Subgroups — C₂×C₆₇×#1977 #30700496

Order 17152.

1566 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	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#100×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1977	C₂	1	8	yes	maximal, direct product factor
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₆₇	—	2	1	no	—
12	C₂²×SD₁₆×C₆₇	—	2	1	no	—
13	C₂²×SD₁₆×C₆₇	—	2	2	no	—
14	C₂²×SD₁₆×C₆₇	—	2	4	no	—
15	C₂²×SD₁₆×C₆₇	C₄	1	8	yes	complemented
16	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₆₇	—	2	4	no	—
23	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
24	#419×C₆₇	—	2	2	no	—
25	#419×C₆₇	C₂²	1	4	yes	complemented
26	#418×C₆₇	—	2	2	no	—
27	#418×C₆₇	C₂²	1	4	yes	—
28	C₂×#103×C₆₇	—	2	2	no	—
29	C₂×#102×C₆₇	—	2	2	no	—
30	C₂×#101×C₆₇	—	2	4	no	—
31	C₂×#101×C₆₇	C₂²	1	4	yes	complemented
32	C₂×#100×C₆₇	—	2	4	no	—
33	C₂×#100×C₆₇	C₂²	1	4	yes	—
34	#405×C₆₇	—	2	2	no	—
35	#405×C₆₇	C₂²	1	4	yes	complemented
36	C₂×#97×C₆₇	—	2	1	no	—
37	C₂×C₄×C₈×C₆₇	—	2	1	no	—
38	C₂⁵×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	—	2	1	no	—
41	C₂²×Q₈×C₆₇	—	2	1	no	—
42	C₂²×Q₈×C₆₇	—	2	1	no	—
43	C₂²×Q₈×C₆₇	—	2	2	no	—
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	2	2	no	—
55	C₂²×D₄×C₆₇	—	2	2	no	—
56	C₂²×D₄×C₆₇	—	2	1	no	—
57	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₆₇	—	2	8	no	—
62	C₂×SD₁₆×C₆₇	—	2	8	no	—
63	C₂×SD₁₆×C₆₇	—	2	16	no	—
64	C₂×SD₁₆×C₆₇	—	2	8	no	—
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	—	2	4	no	—
67	C₂×SD₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
68	C₂×M₁₆×C₆₇	—	2	1	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	—	2	1	no	—
71	C₂×M₁₆×C₆₇	—	2	1	no	—
72	C₂×M₁₆×C₆₇	—	2	2	no	—
73	C₂×M₁₆×C₆₇	—	2	2	no	—
74	C₂×M₁₆×C₆₇	—	2	2	no	—
75	C₂×M₁₆×C₆₇	—	2	2	no	—
76	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
77	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
79	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	—	2	1	no	—
81	C₂²×C₈×C₆₇	—	2	1	no	—
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
84	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	D₄	1	2	yes	—
87	C₂²×C₈×C₆₇	D₄	1	4	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	#117×C₆₇	—	2	1	no	—
90	#117×C₆₇	—	2	1	no	—
91	#117×C₆₇	D₄	1	2	yes	—
92	#117×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	—	2	1	no	—
96	#108×C₆₇	—	2	8	no	—
97	#103×C₆₇	—	2	8	no	—
98	#102×C₆₇	—	2	8	no	—
99	#101×C₆₇	—	2	8	no	—
100	#100×C₆₇	—	2	8	no	—
101	#97×C₆₇	—	2	4	no	—
102	C₄×C₈×C₆₇	—	2	4	no	—
103	C₂⁴×C₆₇	—	2	4	no	—
104	C₂⁴×C₆₇	—	2	2	no	—
105	C₂⁴×C₆₇	—	4	1	no	—
106	C₂⁴×C₆₇	—	4	1	no	—
107	C₂⁴×C₆₇	—	4	2	no	—
108	C₂⁴×C₆₇	—	8	4	no	—
109	C₂⁴×C₆₇	#41	1	1	yes	characteristic
110	C₂×Q₈×C₆₇	—	2	4	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	4	no	—
113	C₂×Q₈×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	—	2	4	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	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	4	1	no	—
121	C₂×Q₈×C₆₇	—	4	1	no	—
122	C₂×Q₈×C₆₇	—	4	2	no	—
123	C₂×Q₈×C₆₇	—	4	2	no	—
124	C₂×Q₈×C₆₇	—	4	2	no	—
125	C₂×Q₈×C₆₇	—	4	2	no	—
126	C₂×Q₈×C₆₇	—	4	2	no	—
127	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	4	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	4	no	—
133	C₂×D₄×C₆₇	—	2	2	no	—
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	2	no	—
138	C₂×D₄×C₆₇	—	4	2	no	—
139	C₂×D₄×C₆₇	—	4	2	no	—
140	C₂×D₄×C₆₇	—	4	1	no	—
141	C₂×D₄×C₆₇	—	4	1	no	—
142	C₂×D₄×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	4	2	no	—
144	C₂×D₄×C₆₇	—	4	2	no	—
145	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	4	1	no	—
150	C₂²×C₄×C₆₇	—	4	1	no	—
151	C₂²×C₄×C₆₇	—	4	2	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	#41	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₆₇	—	2	16	no	—
