Subgroups — D₄×#103×C₆₇ #30701666

Order 17152.

1530 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#103×C₆₇	C₂	1	2	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1892	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1872	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1787	C₂	1	2	yes	maximal, complemented
11	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	D₄×Q₈×C₆₇	—	2	4	no	—
15	D₄×Q₈×C₆₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#103×C₆₇	—	2	8	no	—
28	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#103×C₆₇	C₂²	1	2	yes	—
33	C₂×#103×C₆₇	C₂²	1	2	yes	—
34	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#389×C₆₇	—	2	1	no	—
38	#385×C₆₇	—	2	2	no	—
39	#384×C₆₇	C₂²	1	1	yes	characteristic
40	#380×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	1	yes	characteristic
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	#340×C₆₇	C₂²	1	2	yes	—
45	C₂²×Q₈×C₆₇	—	2	2	no	—
46	C₂²×Q₈×C₆₇	—	2	4	no	—
47	C₂²×Q₈×C₆₇	—	2	4	no	—
48	C₂²×Q₈×C₆₇	—	2	4	no	—
49	C₂²×Q₈×C₆₇	—	4	2	no	—
50	C₂²×Q₈×C₆₇	—	4	2	no	—
51	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	—	2	2	no	—
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	—	2	4	no	—
59	C₂²×C₈×C₆₇	—	2	4	no	—
60	C₂²×C₈×C₆₇	—	4	2	no	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₇	—	2	8	no	—
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
67	C₂²⋊Q₈×C₆₇	—	2	4	no	—
68	C₂²⋊Q₈×C₆₇	—	2	16	no	—
69	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₄×D₄×C₆₇	—	2	8	no	—
73	C₄×D₄×C₆₇	—	4	4	no	—
74	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
75	C₄×D₄×C₆₇	D₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
93	C₂×C₄²×C₆₇	—	2	1	no	—
94	C₂×C₄²×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₇	—	2	2	no	—
97	#103×C₆₇	—	2	4	no	—
98	#103×C₆₇	—	2	2	no	—
99	#103×C₆₇	—	2	4	no	—
100	#103×C₆₇	—	2	4	no	—
101	#103×C₆₇	—	4	2	no	—
102	#103×C₆₇	—	4	2	no	—
103	#103×C₆₇	D₄	1	4	yes	direct product factor
104	C₂²⋊C₈×C₆₇	—	2	4	no	—
105	C₄×C₈×C₆₇	—	2	2	no	—
106	#95×C₆₇	—	2	1	no	—
107	#95×C₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₆₇	—	2	8	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	16	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	4	4	no	—
120	C₂×Q₈×C₆₇	—	4	4	no	—
121	C₂×Q₈×C₆₇	—	4	8	no	—
122	C₂×Q₈×C₆₇	—	4	8	no	—
123	C₂×Q₈×C₆₇	—	4	8	no	—
124	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₆₇	#41	1	2	yes	—
128	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
129	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
130	C₂×D₄×C₆₇	Q₁₆	1	4	yes	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	8	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	4	4	no	—
143	C₂²×C₄×C₆₇	—	4	4	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	—	4	4	no	—
146	C₂²×C₄×C₆₇	—	8	2	no	—
147	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	4	8	no	—
162	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	8	no	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	4	4	no	—
174	C₄⋊C₄×C₆₇	—	4	2	no	—
175	C₄⋊C₄×C₆₇	—	8	1	no	—
176	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	—	4	4	no	—
181	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
182	C₄²×C₆₇	—	2	4	no	—
183	C₄²×C₆₇	—	4	2	no	—
184	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
185	C₄²×C₆₇	C₂×D₄	1	2	yes	—
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
191	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
192	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
193	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	2	no	—
195	Q₈×C₆₇	—	2	2	no	—
196	Q₈×C₆₇	—	2	2	no	—
197	Q₈×C₆₇	—	2	4	no	—
198	Q₈×C₆₇	—	2	4	no	—
199	Q₈×C₆₇	—	4	16	no	—
200	Q₈×C₆₇	—	4	2	no	—
201	Q₈×C₆₇	—	4	2	no	—
202	Q₈×C₆₇	C₄×D₄	1	4	yes	complemented
203	D₄×C₆₇	#103	1	16	yes	direct product factor
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	4	16	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	1	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	1	no	—
225	C₂×C₄×C₆₇	—	4	2	no	—
226	C₂×C₄×C₆₇	—	4	1	no	—
227	C₂×C₄×C₆₇	—	4	1	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	8	8	no	—
232	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
234	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
236	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
237	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
242	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
243	C₈×C₆₇	—	2	2	no	—
244	C₈×C₆₇	—	4	2	no	—
245	C₈×C₆₇	—	4	1	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	2	4	no	—
248	C₂²×C₆₇	—	2	4	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	C₂×#103	1	8	yes	complemented
