	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
11	C₂×#124×C₆₇	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
13	#464×C₆₇	—	2	12	no	—
14	#462×C₆₇	—	2	12	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#103×C₆₇	C₂²	1	6	yes	—
18	C₂×#103×C₆₇	C₂²	1	3	yes	—
19	C₂×#102×C₆₇	C₂²	1	3	yes	—
20	#399×C₆₇	—	2	1	no	—
21	#399×C₆₇	—	2	3	no	—
22	C₄×C₄⋊C₄×C₆₇	—	2	3	no	—
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
24	#376×C₆₇	—	2	1	no	—
25	#376×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
27	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Q₁₆×C₆₇	—	2	12	no	—
30	C₂×Q₁₆×C₆₇	—	2	6	no	—
31	C₂×SD₁₆×C₆₇	—	2	6	no	—
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
34	C₄⋊Q₈×C₆₇	—	2	6	no	—
35	C₄⋊Q₈×C₆₇	—	2	6	no	—
36	#124×C₆₇	—	2	6	no	—
37	#124×C₆₇	—	2	6	no	—
38	C₄×Q₈×C₆₇	—	2	6	no	—
39	C₄×Q₈×C₆₇	D₄	1	12	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
46	C₂×C₄²×C₆₇	—	2	3	no	—
47	C₂×C₄²×C₆₇	—	2	3	no	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	—	2	6	no	—
50	C₄⋊C₈×C₆₇	—	2	6	no	—
51	#103×C₆₇	—	2	12	no	—
52	#103×C₆₇	—	2	6	no	—
53	#102×C₆₇	—	2	6	no	—
54	#95×C₆₇	—	2	3	no	—
55	#95×C₆₇	D₄	1	1	yes	characteristic
56	#95×C₆₇	D₄	1	3	yes	—
57	C₂⁴×C₆₇	—	2	1	no	—
58	C₂×Q₈×C₆₇	—	2	3	no	—
59	C₂×Q₈×C₆₇	—	2	3	no	—
60	C₂×Q₈×C₆₇	—	2	6	no	—
61	C₂×Q₈×C₆₇	—	2	3	no	—
62	C₂×Q₈×C₆₇	—	2	6	no	—
63	C₂×Q₈×C₆₇	C₂×D₄	1	6	yes	—
64	C₂×D₄×C₆₇	—	2	3	no	—
65	C₂×D₄×C₆₇	—	2	3	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
67	C₂²×C₄×C₆₇	—	2	3	no	—
68	C₂²×C₄×C₆₇	—	2	3	no	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	Q₁₆×C₆₇	—	4	12	no	—
73	SD₁₆×C₆₇	—	4	4	no	—
74	C₂×C₈×C₆₇	—	2	3	no	—
75	C₂×C₈×C₆₇	—	2	3	no	—
76	C₂×C₈×C₆₇	—	2	6	no	—
77	C₄⋊C₄×C₆₇	—	2	12	no	—
78	C₄⋊C₄×C₆₇	—	2	3	no	—
79	C₄⋊C₄×C₆₇	—	4	6	no	—
80	C₄⋊C₄×C₆₇	—	4	6	no	—
81	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
82	C₂²⋊C₄×C₆₇	—	4	6	no	—
83	C₂²⋊C₄×C₆₇	—	4	6	no	—
84	C₄²×C₆₇	—	2	12	no	—
85	C₄²×C₆₇	—	8	3	no	—
86	C₄²×C₆₇	C₂×D₄	1	6	yes	—
87	C₄²×C₆₇	#41	1	6	yes	—
88	C₂³×C₆₇	—	4	3	no	—
89	C₂³×C₆₇	—	4	1	no	—
90	C₂³×C₆₇	—	4	3	no	—
91	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₆₇	—	2	6	no	—
93	Q₈×C₆₇	—	4	3	no	—
94	Q₈×C₆₇	—	4	3	no	—
95	Q₈×C₆₇	—	4	6	no	—
96	Q₈×C₆₇	#127	1	4	yes	—
97	D₄×C₆₇	—	4	1	no	—
98	D₄×C₆₇	—	4	3	no	—
99	C₂×C₄×C₆₇	—	2	6	no	—
100	C₂×C₄×C₆₇	—	2	3	no	—
101	C₂×C₄×C₆₇	—	2	6	no	—
102	C₂×C₄×C₆₇	—	2	3	no	—
103	C₂×C₄×C₆₇	—	2	6	no	—
104	C₂×C₄×C₆₇	—	4	3	no	—
105	C₂×C₄×C₆₇	—	4	3	no	—
106	C₂×C₄×C₆₇	—	4	3	no	—
107	C₂×C₄×C₆₇	—	4	6	no	—
108	C₂×C₄×C₆₇	#121	1	6	yes	—
109	C₂×C₄×C₆₇	#121	1	6	yes	—