163	SD₁₆×C₆₇	—	2	16	no	—
164	M₁₆×C₆₇	—	2	4	no	—
165	M₁₆×C₆₇	—	2	2	no	—
166	M₁₆×C₆₇	—	2	2	no	—
167	M₁₆×C₆₇	—	2	2	no	—
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	M₁₆×C₆₇	—	4	2	no	—
171	M₁₆×C₆₇	—	4	2	no	—
172	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
173	C₂×C₈×C₆₇	—	2	4	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	4	2	no	—
178	C₂×C₈×C₆₇	—	4	2	no	—
179	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
180	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
181	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
182	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
183	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
184	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	4	no	—
194	C₂³×C₆₇	—	4	2	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	8	8	no	—
199	C₂³×C₆₇	—	8	4	no	—
200	C₂³×C₆₇	—	8	8	no	—
201	C₂³×C₆₇	—	8	4	no	—
202	C₂³×C₆₇	—	8	4	no	—
203	C₂³×C₆₇	C₄×D₄	1	2	yes	—
204	C₂³×C₆₇	#121	1	1	yes	characteristic
205	C₂³×C₆₇	#121	1	1	yes	characteristic
206	C₂³×C₆₇	#124	1	2	yes	—
207	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
208	Q₈×C₆₇	—	2	4	no	—
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	—	4	2	no	—
211	Q₈×C₆₇	—	4	2	no	—
212	Q₈×C₆₇	—	4	4	no	—
213	Q₈×C₆₇	—	4	4	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	2	4	no	—
216	D₄×C₆₇	—	4	2	no	—
217	D₄×C₆₇	—	4	2	no	—
218	D₄×C₆₇	—	4	4	no	—
219	D₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	4	2	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	4	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	—	4	4	no	—
227	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	#121	1	2	yes	—
233	C₂×C₄×C₆₇	#121	1	2	yes	—
234	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
235	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
236	C₂×C₄×C₆₇	#124	1	2	yes	—
237	C₂×C₄×C₆₇	#127	1	2	yes	—
238	C₂×C₄×C₆₇	#127	1	2	yes	—
239	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
245	C₈×C₆₇	—	2	4	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
249	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	—	8	4	no	—
256	C₂²×C₆₇	—	8	4	no	—
257	C₂²×C₆₇	—	8	8	no	—
258	C₂²×C₆₇	—	8	8	no	—
259	C₂²×C₆₇	—	8	4	no	—
260	C₂²×C₆₇	#390	1	2	yes	—
261	C₂²×C₆₇	#390	1	2	yes	—
262	C₂²×C₆₇	#471	1	4	yes	—
263	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
265	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
266	C₄×C₆₇	—	4	4	no	—
267	C₄×C₆₇	—	4	4	no	—
268	C₄×C₆₇	#390	1	2	yes	—
269	C₄×C₆₇	#390	1	2	yes	—
270	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
271	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
272	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
273	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
274	C₂×#1977	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	2	1	no	—
277	C₂×C₆₇	—	2	2	no	—
278	C₂×C₆₇	—	8	4	no	—
279	C₂×C₆₇	#1977	1	4	yes	direct product factor
280	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
281	C₂×C₆₇	C₂×#471	1	2	yes	—
282	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
283	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
286	C₂²×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₂²×#100	C₂×C₆₇	1	1	yes	characteristic
288	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1977	C₂×C₆₇	1	8	yes	direct product factor
290	C₆₇	C₂×#1977	1	1	yes	characteristic, direct product factor, Sylow
291	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
292	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
293	C₂²×SD₁₆	—	2	1	no	—
294	C₂²×SD₁₆	—	2	1	no	—
295	C₂²×SD₁₆	—	2	2	no	—
296	C₂²×SD₁₆	—	2	4	no	—
297	C₂²×SD₁₆	C₄×C₆₇	1	8	yes	complemented
