Subgroups — C₆₇×#21089 #30700340

Order 17152.

1234 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1936	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1934	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1735	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#418×C₆₇	—	2	4	no	—
18	C₂×#102×C₆₇	C₄	1	8	yes	complemented
19	C₂×#102×C₆₇	C₂²	1	4	yes	—
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	#405×C₆₇	—	2	4	no	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
25	#390×C₆₇	—	2	2	no	—
26	#390×C₆₇	C₂²	1	4	yes	—
27	#389×C₆₇	—	2	2	no	—
28	#389×C₆₇	C₂²	1	4	yes	complemented
29	#380×C₆₇	—	2	2	no	—
30	#379×C₆₇	—	2	2	no	—
31	#340×C₆₇	C₂²	1	2	yes	complemented
32	#340×C₆₇	C₂²	1	2	yes	complemented
33	#340×C₆₇	C₂²	1	4	yes	complemented
34	C₂⁵×C₆₇	—	2	1	no	—
35	C₂²×D₄×C₆₇	—	2	2	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	2	no	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
72	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	#102×C₆₇	—	2	8	no	—
77	#102×C₆₇	C₂×C₄	1	8	yes	—
78	#102×C₆₇	C₂×C₄	1	8	yes	—
79	#97×C₆₇	—	2	4	no	—
80	C₄×C₈×C₆₇	—	2	4	no	—
81	#95×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	4	2	no	—
85	C₂⁴×C₆₇	—	4	2	no	—
86	C₂⁴×C₆₇	—	8	2	no	—
87	C₂⁴×C₆₇	—	8	2	no	—
88	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×D₄×C₆₇	—	2	8	no	—
90	C₂×D₄×C₆₇	—	2	8	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	4	4	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	8	2	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	#41	1	2	yes	—
112	C₂²×C₄×C₆₇	#41	1	2	yes	—
113	C₂²×C₄×C₆₇	#41	1	2	yes	—
114	C₂²×C₄×C₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	2	8	no	—
117	C₂×C₈×C₆₇	—	2	8	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₆₇	—	2	4	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	8	4	no	—
135	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	8	4	no	—
140	C₂²⋊C₄×C₆₇	—	8	4	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	4	2	no	—
144	C₄²×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	4	no	—
151	C₂³×C₆₇	—	4	4	no	—
152	C₂³×C₆₇	—	4	8	no	—
153	C₂³×C₆₇	—	8	4	no	—
154	C₂³×C₆₇	—	8	4	no	—
155	C₂³×C₆₇	—	8	8	no	—
156	C₂³×C₆₇	—	8	8	no	—
157	C₂³×C₆₇	C₄×D₄	1	2	yes	—
158	C₂³×C₆₇	#121	1	2	yes	—
159	C₂³×C₆₇	#124	1	2	yes	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	D₄×C₆₇	—	2	4	no	—
162	D₄×C₆₇	—	2	4	no	—
163	D₄×C₆₇	—	4	2	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	8	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	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	—	8	4	no	—
178	C₂×C₄×C₆₇	#117	1	4	yes	—
179	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₆₇	#121	1	2	yes	—
182	C₂×C₄×C₆₇	#123	1	4	yes	—
183	C₂×C₄×C₆₇	#124	1	2	yes	—
184	C₂×C₄×C₆₇	#126	1	4	yes	—
185	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	2	4	no	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	2	2	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	4	no	—
198	C₂²×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	8	16	no	—
202	C₂²×C₆₇	—	8	4	no	—
203	C₂²×C₆₇	—	8	4	no	—
204	C₂²×C₆₇	#388	1	4	yes	—
205	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
208	C₂²×C₆₇	#576	1	2	yes	—
209	C₂²×C₆₇	#577	1	2	yes	—
210	C₄×C₆₇	—	4	4	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	#388	1	4	yes	—
215	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
216	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
218	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
219	#21089	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	2	4	no	—
221	C₂×C₆₇	—	8	4	no	—
222	C₂×C₆₇	C₂×#388	1	1	yes	characteristic
223	C₂×C₆₇	#2948	1	1	yes	characteristic
224	C₂×C₆₇	#2949	1	1	yes	characteristic
225	C₂×C₆₇	#3062	1	2	yes	—
226	C₂×C₆₇	#3136	1	2	yes	—
227	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
228	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic
229	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#2316	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#2309	C₂×C₆₇	1	1	yes	characteristic
232	#1936	C₂×C₆₇	1	4	yes	complemented
233	#1934	C₂×C₆₇	1	4	yes	complemented
234	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1735	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₆₇	#21089	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
244	#418	—	2	4	no	—
245	C₂×#102	C₄×C₆₇	1	8	yes	complemented
246	C₂×#102	C₂²×C₆₇	1	4	yes	—
247	C₂×#102	C₂²×C₆₇	1	2	yes	—
248	C₂×#102	C₂²×C₆₇	1	2	yes	—
249	#405	—	2	4	no	—
250	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
251	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
252	#390	—	2	2	no	—
253	#390	C₂²×C₆₇	1	4	yes	—
254	#389	—	2	2	no	—
255	#389	C₂²×C₆₇	1	4	yes	complemented
256	#380	—	2	2	no	—
257	#379	—	2	2	no	—
258	#340	C₂²×C₆₇	1	2	yes	complemented
259	#340	C₂²×C₆₇	1	4	yes	complemented
260	#340	C₂²×C₆₇	1	2	yes	complemented
