Subgroups — C₆₇×#21115 #30700366

Order 17152.

934 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1926	C₂	1	2	yes	maximal
5	C₆₇×#1919	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1880	C₂	1	2	yes	maximal
7	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	—	2	1	no	—
17	C₄×SD₁₆×C₆₇	C₄	1	8	yes	complemented
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₆₇	C₂²	1	2	yes	—
21	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#390×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	C₂²	1	2	yes	—
28	#388×C₆₇	—	2	2	no	—
29	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
35	C₄³×C₆₇	—	2	1	no	—
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂×SD₁₆×C₆₇	—	2	4	no	—
46	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	—	2	2	no	—
53	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
56	#117×C₆₇	—	4	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	2	yes	—
74	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#106×C₆₇	—	2	4	no	—
78	#106×C₆₇	C₂×C₄	1	4	yes	—
79	#103×C₆₇	—	2	4	no	—
80	#103×C₆₇	C₂×C₄	1	4	yes	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	C₂×C₄	1	4	yes	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
85	#95×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	1	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₆₇	#41	1	4	yes	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
111	SD₁₆×C₆₇	—	2	8	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	#41	1	4	yes	—
130	C₂²⋊C₄×C₆₇	—	2	4	no	—
131	C₂²⋊C₄×C₆₇	—	4	4	no	—
132	C₂²⋊C₄×C₆₇	—	4	4	no	—
133	C₄²×C₆₇	—	2	8	no	—
134	C₄²×C₆₇	—	2	4	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	4	2	no	—
139	C₄²×C₆₇	—	4	4	no	—
140	C₄²×C₆₇	—	4	2	no	—
141	C₄²×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
143	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
144	C₄²×C₆₇	#41	1	2	yes	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	2	no	—
152	Q₈×C₆₇	—	2	4	no	—
153	Q₈×C₆₇	#117	1	4	yes	complemented
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
180	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
181	C₂×C₄×C₆₇	#135	1	2	yes	—
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	—	2	2	no	—
188	C₈×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	2	no	—
194	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×C₆₇	#525	1	2	yes	—
196	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
198	C₂²×C₆₇	#575	1	2	yes	—
199	C₄×C₆₇	—	2	4	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	—	4	4	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	C₄×SD₁₆	1	4	yes	—
208	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
209	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
210	C₄×C₆₇	#529	1	2	yes	—
211	#21115	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	8	2	no	—
213	C₂×C₆₇	#2290	1	1	yes	characteristic
214	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
215	C₂×C₆₇	#2943	1	1	yes	characteristic
216	C₂×C₆₇	#3102	1	2	yes	—
217	C₂×C₆₇	#3103	1	2	yes	—
218	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
219	#2293	C₂×C₆₇	1	1	yes	characteristic
220	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1926	C₂×C₆₇	1	2	yes	—
222	#1919	C₂×C₆₇	1	2	yes	complemented
223	#1880	C₂×C₆₇	1	2	yes	—
224	#1865	C₂×C₆₇	1	2	yes	complemented
225	C₄×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
228	#1744	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#21115	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#117	—	2	1	no	—
235	C₄×SD₁₆	C₄×C₆₇	1	8	yes	complemented
236	C₂×#106	C₂²×C₆₇	1	2	yes	—
237	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#103	C₂²×C₆₇	1	2	yes	—
239	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#102	C₂²×C₆₇	1	2	yes	—
241	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
244	#390	C₂²×C₆₇	1	2	yes	—
245	#389	C₂²×C₆₇	1	2	yes	—
246	#388	—	2	2	no	—
247	C₄×C₄⋊C₄	—	2	1	no	—
248	C₄×C₄⋊C₄	—	2	2	no	—