298	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
299	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
300	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
301	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
302	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#108	—	2	4	no	—
305	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
306	#419	—	2	2	no	—
307	#419	C₂²×C₆₇	1	4	yes	complemented
308	#418	—	2	2	no	—
309	#418	C₂²×C₆₇	1	4	yes	—
310	C₂×#103	—	2	2	no	—
311	C₂×#102	—	2	2	no	—
312	C₂×#101	—	2	4	no	—
313	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
314	C₂×#100	—	2	4	no	—
315	C₂×#100	C₂²×C₆₇	1	4	yes	—
316	#405	—	2	2	no	—
317	#405	C₂²×C₆₇	1	4	yes	complemented
318	C₂×#97	—	2	1	no	—
319	C₂×C₄×C₈	—	2	1	no	—
320	C₂⁵	—	2	1	no	—
321	C₂²×Q₈	—	2	2	no	—
322	C₂²×Q₈	—	2	1	no	—
323	C₂²×Q₈	—	2	1	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×Q₈	—	2	2	no	—
326	C₂²×Q₈	—	2	1	no	—
327	C₂²×Q₈	—	2	1	no	—
328	C₂²×Q₈	—	2	2	no	—
329	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
330	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	1	no	—
339	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
340	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
341	C₂³×C₄	—	2	1	no	—
342	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×SD₁₆	—	2	8	no	—
344	C₂×SD₁₆	—	2	8	no	—
345	C₂×SD₁₆	—	2	16	no	—
346	C₂×SD₁₆	—	2	4	no	—
347	C₂×SD₁₆	—	2	4	no	—
348	C₂×SD₁₆	—	2	8	no	—
349	C₂×SD₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	2	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	2	no	—
355	C₂×M₁₆	—	2	2	no	—
356	C₂×M₁₆	—	2	2	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
359	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
360	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
361	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₈	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
366	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₈	D₄×C₆₇	1	2	yes	—
369	C₂²×C₈	D₄×C₆₇	1	4	yes	—
370	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	D₄×C₆₇	1	2	yes	—
374	#117	C₂³×C₆₇	1	2	yes	—
375	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₄²	—	2	1	no	—
378	#108	—	2	8	no	—
379	#103	—	2	8	no	—
380	#102	—	2	8	no	—
381	#101	—	2	8	no	—
382	#100	—	2	8	no	—
383	#97	—	2	4	no	—
384	C₄×C₈	—	2	4	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	4	1	no	—
388	C₂⁴	—	4	1	no	—
389	C₂⁴	—	4	2	no	—
390	C₂⁴	—	8	4	no	—
391	C₂⁴	#41×C₆₇	1	1	yes	characteristic
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	4	1	no	—
403	C₂×Q₈	—	4	1	no	—
404	C₂×Q₈	—	4	2	no	—
405	C₂×Q₈	—	4	2	no	—
406	C₂×Q₈	—	4	2	no	—
407	C₂×Q₈	—	4	2	no	—
408	C₂×Q₈	—	4	2	no	—
409	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	4	2	no	—
421	C₂×D₄	—	4	2	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	—	4	1	no	—
424	C₂×D₄	—	4	2	no	—
425	C₂×D₄	—	4	2	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	4	1	no	—
432	C₂²×C₄	—	4	1	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
441	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	#41×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
444	SD₁₆	—	2	16	no	—
445	SD₁₆	—	2	16	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	2	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	4	2	no	—
453	M₁₆	—	4	2	no	—
454	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
455	C₂×C₈	—	2	4	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	4	2	no	—
460	C₂×C₈	—	4	2	no	—
461	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
462	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
463	C₂×C₈	C₂²⋊C₄×C₆₇	1	8	yes	—
464	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
465	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
