Subgroups — C₂×C₆₇×#3375 #30740562

Order 17152.

1906 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3375	C₂	1	16	yes	maximal, direct product factor
4	C₂×#489×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#474×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#464×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#440×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	#559×C₆₇	—	2	4	no	—
16	#559×C₆₇	C₂²	1	8	yes	—
17	Q₈²×C₆₇	—	2	4	no	—
18	Q₈²×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#124×C₆₇	—	2	1	no	—
25	C₂×#124×C₆₇	C₂²	1	2	yes	—
26	C₂×#123×C₆₇	—	2	2	no	—
27	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	—	2	1	no	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	#489×C₆₇	C₂²	1	16	yes	complemented
35	#486×C₆₇	C₂²	1	8	yes	—
36	#477×C₆₇	C₂²	1	8	yes	complemented
37	#474×C₆₇	C₂²	1	16	yes	complemented
38	#464×C₆₇	C₂²	1	16	yes	complemented
39	#460×C₆₇	C₂²	1	8	yes	—
40	#446×C₆₇	C₂²	1	8	yes	complemented
41	#440×C₆₇	C₂²	1	16	yes	complemented
42	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#107×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#102×C₆₇	C₂²	1	2	yes	—
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	2	yes	—
53	C₂×#97×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	—	2	2	no	—
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂×Q₁₆×C₆₇	—	2	4	no	—
65	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×SD₁₆×C₆₇	—	2	8	no	—
67	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	—	2	8	no	—
72	C₄⋊Q₈×C₆₇	—	2	16	no	—
73	C₄⋊Q₈×C₆₇	—	2	4	no	—
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	#127×C₆₇	—	2	2	no	—
78	#127×C₆₇	C₂³	1	4	yes	—
79	#124×C₆₇	—	2	8	no	—
80	#124×C₆₇	—	2	4	no	—
81	#124×C₆₇	C₂³	1	8	yes	—
82	#123×C₆₇	—	2	16	no	—
83	C₄×Q₈×C₆₇	—	2	8	no	—
84	C₄×Q₈×C₆₇	—	2	4	no	—
85	C₄×Q₈×C₆₇	—	2	2	no	—
86	C₄×Q₈×C₆₇	—	2	2	no	—
87	C₄×Q₈×C₆₇	D₄	1	4	yes	—
88	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
91	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	—	2	8	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	C₄×D₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	—	2	2	no	—
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	#107×C₆₇	C₂³	1	4	yes	—
114	#107×C₆₇	C₂³	1	8	yes	—
115	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
117	#103×C₆₇	C₂³	1	8	yes	—
118	#103×C₆₇	C₂³	1	4	yes	—
119	#103×C₆₇	C₂³	1	4	yes	—
120	#102×C₆₇	C₂³	1	8	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	8	yes	—
123	#97×C₆₇	C₂³	1	8	yes	—
124	C₄×C₈×C₆₇	C₂³	1	4	yes	—
125	C₂⁴×C₆₇	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	8	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	2	2	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	8	4	no	—
135	C₂×Q₈×C₆₇	—	8	2	no	—
136	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
137	C₂×Q₈×C₆₇	#41	1	4	yes	—
138	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
139	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	4	no	—
145	C₂×D₄×C₆₇	—	4	4	no	—
146	C₂×D₄×C₆₇	—	8	2	no	—
147	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	8	4	no	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₆₇	—	2	8	no	—
161	SD₁₆×C₆₇	—	2	16	no	—
162	C₂×C₈×C₆₇	—	2	4	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
166	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
167	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₆₇	—	2	8	no	—
182	C₂²⋊C₄×C₆₇	—	2	8	no	—
183	C₂²⋊C₄×C₆₇	—	2	8	no	—
184	C₄²×C₆₇	—	2	4	no	—
185	C₄²×C₆₇	—	2	8	no	—
186	C₄²×C₆₇	—	2	4	no	—
187	C₄²×C₆₇	C₂×D₄	1	4	yes	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₄²×C₆₇	C₂⁴	1	4	yes	—
190	C₄²×C₆₇	C₂⁴	1	2	yes	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	—	4	4	no	—
194	C₂³×C₆₇	—	4	4	no	—
195	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₆₇	—	2	4	no	—
197	Q₈×C₆₇	—	2	8	no	—
198	Q₈×C₆₇	—	2	2	no	—
199	Q₈×C₆₇	—	2	2	no	—
200	Q₈×C₆₇	—	2	2	no	—
201	Q₈×C₆₇	—	4	2	no	—
202	Q₈×C₆₇	—	8	8	no	—
203	Q₈×C₆₇	—	8	4	no	—
204	Q₈×C₆₇	C₂×#41	1	4	yes	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	2	4	no	—
207	D₄×C₆₇	—	4	4	no	—
208	D₄×C₆₇	—	4	8	no	—
209	D₄×C₆₇	—	8	4	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	4	no	—
223	C₂×C₄×C₆₇	—	8	8	no	—
224	C₂×C₄×C₆₇	#137	1	4	yes	—
225	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
231	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
232	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
233	C₈×C₆₇	—	2	4	no	—
234	C₈×C₆₇	—	2	2	no	—