249	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
250	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₄×C₂²⋊C₄	—	2	1	no	—
253	C₄³	—	2	1	no	—
254	C₄³	C₂²×C₆₇	1	1	yes	characteristic
255	#340	C₂²×C₆₇	1	2	yes	—
256	#340	C₂²×C₆₇	1	2	yes	—
257	#340	C₂²×C₆₇	1	2	yes	—
258	#336	C₂²×C₆₇	1	2	yes	—
259	#336	C₂²×C₆₇	1	2	yes	—
260	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
262	C₂³×C₄	—	2	1	no	—
263	C₂×SD₁₆	—	2	4	no	—
264	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
268	C₄×Q₈	—	2	4	no	—
269	C₄×Q₈	—	2	4	no	—
270	C₄×Q₈	—	2	2	no	—
271	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
274	#117	—	4	4	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	D₄×C₆₇	1	2	yes	—
292	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	#106	—	2	4	no	—
296	#106	C₂×C₄×C₆₇	1	4	yes	—
297	#103	—	2	4	no	—
298	#103	C₂×C₄×C₆₇	1	4	yes	—
299	#102	—	2	4	no	—
300	#102	C₂×C₄×C₆₇	1	4	yes	—
301	C₄×C₈	—	2	2	no	—
302	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	1	no	—
306	#95	—	2	1	no	—
307	C₂⁴	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
311	C₂×Q₈	#41×C₆₇	1	4	yes	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
329	SD₁₆	—	2	8	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
346	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
347	C₄⋊C₄	#41×C₆₇	1	4	yes	—
348	C₂²⋊C₄	—	2	4	no	—
349	C₂²⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₄²	—	2	8	no	—
352	C₄²	—	2	4	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	4	2	no	—
357	C₄²	—	4	4	no	—
358	C₄²	—	4	2	no	—
359	C₄²	—	4	2	no	—
360	C₄²	SD₁₆×C₆₇	1	4	yes	complemented
361	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
362	C₄²	#41×C₆₇	1	2	yes	—
363	C₂³	—	2	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	2	no	—
368	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	Q₈	—	2	2	no	—
370	Q₈	—	2	4	no	—
371	Q₈	#117×C₆₇	1	4	yes	complemented
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	8	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
397	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
398	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
399	C₂×C₄	#135×C₆₇	1	2	yes	—
400	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
404	C₈	—	2	2	no	—
405	C₈	—	2	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
413	C₂²	#525×C₆₇	1	2	yes	—
414	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
415	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
416	C₂²	#575×C₆₇	1	2	yes	—
417	C₄	—	2	4	no	—
418	C₄	—	2	4	no	—
419	C₄	—	2	2	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	2	no	—
424	C₄	—	8	2	no	—
425	C₄	C₄×SD₁₆×C₆₇	1	4	yes	—
426	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
427	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
428	C₄	#529×C₆₇	1	2	yes	—
429	C₂	—	8	2	no	—
430	C₂	C₆₇×#2290	1	1	yes	characteristic
431	C₂	C₂×C₄×SD₁₆×C₆₇	1	1	yes	characteristic
432	C₂	C₆₇×#2943	1	1	yes	characteristic
433	C₂	C₆₇×#3102	1	2	yes	—
434	C₂	C₆₇×#3103	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄²×Q₈×C₆₇	C₂	semidirect
5. C₆₇×#1919	C₂	semidirect
7. C₆₇×#1865	C₂	semidirect
8. C₄×#106×C₆₇	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
11. C₆₇×#1744	C₂	semidirect
17. C₄×SD₁₆×C₆₇	C₄	semidirect
142. C₄²×C₆₇	SD₁₆	semidirect
153. Q₈×C₆₇	#117	semidirect
211. #21115	C₆₇	direct (N × H)
220. C₄²×Q₈	C₂×C₆₇	semidirect
222. #1919	C₂×C₆₇	semidirect
224. #1865	C₂×C₆₇	semidirect
225. C₄×#106	C₂×C₆₇	semidirect
226. C₄×#103	C₂×C₆₇	semidirect
228. #1744	C₂×C₆₇	semidirect
229. C₆₇	#21115	direct (N × H)
235. C₄×SD₁₆	C₄×C₆₇	semidirect
360. C₄²	SD₁₆×C₆₇	semidirect
371. Q₈	#117×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#2293	2
C₄²×Q₈×C₆₇	2
C₆₇×#1926	2
C₆₇×#1919	2
C₆₇×#1880	2
C₆₇×#1865	2
C₄×#106×C₆₇	2
C₄×#103×C₆₇	2
C₄×#102×C₆₇	2
C₆₇×#1744	2
#21115	67

Subgroups — C67×#21115 #30700366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21115 #30700366