466	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	2	no	—
472	C₄²	—	2	2	no	—
473	C₄²	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	4	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	8	4	no	—
481	C₂³	—	8	8	no	—
482	C₂³	—	8	4	no	—
483	C₂³	—	8	8	no	—
484	C₂³	—	8	4	no	—
485	C₂³	C₄×D₄×C₆₇	1	2	yes	—
486	C₂³	#121×C₆₇	1	1	yes	characteristic
487	C₂³	#121×C₆₇	1	1	yes	characteristic
488	C₂³	#124×C₆₇	1	2	yes	—
489	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
490	Q₈	—	2	4	no	—
491	Q₈	—	2	4	no	—
492	Q₈	—	4	2	no	—
493	Q₈	—	4	2	no	—
494	Q₈	—	4	4	no	—
495	Q₈	—	4	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	4	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	4	no	—
501	D₄	—	4	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	4	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
510	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
511	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
513	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
514	C₂×C₄	#121×C₆₇	1	2	yes	—
515	C₂×C₄	#121×C₆₇	1	2	yes	—
516	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
517	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
518	C₂×C₄	#124×C₆₇	1	2	yes	—
519	C₂×C₄	#127×C₆₇	1	2	yes	—
520	C₂×C₄	#127×C₆₇	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
526	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
527	C₈	—	2	4	no	—
528	C₈	—	4	2	no	—
529	C₈	—	4	2	no	—
530	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
531	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	8	4	no	—
538	C₂²	—	8	4	no	—
539	C₂²	—	8	8	no	—
540	C₂²	—	8	4	no	—
541	C₂²	—	8	8	no	—
542	C₂²	#390×C₆₇	1	2	yes	—
543	C₂²	#390×C₆₇	1	2	yes	—
544	C₂²	#471×C₆₇	1	4	yes	—
545	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
546	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
547	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
548	C₄	—	4	4	no	—
549	C₄	—	4	4	no	—
550	C₄	#390×C₆₇	1	2	yes	—
551	C₄	#390×C₆₇	1	2	yes	—
552	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
553	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
554	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
555	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	2	2	no	—
559	C₂	—	8	4	no	—
560	C₂	C₆₇×#1977	1	4	yes	direct product factor
561	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
562	C₂	C₂×#471×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
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 ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₆₇	C₂	semidirect
3. C₂×#419×C₆₇	C₂	semidirect
5. C₂²×#101×C₆₇	C₂	semidirect
7. C₂×#405×C₆₇	C₂	semidirect
8. C₆₇×#1977	C₂	direct (N × H)
15. C₂²×SD₁₆×C₆₇	C₄	semidirect
23. C₂×#108×C₆₇	C₂²	semidirect
25. #419×C₆₇	C₂²	semidirect
31. C₂×#101×C₆₇	C₂²	semidirect
35. #405×C₆₇	C₂²	semidirect
67. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
274. C₂×#1977	C₆₇	direct (N × H)
279. C₂×C₆₇	#1977	direct (N × H)
283. C₂²×#108	C₂×C₆₇	semidirect
284. C₂×#419	C₂×C₆₇	semidirect
286. C₂²×#101	C₂×C₆₇	semidirect
288. C₂×#405	C₂×C₆₇	semidirect
289. #1977	C₂×C₆₇	direct (N × H)
290. C₆₇	C₂×#1977	direct (N × H)
297. C₂²×SD₁₆	C₄×C₆₇	semidirect
305. C₂×#108	C₂²×C₆₇	semidirect
307. #419	C₂²×C₆₇	semidirect
313. C₂×#101	C₂²×C₆₇	semidirect
317. #405	C₂²×C₆₇	semidirect
349. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
560. C₂	C₆₇×#1977	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#419×C₆₇	2
C₂×#418×C₆₇	2
C₂²×#101×C₆₇	2
C₂²×#100×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#1977	2
C₂×#1977	67

Subgroups — C2×C67×#1977 #30700496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1977 #30700496