110	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
111	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
112	C₈×C₆₇	—	4	1	no	—
113	C₈×C₆₇	—	4	3	no	—
114	C₂²×C₆₇	—	8	1	no	—
115	C₂²×C₆₇	—	8	3	no	—
116	C₂²×C₆₇	—	8	3	no	—
117	C₂²×C₆₇	C₂×#121	1	3	yes	—
118	C₂²×C₆₇	#577	1	1	yes	characteristic
119	C₂²×C₆₇	#578	1	3	yes	—
120	C₄×C₆₇	—	2	4	no	—
121	C₄×C₆₇	—	2	6	no	—
122	C₄×C₆₇	—	2	6	no	—
123	C₄×C₆₇	—	4	6	no	—
124	C₄×C₆₇	—	8	3	no	—
125	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
126	C₄×C₆₇	#532	1	3	yes	—
127	#33169	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂×C₆₇	—	16	1	no	—
129	C₂×C₆₇	#2856	1	1	yes	characteristic
130	C₂×C₆₇	#3044	1	3	yes	—
131	C₂×C₆₇	#3046	1	3	yes	—
132	C₂×#464	C₂×C₆₇	1	3	yes	—
133	C₂×#462	C₂×C₆₇	1	3	yes	complemented
134	#2658	C₂×C₆₇	1	1	yes	characteristic
135	#2276	C₂×C₆₇	1	1	yes	characteristic, complemented
136	#1983	C₂×C₆₇	1	3	yes	—
137	#1981	C₂×C₆₇	1	3	yes	complemented
138	#1843	C₂×C₆₇	1	1	yes	characteristic, complemented
139	C₆₇	#33169	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×Q₁₆	C₂²×C₆₇	1	3	yes	—
141	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
142	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
143	C₂×#124	C₂²×C₆₇	1	3	yes	—
144	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
145	#464	—	2	12	no	—
146	#462	—	2	12	no	—
147	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
148	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	—
149	C₂×#103	C₂²×C₆₇	1	3	yes	—
150	C₂×#103	C₂²×C₆₇	1	6	yes	—
151	C₂×#102	C₂²×C₆₇	1	3	yes	—
152	#399	—	2	1	no	—
153	#399	—	2	3	no	—
154	C₄×C₄⋊C₄	—	2	3	no	—
155	C₄×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
156	#376	—	2	1	no	—
157	#376	C₂²×C₆₇	1	1	yes	characteristic
158	C₂²×Q₈	C₂³×C₆₇	1	3	yes	—
159	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
160	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
161	C₂×Q₁₆	—	2	12	no	—
162	C₂×Q₁₆	—	2	6	no	—
163	C₂×SD₁₆	—	2	6	no	—
164	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
166	C₄⋊Q₈	—	2	6	no	—
167	C₄⋊Q₈	—	2	6	no	—
168	#124	—	2	6	no	—
169	#124	—	2	6	no	—
170	C₄×Q₈	—	2	6	no	—
171	C₄×Q₈	D₄×C₆₇	1	12	yes	—
172	C₂×C₄⋊C₄	—	2	3	no	—
173	C₂×C₄⋊C₄	—	2	3	no	—
174	C₂×C₄⋊C₄	—	2	3	no	—
175	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
176	C₂×C₂²⋊C₄	—	2	3	no	—
177	C₂×C₂²⋊C₄	—	2	3	no	—
178	C₂×C₄²	—	2	3	no	—
179	C₂×C₄²	—	2	3	no	—
180	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
181	C₄⋊C₈	—	2	6	no	—
182	C₄⋊C₈	—	2	6	no	—
183	#103	—	2	12	no	—
184	#103	—	2	6	no	—
185	#102	—	2	6	no	—
186	#95	—	2	3	no	—
187	#95	D₄×C₆₇	1	1	yes	characteristic
188	#95	D₄×C₆₇	1	3	yes	—