235	C₈×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	8	8	no	—
240	C₂²×C₆₇	#548	1	2	yes	—
241	C₂²×C₆₇	C₂×#137	1	2	yes	—
242	C₂²×C₆₇	#576	1	2	yes	—
243	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	2	4	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	2	no	—
249	C₄×C₆₇	—	4	2	no	—
250	C₄×C₆₇	—	4	2	no	—
251	C₄×C₆₇	—	4	4	no	—
252	C₄×C₆₇	#548	1	2	yes	—
253	C₄×C₆₇	C₂×#137	1	4	yes	—
254	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
255	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
256	C₂×#3375	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	8	4	no	—
258	C₂×C₆₇	#3375	1	4	yes	direct product factor
259	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
260	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
261	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
262	C₂×#559	C₂×C₆₇	1	1	yes	characteristic
263	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic, complemented
264	#3375	C₂×C₆₇	1	16	yes	direct product factor
265	C₂×#489	C₂×C₆₇	1	2	yes	complemented
266	C₂×#486	C₂×C₆₇	1	1	yes	characteristic
267	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
268	C₂×#474	C₂×C₆₇	1	2	yes	complemented
269	C₂×#464	C₂×C₆₇	1	2	yes	complemented
270	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
271	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#440	C₂×C₆₇	1	2	yes	complemented
273	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₆₇	C₂×#3375	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
277	#559	—	2	4	no	—
278	#559	C₂²×C₆₇	1	8	yes	—
279	Q₈²	—	2	4	no	—
280	Q₈²	C₂²×C₆₇	1	8	yes	complemented
281	C₂×C₄⋊Q₈	—	2	1	no	—
282	C₂×C₄⋊Q₈	—	2	2	no	—
283	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
285	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#124	—	2	1	no	—
287	C₂×#124	C₂²×C₆₇	1	2	yes	—
288	C₂×#123	—	2	2	no	—
289	C₂×C₄×Q₈	—	2	1	no	—
290	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×C₄×D₄	—	2	1	no	—
295	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
296	#489	C₂²×C₆₇	1	16	yes	complemented
297	#486	C₂²×C₆₇	1	8	yes	—
298	#477	C₂²×C₆₇	1	8	yes	complemented
299	#474	C₂²×C₆₇	1	16	yes	complemented
300	#464	C₂²×C₆₇	1	16	yes	complemented
301	#460	C₂²×C₆₇	1	8	yes	—
302	#446	C₂²×C₆₇	1	8	yes	complemented
303	#440	C₂²×C₆₇	1	16	yes	complemented
304	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
305	C₂×#107	C₂²×C₆₇	1	2	yes	—
306	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
309	C₂×#103	C₂²×C₆₇	1	2	yes	—
310	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×#102	C₂²×C₆₇	1	2	yes	—
313	C₂×#102	C₂²×C₆₇	1	2	yes	—
314	C₂×#102	C₂²×C₆₇	1	2	yes	—
315	C₂×#97	C₂²×C₆₇	1	2	yes	—
316	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₂²×Q₈	—	2	2	no	—
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
320	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
325	C₂³×C₄	—	2	2	no	—
326	C₂×Q₁₆	—	2	4	no	—
327	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
328	C₂×SD₁₆	—	2	8	no	—
329	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
333	C₄⋊Q₈	—	2	8	no	—
334	C₄⋊Q₈	—	2	16	no	—
335	C₄⋊Q₈	—	2	4	no	—
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
338	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
339	#127	—	2	2	no	—
340	#127	C₂³×C₆₇	1	4	yes	—
341	#124	—	2	8	no	—
342	#124	—	2	4	no	—
343	#124	C₂³×C₆₇	1	8	yes	—
344	#123	—	2	16	no	—
345	C₄×Q₈	—	2	8	no	—
346	C₄×Q₈	—	2	4	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	D₄×C₆₇	1	4	yes	—
350	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
351	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
352	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
353	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	C₂³×C₆₇	1	8	yes	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
367	C₂×C₂²⋊C₄	—	2	2	no	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	—	2	2	no	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
374	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
375	#107	C₂³×C₆₇	1	4	yes	—
376	#107	C₂³×C₆₇	1	8	yes	—
377	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
378	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
379	#103	C₂³×C₆₇	1	4	yes	—
380	#103	C₂³×C₆₇	1	4	yes	—
381	#103	C₂³×C₆₇	1	8	yes	—
382	#102	C₂³×C₆₇	1	8	yes	—
383	#102	C₂³×C₆₇	1	8	yes	—
384	#102	C₂³×C₆₇	1	8	yes	—
385	#97	C₂³×C₆₇	1	8	yes	—
386	C₄×C₈	C₂³×C₆₇	1	4	yes	—
387	C₂⁴	—	2	2	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	2	8	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	8	4	no	—