253	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²×C₆₇	D₄²	1	1	yes	characteristic
255	C₂²×C₆₇	#546	1	1	yes	characteristic
256	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
257	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
258	C₂²×C₆₇	#577	1	1	yes	characteristic
259	C₂²×C₆₇	#578	1	1	yes	characteristic
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	2	4	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	4	4	no	—
264	C₄×C₆₇	—	4	2	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	8	2	no	—
267	C₄×C₆₇	—	8	1	no	—
268	C₄×C₆₇	C₂×#103	1	4	yes	complemented
269	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
270	C₄×C₆₇	#518	1	1	yes	characteristic
271	C₄×C₆₇	D₄²	1	1	yes	characteristic
272	C₄×C₆₇	#546	1	1	yes	characteristic
273	D₄×#103	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₆₇	—	2	8	no	—
275	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
277	C₂×C₆₇	#2898	1	1	yes	characteristic
278	C₂×C₆₇	D₄×SD₁₆	1	1	yes	characteristic
279	C₂×C₆₇	D₄×Q₁₆	1	1	yes	characteristic
280	C₂×C₆₇	#3298	1	1	yes	characteristic
281	C₂×C₆₇	#3299	1	1	yes	characteristic
282	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
283	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
284	C₂²×#103	C₂×C₆₇	1	2	yes	complemented
285	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
286	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1892	C₂×C₆₇	1	2	yes	complemented
289	#1872	C₂×C₆₇	1	2	yes	complemented
290	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#1787	C₂×C₆₇	1	2	yes	complemented
292	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
293	C₆₇	D₄×#103	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
295	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
296	D₄×Q₈	—	2	4	no	—
297	D₄×Q₈	C₄×C₆₇	1	8	yes	complemented
298	C₂×C₄⋊Q₈	—	2	1	no	—
299	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₂²⋊Q₈	—	2	2	no	—
301	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×D₄	—	2	1	no	—
304	C₂×C₄×D₄	—	2	1	no	—
305	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
306	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
307	C₈×D₄	—	2	4	no	—
308	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#103	—	2	8	no	—
310	C₂×#103	C₂²×C₆₇	1	2	yes	—
311	C₂×#103	C₂²×C₆₇	1	2	yes	—
312	C₂×#103	C₂²×C₆₇	1	2	yes	—
313	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#103	C₂²×C₆₇	1	2	yes	—
315	C₂×#103	C₂²×C₆₇	1	2	yes	—
316	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
319	#389	—	2	1	no	—
320	#385	—	2	2	no	—
321	#384	C₂²×C₆₇	1	1	yes	characteristic
322	#380	C₂²×C₆₇	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
324	#340	C₂²×C₆₇	1	2	yes	—
325	#340	C₂²×C₆₇	1	1	yes	characteristic
326	#340	C₂²×C₆₇	1	1	yes	characteristic
327	C₂²×Q₈	—	2	2	no	—
328	C₂²×Q₈	—	2	4	no	—
329	C₂²×Q₈	—	2	4	no	—
330	C₂²×Q₈	—	2	4	no	—
331	C₂²×Q₈	—	4	2	no	—
332	C₂²×Q₈	—	4	2	no	—
333	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
334	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
335	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
337	C₂³×C₄	—	2	2	no	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
340	C₂²×C₈	—	2	4	no	—
341	C₂²×C₈	—	2	4	no	—
342	C₂²×C₈	—	4	2	no	—
343	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
344	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₄⋊Q₈	—	2	8	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	16	no	—
351	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
352	C₄×Q₈	—	2	2	no	—
353	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	—	4	4	no	—
356	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
357	C₄×D₄	D₄×C₆₇	1	4	yes	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	—	2	4	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	4	2	no	—
367	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
378	C₄⋊C₈	—	2	2	no	—
379	#103	—	2	4	no	—
380	#103	—	2	2	no	—
381	#103	—	2	4	no	—
382	#103	—	2	4	no	—
383	#103	—	4	2	no	—
384	#103	—	4	2	no	—
385	#103	D₄×C₆₇	1	4	yes	direct product factor
386	C₂²⋊C₈	—	2	4	no	—
387	C₄×C₈	—	2	2	no	—
388	#95	—	2	1	no	—
389	#95	—	2	2	no	—
390	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
391	C₂×Q₈	—	2	8	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	16	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	4	4	no	—
402	C₂×Q₈	—	4	4	no	—
403	C₂×Q₈	—	4	8	no	—
404	C₂×Q₈	—	4	8	no	—
405	C₂×Q₈	—	4	8	no	—
406	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
407	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
409	C₂×Q₈	#41×C₆₇	1	2	yes	—
410	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