261	C₂⁵	—	2	1	no	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	2	no	—
265	C₂²×D₄	—	2	2	no	—
266	C₂²×D₄	—	2	2	no	—
267	C₂²×D₄	—	2	2	no	—
268	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
269	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
270	C₂³×C₄	—	2	1	no	—
271	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	2	2	no	—
277	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
278	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
281	#117	—	2	2	no	—
282	#117	—	2	2	no	—
283	#117	—	2	2	no	—
284	#117	—	2	2	no	—
285	#117	—	2	2	no	—
286	#117	—	2	2	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	4	no	—
290	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
291	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₂²⋊C₄	—	2	4	no	—
298	C₂×C₂²⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
303	#102	—	2	8	no	—
304	#102	C₂×C₄×C₆₇	1	8	yes	—
305	#102	C₂×C₄×C₆₇	1	8	yes	—
306	#97	—	2	4	no	—
307	C₄×C₈	—	2	4	no	—
308	#95	—	2	4	no	—
309	C₂⁴	—	2	2	no	—
310	C₂⁴	—	2	4	no	—
311	C₂⁴	—	4	2	no	—
312	C₂⁴	—	4	2	no	—
313	C₂⁴	—	8	2	no	—
314	C₂⁴	—	8	2	no	—
315	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	8	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	—	4	4	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	—	4	4	no	—
328	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	8	2	no	—
334	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	#41×C₆₇	1	2	yes	—
339	C₂²×C₄	#41×C₆₇	1	2	yes	—
340	C₂²×C₄	#41×C₆₇	1	2	yes	—
341	C₂²×C₄	#41×C₆₇	1	2	yes	—
342	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
343	M₁₆	—	2	8	no	—
344	C₂×C₈	—	2	8	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
352	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	8	4	no	—
362	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	8	4	no	—
367	C₂²⋊C₄	—	8	4	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	4	2	no	—
371	C₄²	—	4	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	4	no	—
378	C₂³	—	4	4	no	—
379	C₂³	—	4	8	no	—
380	C₂³	—	8	4	no	—
381	C₂³	—	8	4	no	—
382	C₂³	—	8	8	no	—
383	C₂³	—	8	8	no	—
384	C₂³	C₄×D₄×C₆₇	1	2	yes	—
385	C₂³	#121×C₆₇	1	2	yes	—
386	C₂³	#124×C₆₇	1	2	yes	—
387	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
388	D₄	—	2	4	no	—
389	D₄	—	2	4	no	—
390	D₄	—	4	2	no	—
391	D₄	—	4	2	no	—
392	D₄	—	4	8	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	8	4	no	—
405	C₂×C₄	#117×C₆₇	1	4	yes	—
406	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
407	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
408	C₂×C₄	#121×C₆₇	1	2	yes	—
409	C₂×C₄	#123×C₆₇	1	4	yes	—
410	C₂×C₄	#124×C₆₇	1	2	yes	—
411	C₂×C₄	#126×C₆₇	1	4	yes	—
412	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	4	no	—
421	C₂²	—	2	4	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	8	16	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	8	4	no	—
431	C₂²	#388×C₆₇	1	4	yes	—
432	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
433	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
434	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
435	C₂²	#576×C₆₇	1	2	yes	—
436	C₂²	#577×C₆₇	1	2	yes	—
437	C₄	—	4	4	no	—
438	C₄	—	4	4	no	—
439	C₄	—	8	2	no	—
440	C₄	—	8	2	no	—
441	C₄	#388×C₆₇	1	4	yes	—
442	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
443	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
444	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
445	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
446	C₂	—	2	4	no	—
447	C₂	—	8	4	no	—
448	C₂	C₂×#388×C₆₇	1	1	yes	characteristic
449	C₂	C₆₇×#2948	1	1	yes	characteristic
450	C₂	C₆₇×#2949	1	1	yes	characteristic
451	C₂	C₆₇×#3062	1	2	yes	—
452	C₂	C₆₇×#3136	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#405×C₆₇	C₂	semidirect
4. C₆₇×#2316	C₂	semidirect
6. C₆₇×#1936	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1736	C₂	semidirect
9. C₆₇×#1735	C₂	semidirect
18. C₂×#102×C₆₇	C₄	semidirect
23. C₂×#97×C₆₇	C₂²	semidirect
24. C₂×C₄×C₈×C₆₇	C₂²	semidirect
28. #389×C₆₇	C₂²	semidirect
31. #340×C₆₇	C₂²	semidirect
32. #340×C₆₇	C₂²	semidirect
33. #340×C₆₇	C₂²	semidirect
219. #21089	C₆₇	direct (N × H)
229. C₂×#405	C₂×C₆₇	semidirect
230. #2316	C₂×C₆₇	semidirect
232. #1936	C₂×C₆₇	semidirect
233. #1934	C₂×C₆₇	semidirect
234. #1736	C₂×C₆₇	semidirect
235. #1735	C₂×C₆₇	semidirect
236. C₆₇	#21089	direct (N × H)
245. C₂×#102	C₄×C₆₇	semidirect
250. C₂×#97	C₂²×C₆₇	semidirect
251. C₂×C₄×C₈	C₂²×C₆₇	semidirect
255. #389	C₂²×C₆₇	semidirect
258. #340	C₂²×C₆₇	semidirect
259. #340	C₂²×C₆₇	semidirect
260. #340	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#418×C₆₇	2
C₂²×#102×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2316	2
C₆₇×#2309	2
C₆₇×#1936	2
C₆₇×#1934	2
C₆₇×#1736	2
C₆₇×#1735	2
#21089	67

Subgroups — C67×#21089 #30700340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21089 #30700340