397	C₂×Q₈	—	8	2	no	—
398	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
399	C₂×Q₈	#41×C₆₇	1	4	yes	—
400	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
401	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	8	2	no	—
409	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	8	4	no	—
417	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
418	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
422	Q₁₆	—	2	8	no	—
423	SD₁₆	—	2	16	no	—
424	C₂×C₈	—	2	4	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
428	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
429	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
439	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
440	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
441	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
442	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	8	no	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₄²	—	2	4	no	—
447	C₄²	—	2	8	no	—
448	C₄²	—	2	4	no	—
449	C₄²	C₂×D₄×C₆₇	1	4	yes	—
450	C₄²	C₂×D₄×C₆₇	1	2	yes	—
451	C₄²	C₂⁴×C₆₇	1	2	yes	—
452	C₄²	C₂⁴×C₆₇	1	4	yes	—
453	C₂³	—	2	4	no	—
454	C₂³	—	4	4	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	4	no	—
457	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
458	Q₈	—	2	4	no	—
459	Q₈	—	2	8	no	—
460	Q₈	—	2	2	no	—
461	Q₈	—	2	2	no	—
462	Q₈	—	2	2	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	8	8	no	—
465	Q₈	—	8	4	no	—
466	Q₈	C₂×#41×C₆₇	1	4	yes	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	8	no	—
471	D₄	—	8	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	8	8	no	—
486	C₂×C₄	#137×C₆₇	1	4	yes	—
487	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
488	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
489	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
492	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
493	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
494	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
495	C₈	—	2	4	no	—
496	C₈	—	2	2	no	—
497	C₈	—	2	2	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	8	8	no	—
502	C₂²	#548×C₆₇	1	2	yes	—
503	C₂²	C₂×#137×C₆₇	1	2	yes	—
504	C₂²	#576×C₆₇	1	2	yes	—
505	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
506	C₄	—	2	4	no	—
507	C₄	—	2	4	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	4	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	2	no	—
514	C₄	#548×C₆₇	1	2	yes	—
515	C₄	C₂×#137×C₆₇	1	4	yes	—
516	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
517	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
518	C₂	—	8	4	no	—
519	C₂	C₆₇×#3375	1	4	yes	direct product factor
520	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
521	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
522	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈²×C₆₇	C₂	semidirect
3. C₆₇×#3375	C₂	direct (N × H)
4. C₂×#489×C₆₇	C₂	semidirect
6. C₂×#477×C₆₇	C₂	semidirect
7. C₂×#474×C₆₇	C₂	semidirect
8. C₂×#464×C₆₇	C₂	semidirect
10. C₂×#446×C₆₇	C₂	semidirect
11. C₂×#440×C₆₇	C₂	semidirect
12. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
18. Q₈²×C₆₇	C₂²	semidirect
34. #489×C₆₇	C₂²	semidirect
36. #477×C₆₇	C₂²	semidirect
37. #474×C₆₇	C₂²	semidirect
38. #464×C₆₇	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
41. #440×C₆₇	C₂²	semidirect
42. C₄×Q₁₆×C₆₇	C₂²	semidirect
256. C₂×#3375	C₆₇	direct (N × H)
258. C₂×C₆₇	#3375	direct (N × H)
263. C₂×Q₈²	C₂×C₆₇	semidirect
264. #3375	C₂×C₆₇	direct (N × H)
265. C₂×#489	C₂×C₆₇	semidirect
267. C₂×#477	C₂×C₆₇	semidirect
268. C₂×#474	C₂×C₆₇	semidirect
269. C₂×#464	C₂×C₆₇	semidirect
271. C₂×#446	C₂×C₆₇	semidirect
272. C₂×#440	C₂×C₆₇	semidirect
273. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
274. C₆₇	C₂×#3375	direct (N × H)
280. Q₈²	C₂²×C₆₇	semidirect
296. #489	C₂²×C₆₇	semidirect
298. #477	C₂²×C₆₇	semidirect
299. #474	C₂²×C₆₇	semidirect
300. #464	C₂²×C₆₇	semidirect
302. #446	C₂²×C₆₇	semidirect
303. #440	C₂²×C₆₇	semidirect
304. C₄×Q₁₆	C₂²×C₆₇	semidirect
519. C₂	C₆₇×#3375	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×#559×C₆₇	2
C₂×Q₈²×C₆₇	2
C₆₇×#3375	2
C₂×#489×C₆₇	2
C₂×#486×C₆₇	2
C₂×#477×C₆₇	2
C₂×#474×C₆₇	2
C₂×#464×C₆₇	2
C₂×#460×C₆₇	2
C₂×#446×C₆₇	2
C₂×#440×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×#3375	67

Subgroups — C2×C67×#3375 #30740562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3375 #30740562