411	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
412	C₂×D₄	Q₁₆×C₆₇	1	4	yes	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	4	4	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	—	4	4	no	—
428	C₂²×C₄	—	8	2	no	—
429	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	4	8	no	—
444	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	4	4	no	—
456	C₄⋊C₄	—	4	2	no	—
457	C₄⋊C₄	—	8	1	no	—
458	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
459	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
460	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	4	4	no	—
463	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
464	C₄²	—	2	4	no	—
465	C₄²	—	4	2	no	—
466	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
467	C₄²	C₂×D₄×C₆₇	1	2	yes	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
473	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
474	C₂³	C₂×Q₁₆×C₆₇	1	4	yes	—
475	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
476	Q₈	—	2	2	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	2	2	no	—
479	Q₈	—	2	4	no	—
480	Q₈	—	2	4	no	—
481	Q₈	—	4	16	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	C₄×D₄×C₆₇	1	4	yes	complemented
485	D₄	#103×C₆₇	1	16	yes	direct product factor
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	4	16	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	8	8	no	—
514	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
515	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
516	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
517	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
518	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
523	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
524	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
525	C₈	—	2	2	no	—
526	C₈	—	4	2	no	—
527	C₈	—	4	1	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	C₂×#103×C₆₇	1	8	yes	complemented
535	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
536	C₂²	D₄²×C₆₇	1	1	yes	characteristic
537	C₂²	#546×C₆₇	1	1	yes	characteristic
538	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
539	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
540	C₂²	#577×C₆₇	1	1	yes	characteristic
541	C₂²	#578×C₆₇	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	2	4	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	2	no	—
548	C₄	—	8	2	no	—
549	C₄	—	8	1	no	—
550	C₄	C₂×#103×C₆₇	1	4	yes	complemented
551	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
552	C₄	#518×C₆₇	1	1	yes	characteristic
553	C₄	D₄²×C₆₇	1	1	yes	characteristic
554	C₄	#546×C₆₇	1	1	yes	characteristic
555	C₂	—	2	8	no	—
556	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
557	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic
558	C₂	C₆₇×#2898	1	1	yes	characteristic
559	C₂	D₄×SD₁₆×C₆₇	1	1	yes	characteristic
560	C₂	D₄×Q₁₆×C₆₇	1	1	yes	characteristic
561	C₂	C₆₇×#3298	1	1	yes	characteristic
562	C₂	C₆₇×#3299	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂²×#103×C₆₇	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
6. C₆₇×#1956	C₂	semidirect
7. C₆₇×#1892	C₂	semidirect
8. C₆₇×#1872	C₂	semidirect
9. C₆₇×#1803	C₂	semidirect
10. C₆₇×#1787	C₂	semidirect
11. C₄×#103×C₆₇	C₂	semidirect
15. D₄×Q₈×C₆₇	C₄	semidirect
51. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
66. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
69. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
71. C₄×Q₈×C₆₇	C₂×C₄	semidirect
103. #103×C₆₇	D₄	direct (N × H)
202. Q₈×C₆₇	C₄×D₄	semidirect
203. D₄×C₆₇	#103	direct (N × H)
252. C₂²×C₆₇	C₂×#103	semidirect
268. C₄×C₆₇	C₂×#103	semidirect
273. D₄×#103	C₆₇	direct (N × H)
284. C₂²×#103	C₂×C₆₇	semidirect
286. #1981	C₂×C₆₇	semidirect
287. #1956	C₂×C₆₇	semidirect
288. #1892	C₂×C₆₇	semidirect
289. #1872	C₂×C₆₇	semidirect
290. #1803	C₂×C₆₇	semidirect
291. #1787	C₂×C₆₇	semidirect
292. C₄×#103	C₂×C₆₇	semidirect
293. C₆₇	D₄×#103	direct (N × H)
297. D₄×Q₈	C₄×C₆₇	semidirect
333. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
348. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
351. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
353. C₄×Q₈	C₂×C₄×C₆₇	semidirect
385. #103	D₄×C₆₇	direct (N × H)
484. Q₈	C₄×D₄×C₆₇	semidirect
485. D₄	#103×C₆₇	direct (N × H)
534. C₂²	C₂×#103×C₆₇	semidirect
550. C₄	C₂×#103×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#103×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1981	2
C₆₇×#1956	2
C₆₇×#1892	2
C₆₇×#1872	2
C₆₇×#1803	2
C₆₇×#1787	2
C₄×#103×C₆₇	2
D₄×#103	67

Subgroups — D4×#103×C67 #30701666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#103×C₆₇ #30701666