189	C₂⁴	—	2	1	no	—
190	C₂×Q₈	—	2	3	no	—
191	C₂×Q₈	—	2	3	no	—
192	C₂×Q₈	—	2	6	no	—
193	C₂×Q₈	—	2	3	no	—
194	C₂×Q₈	—	2	6	no	—
195	C₂×Q₈	C₂×D₄×C₆₇	1	6	yes	—
196	C₂×D₄	—	2	3	no	—
197	C₂×D₄	—	2	3	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	3	no	—
200	C₂²×C₄	—	2	3	no	—
201	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
202	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
203	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
204	Q₁₆	—	4	12	no	—
205	SD₁₆	—	4	4	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	2	6	no	—
209	C₄⋊C₄	—	2	12	no	—
210	C₄⋊C₄	—	2	3	no	—
211	C₄⋊C₄	—	4	6	no	—
212	C₄⋊C₄	—	4	6	no	—
213	C₄⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
214	C₂²⋊C₄	—	4	6	no	—
215	C₂²⋊C₄	—	4	6	no	—
216	C₄²	—	2	12	no	—
217	C₄²	—	8	3	no	—
218	C₄²	C₂×D₄×C₆₇	1	6	yes	—
219	C₄²	#41×C₆₇	1	6	yes	—
220	C₂³	—	4	3	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	3	no	—
223	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
224	Q₈	—	2	6	no	—
225	Q₈	—	4	3	no	—
226	Q₈	—	4	3	no	—
227	Q₈	—	4	6	no	—
228	Q₈	#127×C₆₇	1	4	yes	—
229	D₄	—	4	1	no	—
230	D₄	—	4	3	no	—
231	C₂×C₄	—	2	6	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	3	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	4	3	no	—
237	C₂×C₄	—	4	3	no	—
238	C₂×C₄	—	4	3	no	—
239	C₂×C₄	—	4	6	no	—
240	C₂×C₄	#121×C₆₇	1	6	yes	—
241	C₂×C₄	#121×C₆₇	1	6	yes	—
242	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
243	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
244	C₈	—	4	1	no	—
245	C₈	—	4	3	no	—
246	C₂²	—	8	3	no	—
247	C₂²	—	8	3	no	—
248	C₂²	—	8	1	no	—
249	C₂²	C₂×#121×C₆₇	1	3	yes	—
250	C₂²	#577×C₆₇	1	1	yes	characteristic
251	C₂²	#578×C₆₇	1	3	yes	—
252	C₄	—	2	6	no	—
253	C₄	—	2	6	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	6	no	—
256	C₄	—	8	3	no	—
257	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
258	C₄	#532×C₆₇	1	3	yes	—
259	C₂	—	16	1	no	—
260	C₂	C₆₇×#2856	1	1	yes	characteristic
261	C₂	C₆₇×#3044	1	3	yes	—
262	C₂	C₆₇×#3046	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#462×C₆₇	C₂	semidirect
4. C₆₇×#2276	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1843	C₂	semidirect
127. #33169	C₆₇	direct (N × H)
133. C₂×#462	C₂×C₆₇	semidirect
135. #2276	C₂×C₆₇	semidirect
137. #1981	C₂×C₆₇	semidirect
138. #1843	C₂×C₆₇	semidirect
139. C₆₇	#33169	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×#464×C₆₇	2
C₂×#462×C₆₇	2
C₆₇×#2658	2
C₆₇×#2276	2
C₆₇×#1983	2
C₆₇×#1981	2
C₆₇×#1843	2
#33169	67

Subgroups — C67×#33169 #30712420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33169 #30712420